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DIFFERENTIAL OPERATORS ON RIEMANNIAN MANIFOLDS1).
by N. ARONSZAJN and A. N, MIL6RAM
INTRODUCTION. Most investigations involving partial differential equations in a domain of Euclidean space lead quickly to problems involving boundary conditions. Even the assumption that the domain has a smooth boundary by no means makes trivial the inter-relation between the various kinds of boundary conditions, in the natural course of such an investigation, differential operators intrinsic to manifolds arise. The problem of describing the adjoint of a given operator, intrinsically within a manifold, at once presents itself, together with a galaxy of questions involving boundary integrals and associated forms, it seemed to us that most of these questions have either not been discussed or at best examined in a very cursory manner.
This paper is concerned with precisely such problems. It is our aim to set up an apparatus which will facilitate the investigation of differential problems on Riemannian manifolds. Our motivation is not only the needs of boundary and eigenvalue problems for a domain in Euclidean space. The intrinsic interest which Riemannian manifolds possess would provide its own justification.
In the first part we review briefly some known definitions and results on manifolds, tensors, covariant derivatives, etc. it is hoped that this will serve as a reminder to the reader who may have forgotten some of these, as well as to establish notations to be used in the sequel.
In the second part we define differential operators on Riemannian manifolds- bilinear differential operators and forms, and establish the existence and unique- ness of adjoint operators. Operator-forms are discussed, connecting exterior differ- ential forms with bilinear differential operators, and a useful and somewhat unexpected theorem is obtained showing that all such forms which are closed are also of necessity exact differentials of forms of the same kind.
!) Paper written under contract with the Office of Naval Research, Nonr 63900.
DIFFERENTIAL OPERATORS ON RIEMANNIAN MANIFOLDS 2 ~
Part three is devoted to boundary operators. Various systems of boundary operators are described, and their inter-relation deduced. Boundary conditions attached to elliptic, positive, self-adjoint, and indeed much more general opera- tors, are discussed after a close study of restrictions of bilinear exterior differ- ential forms to submanifolds. A complete description of all adjoint, self-adjoint and positive systems are obtained among all normal systems of boundary con- ditions.
For the case of plane domains the results in Part Ill were established previously ~ (but no complete proofs were published). Problems involved in approximation methods indicated the need of considering boundary conditions at least as general as those treated here'. Applications of our results to problems in approximation methods will be carried out at a later date3.
PART I - - PRELIMINARY DEFINITIONS AND NOTATIONS; REVIEW OF PROPERTIES OF DIFFERENTIABLE AND RIEMANNIAN MANIFOLDS
1. - - Differentiable manifolds. An n-dimensional manifold Mn is a connected topological space which can be covered by a collection {U} of open sets, homeomorphic to open intervals of the n-dimensional Euclidean space E,~. We shall restrict attention only to those manifolds for which {U} can be taken to be denumerable.
An open interval of E,, is the set of all x , x : (x,, x , . . . , x,,), satisfying the inequalities x~ < x k < x~, k : 1, 2 , . . . , n, for some real numbers x~: and x~.
Let U' and U" be two open sets of M,~ which are transformed by two homeomorphisms H ' and H" onto intervals 1' and I" if U' U" :[: 0, then
H " H "-1 is a homeomorphism between two open subsets of E n, H" (U" U")
and H" (U' U"). Consider now a class ~Z'of homeomorphisms between open sets in E~ with the
following properties: 1 ~ with any homeomorphism T, ,~ contains the in-
verse: T-Z; 2 ~ if T and T~ belong to ~ f a n d T 7"1 has a meaning, then T 7"t
1). See N. Aronszajn, Studies in Eigenvalue Problems, Report 11, Office of Naval Research, N 9 onr 85101, 1950, and also Oreen's functions and reproducing kernels, Proceedings of the Symposium on Spectral theory and Differential Problems; Oklahoma A. and M. College, 1951.
2} lbid See footnote I. 3) This will be done in a paper now under preparation by N. Aronszajn, 'Differential
problems, variational problems, and functional completion'.
~ 8 N. A R O N S Z A J N , A. N. M I L G R A M
also belongs to o-~., 30 T belongs to ~ i f and only if for any point in the
domain of T there exists a neighborhood V such thai the restriction of T to V
belongs to ~ The last condition means that the homeomorphisms of class
are characterised by local properties. As a consequence of 3 o, the restriction of
any T of .~ an open set also belongs to
The most important examples of such classes ~ are the following: the
class o-~c0 of all homeomorphisms between open sets; the class :~f" , r --- 1, 2 , . . .
of all homeomorphisms which together with their inverses are differentiable up to
order r; the class ~ - ' * of all T, such that T and T -] are infinitely differentia-
ble; the class 2 ~ c(u') of all T, which with T -1, are analytic in n real variables;
the class .~cz of all identical mappings of open sets onto themselves.
If the open sets U covering M,, and the corresponding homeomorphisms H
are chosen in such a way that for any two U' and U" the homeomorphism
H" H '-~ of H'(U" U") onto H"(U" U") belongs to the class .~ then we
say that a structure of class ~Tzc(:~C-structure) is determined in the manifold M,,.
If another covering of M~ by open sets 0 with corresponding homeomorphisms
/-t onto open intervals also determines an ~~ we say that it determines
the same structure as the covering by the sets U if for any U and 0 of the two
coverings, the homeomorphism /-) H -~ of H (U U) onto /q'(U U) belongs to the class
in particular, if .~ o--~-,', 1 "< r < co, M,, is called a differentiable mani-
fold of class C' . If ~ t c - - ~ (u,), M~ is called an analytic manifold. If ~ : ~,z-z,
M~ is said to be over E~ (the homeomorphism H then gives the projection of
M~ on E~).
If ~ D o-~c,, every covering of M, which determines an ~Z"-structure also
determines an ~Z'-structure and two such coverings which determine the same
structure of class J Z " also determine the same structure of class ..:~F. In this
sense we say that an 5~"-structure determines a unique 2~C-structure (but diffe-
rent structures of the class o--~f, may determine the same ~zC-structure).
Let the covering M~ - : U U, and the corresponding homeomorphisms H
determine an ~E-structure. An open set U ~ C M,, with a fixed homeomorphism
H ~ of U ~ onto an interval I will be called a coordinate patch (c-patch) in the
structure if for any set U of the covering the homeomorphism H ~ H - t of H (U U ~
D I F F E R E N T I A L OPERATORS ON R I E M A N N I A N M A N I F O L D S 2 0 0
onto H 0 (U U ~ belongs to class ~,c. Clearly every U of the covering with its corresponding homeomorphism H is a c-patch; also if another covering by sets U' and corresponding H' determines the same structure, then every system {U', H'I is a c-patch. Each point P in a c-patch U ~ is completely determined by the coor- dinates xk of the point H ~ (P) in the interval I : H ~ (U~ Often it is convenient to speak of these coordinates as "the coordinates of P in U ~ (without mentio- ning the homeomorphism H~ But the same point P in different c-patches may have different coordinates.
A covering M, = U U by open sets is called locally finite if every compact subset A C M~ has points in common with only a finite number of the U' s.
If M,~ has an ~r-f-structure, and if M , - - U U is a covering by c-patches, m
there will exists an enumerable locally finite covering by c-patches, M . = U Uk k = l
such that for each U~ and some U, Uk C U, the closureU~ is compact and of the corresponding homeomorphisms Hk and H the first is a restriction of the second.
Let M, be a manifold with an :~r-f-structure, and S a homeomorphism of M, onto M~,. S will induce an ~/Z'-structure on M n given as follows: Let
M~ = U U be a covering by c-patches, and let H be the corresponding ho-
meomorphisms. The open sets S (U), with homeomorphisms H S -] , determine an ~Tf-structure- the induced structure in M~,. Suppose that M~, has its own
structure of class ~ S is an isomorphism (~Z'-isomorphism)and the manifolds M,, and M~ are isomorphic, if in M~, the induced structure is the same as the
original one. From the point of view of the theory of structures of class ~ , isomorphic manifolds will have identical properties and in many instances can be interchanged.
We turn now to ~C'-structures, i. e. to differentiable manifolds of class C'(i).
The homeomorphisms of class ~z - , have clearly non-vanishing Jacobians. We de-
note by ~ + the subclass of ~ c , formed by homeomorphisms with positive
Jacobians. A manifold with an ~~ is called a differentiable orientable
manifold of class C r. It is known that in order that a manifold admit of an
~-Z+-structure, it is necessary and sufficient that it admit of an ~ r - s t r uc tu r e
and be orientable in the topological sense.
(i) For a complete discussion of differentiable manifolds see O. Veblen and J. H. C. Whitehead, Foundations of Differential Oeometry, Cambridge, 1932.
x8 - R e n d . Circ . M a t c m . P a l e r m o , ~ s e r i e I I - t o m o I I - a n n o ~953
2 7 0 ZN. AB. O N S Z A J N. A. N. M I L G R A M
From now on we shall limit ourselves to differentiable orientable manifolds
of class C'*. The restriction to C'* will simplify some of the developments,
but it will be clear from the context that our results are valid also for C r with
finite r, as long as r is sufficiently large. Consider a real or complex-valued function f (P) defined on M~ (or on an
open set g C M~). In any c-patch U, f becomes a function f ( x ~ , . . . , x~) of the n coordinates x~, representing P in the interval I ~--- H (U). We say that f is of class C ~ if in each c-patch, f as function of the n coordinates is of class C m. For f to be of class C ~ it is enough that f be of classC ~ in every c-patch of a covering M~--- O U (~.
Let V a n d V" be two open sets in M~ such that V C V C V" and let the
closure V be compact. It is well known that there exist functions ~ (P) of class
C" defined on M~ such that ? (P) - - 1 for P E - V , ? (P) - - 0 for P E M~ - - V'
and 0_< ? ( P ) = < 1 for P E V ' - - V.
Consider now a locally finite covering by c-patches Uk, M,, = U Uk.
A sequence of functions {~r of class C" on M,, forms a corresponding partition of unity
if 0 <_ ?k(P) ~< 1 for all k; ~r ~ 0 for PE Uk, ?k(P) - - 0 for P E M ~ - - Uk;
' ~ ? ~ ( P ) = 1 for all P E M~. It is well known that such partitions of unity k~---1
always exist. They are a powerful tool in the study of differentiable manifolds. i 1 . . . i r
The classical notion of a tensor field a , . . .y~ over M~ can be defined as a
system of functions of the point P E U assigned to each c-patch U,
oo) _(o (p , U) . ( j ) ~ u ( j )
If P belongs to two c-patches U and 0 having in them the coordinates x~ and
_(i) in U at the point ~(~) in 0 are connected with those of uo~ xk, then the values all ~
P by the classical tensor transformation. A tensor is called of class C r~ if in each
c-patch all its components are functions of class C m in the coordinates of
the patch.
We shall sometimes consider tensors a~!) in M n defined only on a subset
t
(l) If we consider manifolds of class C" with finite r this property will ~tot hold in general for m ~ r.
D I F F E R E N T I A L OPEBATOFtS ON I I I E M A N N I A N M A N I F O L D S 271
(1) - ( 0 A of M,,. This means that an ) =-- u(j~ (P, U) are only defined for P E A U, the
components for the same point P E A and two different c-patches still being
connected by the usual tensor-transformation on M,,.
The tensor is completely determined if it is known in each c-patch of a
covering M,, ~ U U. In all the other c-patches it is determined by the tensor
transformation, if all the components of a tensor vanish at a point p in one
c-patch U, they will vanish at p in all c-patches containing p. This remark leads
to the following construction of the most general tensor on the manifold, using
partitions of unity.
We consider a locally finite covering M, , - - U U~ by c-patches U~ and a
corresponding partition of unity I~0~l. in each c-patch Uk we choose arbitrarily
the components of a tensor a ~i)~j). We then define a tensor v~j) over M, by putting
~<0 __ % u,y,k<i' in U h and ~(i)~(j) = 0 outside of U k. For any c-patch U the com-
ponents of this tensor in U U k are then determined by the tensor transformation and
in U - - O k are identically 0. It is then clear that X/~cc~> is a tensor over M. . k
(lJ On the other hand, if we have an arbitrary tensor a <j~ over M, and if we choose
for at< O) the components of this tensor in Uk, then, clearly,/~,i>(j) . -_ ~%a(j)(i) and o ti),j)-- k
~(i) h(i)
k
If the components -*(~) in U k are chosen so that as functions of the coordi- u(j)
nates in the corresponding interval I~ they are of class C m Jn the closed interval I~,
then each tensor ~,<~) is of class C" ~<i) is vcj ~ over M, and the resulting tensor ~ ,.(j~ k
also of class C m.
A function defined on M,~ is considered as a tensor of rank 0. Addition, multi-
plication, or contraction of a finite number of tensors is considered in the usual
manner. If all the tensors are of class C ' , the result will also be of class C m.
As usual, the tensor is called symmetric if in each c-patch the components
do not change when the orders of their contravariant indices i~ or covariant
indices jt are changed. The tensor is skew-symmetric if in each c-patch a per-
(o leaves mutation of their contravariant or covariant indices in a component a,j~
272 N. , n o N s z A j ~ . A. N. M~LCRA~
the component invariant except for a factor + 1, + 1 if the permutation is even,
- - 1 if it is odd. Therefore every component of a skew-symmetric tensor with two
equal covariant or contravariant indices is equal O. The tensor vanishes identically
if its contravariant rank r or covariant rank s is > n.
If in the above construction of a tensor by a partition of unity {9~1 we
choose each tensor -k~i) ,~j) as symmetric or skew-symmetric, then the resulting tensor
over M. is also symmetric or skew-symmetric.
The consideration of covariant skew-symmetric tensors ac~ ) is equivalent to
the consideration of the corresponding exterior differential forms which, in each
c-patch U, are represented by the expression
- -1- a / ' .
" ' - s ! aj , . . j , . .
where the summation is extended to all systems of indices ! _< j, < n. (~).
As usual, for the Orassmann products dx j~ . . . d x J= we have the rules:
1 o - when the indices j~ . . . f s are a permutation of the indices J l . . - i s ,
dxJ~.., dxJs--_+_ dxJ~.., dxJs, with the plus sign for even and minus sign for
odd permutations; 20 - - if any two indices jk are equal, dxi~.., dx Js --O. We can then represent the form =, by
d x . % - - a j ~ . . . Ys """
Jl < J2 < . . . < i s
The correspondence between the skew-symmetric tensor at~l and the form =, is
one-to-one. The sum of skew-symmetric tensors corresponds to the sum of forms.
A product of two skew-symmetric tensors is not in general skew-symmetric
However, each covariant tensor bj~. . . j , determines a covariant skew-symmetric
tensor (the correspondence being many-to-one) given in each c-patch by the formula :
�9 - - . t ,
(~) For the theory of exterior differential forms see Elie Carton, Les systimes differentiels extdrieurs et lettrs applications gdomftriqaes, Paris, Hermann 1945. See also W. V. D. Hodge, Harmonic Integrals. Cambridge, 1941.
DIFFERENTIAL OPERATORS ON RIEMANNIAN MKNIFOLD$ 273
where the sequence j ; . . . j~ varies through all the permutations of Jl...Js, the
factor being + 1 or -- 1 (depending on whether the permutation is even or odd)
and the component cy~...y, - - O if two of its indices are equal. The product of
two forms % and ~q is formed by taking the product of the skew-symmetric
tensors aoand br ). This product determines a skew-symmetric tensor cj of rank
p + q, and the corresponding form ~'p+q is defined as the product % ~q. The
product % 13q can also be written directly in each c-patch as
' E p; qt J l ' ' ' J p " J l ' " J q
%.. . j , t~j;...jq J ' ...ax j, d~J~.., d/~,,
where this expression is simplified by using the properties of Orassmann products.
The differential d ~, is defined in each c-patch by the expression
0 ) d o,, --- s--~. ~_~ aj~...js ,~:xj dxYa . . . dxJ, . J I ' ' ' J s j : l OXj
Here we develop formally the right-hand member by using the rules concerning
product dxY~.. , dx is. W e obtain a form [~s+l of rank s q- 1
1 . . dx . l l dxJS+l, ~'+~ - ( s + 1)! L bj~ .j,+, ...
where the skew-symmetric tensor b~y} is determined for j~ < Jz < . . . <J ,+, by
s+l C~ a j l . , "Jk--I Jk+l " " " Js--[-1 bjl . . .J$.-~l = E ( - l )k-- ,
k=, 0 xjk
The following formulas for differentials will be of constant use:
d(d~ ,3 : O,
d (0,. IL) = (d ,,.) i3, + ( - - 1)' ~,, (d ~,).
if % = d ~la_~, % is called an exact differential. If d ~p = 0, ~p is called a
closed form. A sum %-+-IBp and a product % ~r of two exact differentials is an
exact differential. Similarly, a sum and a product of two closed forms is a closed
274 N . A R O N S Z A 3 N , A . N . M I L G R A M
form. Every exact differential is a closed form, but the converse is not true in
general. The existence of closed forms which are not exact differentials over the
whole manifold M~ depends on the topological properties of M~ (the non-vanishing
of the corresponding Betti numbers).
We shall consider submanifolds of M~. Let Mp be a differentiable manifold
of class C" and dimension p <_ n contained in M~.
A c-patch U in M~ is said to agree with ~I, if the points in the intersection o U ~/p are characterised by the n - - p equations x k ~ xk, k > p, with some con-
slants x~ and if U Mp is a c-patch in ,~/p with coordinates equal to the first
p-coordinates in U.
We shall say that Mp is a snbmaMfold of M~ if it can be covered by
c-patches in M~ which agree with it.
If ~/p is an orientable manifold and is a submanifold of M~, then it is called
an orientable submanifold.
By similar definitions we introduce submanifolds of class C m, but we must
consider M~ itself as a manifold of class C m, i. e. we must consider in M~ the
~,Cm-structure corresponding to the ~F+-s t ruc ture given in M.
Remark. We can introduce similarly the notion of submanifolds for a general
~,C-structure given in M~, but our definition gives in some cases a very restricted
type of submanifold. For instance, if M~ has a structure of being over E~, the
projection of a submanifold of dimension p will be contained in one p-dimensional O hyperplane defined by x k - - x k for k > p, and x ~ k constants. For differentiable
manifolds, however, our definition gives a sufficiently large class of submanifolds.
The submanifolds of dimension n are given by all the open connected sub-
sets G of M~. As c-patch in G we consider all c-patches of M~ which are con-
tained in g.
if Mq is a submanifold of Mp, and Mp a submanifold of M~, then Mq is a
submanifold of M~. Also, if ~rp and /{/~ are submanifolds of M~ and/~/q C 217/p, then
M~ is a submanifold of Mp.
The same statements remain true if we replace the term ~ submanifold ~ by
"orientable submanifo ld , or ~ submanifold of class C m ~. Unless otherwise
.~tated, we shall consider only orientab!e submanifolds of class C".
DII,'FEIILSTIAL OPI..RAIORS Ogi RIEMANNIAN MANIFOLDS ~ . 7 5
An open set containing a submanifold /~/~, (p < n) is called a coordinate neighborhood (c-neighborhood) of .Q, if it can be covered by c-patches agreeing
with M,. A sufficiently small c-neighborhood I, of a ( n - 1)-dimensional submanifold
~/,,_~ decomposes into two connected paris V~ and V --the positive and ne-
gative neighborhoods--lhe intersection of which is M,_~. Let V = L; U k, where
U ~ are c-patches agreeing with 20'_~, with coordinates x~. Further, in each U k, let
s U k the condition x~ - -~k represent U k ~_~. If we then denote by U+ and
k > ~S, and s < ~s the parts of U s which are given by lhe conditions x n =__ x~ =
respectively, we have V+ -- t.) U_~ k_ and V._ - - U U s._ if we change the orientation
of &/n_~ into the opposite one, V remains a c-neighborhood, but V+ and V_
are interchanged.
Let a~/)be a covariant tensor defined in a whole c-neighborhood V of /flp.
Consider this tensor in a c-patch U' C V agreeing with Mp. If we take only
components of the tensor with indices j ~ p, they will form components of a
tensor in the c-patch U'~4p of Mp of the same covariant rank as the original
one. For any two such patches U' ,,17/p and U" Mp, the two tensors so obtained
are connected by the usual tensor transformation on the manifold .,f/p. This means
that by restricting the indices to the first p (in c-patches agreeing with Mp) and
restricting the domain of definition to Mp, we obtain from a given tensor in V
a tensor on if/v of the same covariant rank. This tensor will be called the
restrictioa a~/~ of au~ to M~. It should be noted that the restriction au) is not a tensor
in M~ defined on A! v in the sense described previously (the indices range only
from 1 to p instead of 1 to n). If the original tensor is symmetric or skew-symmetric its restriction is also
of the same character. This leads immediately to the restriction of an exterior
differential form ~. The restriction ~ can be obtained also by developing ~ in
~he coordinates of a c-patch agreeing with A4p and putting 0 for every dx ~ for A
j > p. Obviously, if q > p, the restriction ac~ is identically 0,
276 ~ . A B O N S Z A J N , A. N. M 1 L 6 R A M
It is immediately verified that if y~ = ~, + i8,, then % - - &~-~- ~. If
= then = ;,, L
If we consider d4,, as a manifold in itself and denote by d the differential
on this manifold, it is immediate that if ~ , - - d % _ , then #,, = ~p__~. It follows
that the restriction of an exact differential in M~ is an exact differential in fi4~,,
and the restriction of a closed form in M,, is a closed form in .,i,/,.
In the usual manner we now introduce the integrals ] . ,% the rank of gp
d ,v / P
being the same as the dimension of the submanifold ~ , . We first define the
integral over a subset R of a c-patch l ) - - UMp where U is a c-patch in M,,
agreeing with 3;/p. This is given by the formula
/ OLp ~ ; !~ 01 '2 . . . . P d x t " '" dxp '
where a~...jp denotes in general the components of the skew-symmetric tensor
determining the form %. If /? is a subset of another c-patch of the same kind, the value of the integral is the same. Here the integral is taken in the Lebesgue sense. We shall deal only with sets R of the first two Borel classes and the
form % will be assumed continuous on b/p.
We now take a locally finite covering of Mp by c-patches of this kind, k- - I
f.)~ = U' M, and put / ~ = 0 ~ - - U 0 z. Hence/~ c /) ' and ,r - - U /~'~. We 1=1 k
then define
If Mp is contained in one c-patch U of M. (not necessarily agreeing with ~/p) the expression of the integral can be written in the classical form in terms of coordinates of U:
O~p ~ - - ajl . . "jp dxJl . . . dxJp = Oil... jp P ! A . . . l~_Jl < . . . < / p ~ _ n
d x ~ . . . dxJp.
D I F F E B E N ] ' I A L OPEBATOBS ON B I E M A N N I A N M A N I F O L D S 277
These definitions lead immediately to the following remarks.
1) As a particular case of the above definition we obtain the integral of an
n-form ,~ over the manifold M~ or over any open and connected subset of it.
2) In the same way we obtain the definition of the integral over an arbi-
trary subset R of M~ even if it does not lie in one c-patch U.
L 3) We have ~p----. ~,, P P
p-form on the p-manifold ~/p.
~ being the restriction of ~ to 2~,, i .e . a
4) If we change the orientation of .,i/p into the opposite one, we denote the
resulting submanifold as - - ~/'~, and we have ] % - - - - %. J ~p ~p
We shall make frequent use of Stokes theorem, though we shall need it only for very simple cases; we state it only for these cases.
We consider a c-patch U in M,, and the corresponding coordinate interval
I - - ! , : x ~ < x k ( x~', k = 1 , . . . , n . Let in be an interval completely interior
to /,, given by the inequalities x~ < .~ < x k < x'~ < x'~'.
The boundary of the interval ]~ is composed of a finite union of at most
(n -- 2)-dimensional edges and of 2 n (n - - 1)-dimensional intervals i~,k_ ~ and/~'k_ 1
k = 1 , 2 , . . . , n lying on the hyperplanes x k : .~ or xk--~x~'respectively. On the
;k ~;,k manifold M. the images of /~, and of 1,_ 1 and /,,_, form a domain U,~ and its
boundary m a n i f o l d s / ~ and /~:sx respectively.
The orientation on the boundary manifolds /~,-t should be chosen in such
a way that U~ + /3~_~ is a positive c-neighborhood of /~_t. Stokes theorem
can now be stated as follows:
If the (n - - l)-form ~ _ t is of class C -~, then
k=l J Bn--!
278 N . A R O N ~ Z A J N , A. N . ~ | L G R A - 1~
2 . R i e m a n n i a n M a n i f o l d s (~). A Riemannian manifold is a differentiable manifold on which is given a covariant symmetric tensor g,j which forms a positive definite matrix at each point of each c-patch of the manifold. The tensor go is called the metric tensor of the manifold, if g~j is of class C .... ~ the Riemannian manifold is said to be of class C".
Every differentiable manifold (of class C+) can be made into a Riemannian
manifold of class C" by defining the tensor g,~ in the same way as a general tensor was constructed previously by the use of a partition of unity. To this end, in each c-patch of a covering we take a covariant tensor of rank 2, forming a positive definite matrix and obtain a tensor on the whole manifold which has the same properties.
As usual, we denote by g the determinant of the matrix {gol, and by {g'J/ its inverse, go is a contravariant symmetric positive definite tensor and its deter- minant is the reciprocal of g. The expression,
~ g d x t d x 2 . . . d x ~ ,
represents an n-form in each c-patch which is an n-form over the whole mani- fold. It gives the invariant volume element in M~. The corresponding skew- symmetric tensor has, in each c-patch, the components
1 n~V. ~ g E , , . . . ~ ,
where E3~...in : 0,-[- 1, or - - 1, depending on whether two indicesj are equal,
or whether the sequence J t . . . J~ is an even or odd permutation of 1 , . . . , n . For a p-form % on M~ we define the star operator as follows. Consider
the skew-symmetric tensor aol corresponding to ~, contracted with the product
g;L. J~ gt2.j~ ...giP'JP. The resulting contravariant skew-symmetric tensor of rank p,
gll ,Jl . gip, jp b ~m . . . ip = ~ aa . . . J, . .
y , . . . / p
is then contracted with the above defined tensor }/g Eil...ip, jt...Jn_p. The result
(t) For the theory of Riemannian manifolds we refer the reader to the classical textbooks of T. Levi-Civita The absolute differential calculus [translated from the Italian) Blackie and Son, Ltd" London and Olas~ow, 1927 and L. P. Eisenhart, Rie~annian Oeometry Princeton Univ. Press, 1920.
DIFFERENTIA| , OPERATORS OH R | E M A N N I A H MANIFOLDS ~ 7 0
is a skew-symmetric covariant tensor of rank n -- p which, divided by p!, gives rise to an ( n - p)-form which is the star of % written (~)* or ,~,.
We have the general formula �9 �9 ~,, - - ( - - 1) ~/~-~/~r; for two p-forms ~ and
~p, ~p (~)" ----- ~ (~,,)" = an n-form. For our purposes the most important cases will be when p - - 0 or p = n,
i. e. when we consider a function f on the manifold or an n-form a d x * . . , dx ~.
in these cases the formulas are simple:
f - / V g ax*...dx ", fl
. (a d x t . . , dx") - - -~g.
We recall briefly a few properties of geodesics. The metric tensor go will be assumed of class C" for m--> 2. The manifold M, can be covered by c-patches U in which every two distinct points can be joined by a unique geodesic line. The components of the intersection of a geodesic line with such a c-patch form one-dimensional submanifolds if the metric tensor go is of class C | If the tensor is of class C ~, the geodesic line is a submanifold of class C"-+L A one-dimensional submanifold determines a tangential direction at each of its points. A geodesic line is completely determined by each of its points and its tangential direction at this point.
tn Remark . If we take an ~ r+ - s t ruc tu re on the manifold, every tensor of po-
sitive rank, and in particular the metric tensor g~ will be at most of class C "-~. Thus the class of geodesic lines will not exceed the class of the manifold as should be expected.
From now on we shall suppose, for simplicity, that go is of class C".
Consider now a submanifold ~f~,. The restriction go of the tensor g~j to -~v
is a metric tensor on My which gives the induced metric on M,.
In particular we shall consider the case of an ( n - l)-manifold, M,_~. A
unique normal geodesic passes through each point P of M~-r It is determined
by the point P and by the normal direction at P. In a c-patch agreeing with/~/~
and containing P the normal direction is given by the straight line x k - - x ~ ~--- o - - ~k (x~ - - x~ k --- 1 , . . . , n -- I, where x~ are the coordinates of P and the
~k's are determined by the system of linear equations
rt 1
~_~ g,~ (P) ~j -p g,~ (P) = O, i --- 1 , . . . , n - - 1. j = l
2 8 0 N. A R O N S Z A J N , A. N. M 1 L G R A M
The submanifold M,,-t can be covered by c-patches U in M,, with the following properties :
1 ~ U agrees with M.-t.
2 ~ Each straight line in the coordinate interval parallel to the x~-axis represents
a geodesic normal to M~_, (more precisely, the portion of this geodesic lying in U).
3 ~ For any two points on such a normal with x -coordinates x~ and xZ., the
geodesic distance between the two points is given by I x~. - - xl.]. A c-patch U of this kind will be called normal to M,- I . For any two normal
c-patches the transformation of coordinates in their common part leaves the n-th coordinate invariant except perhaps for an additive constant. It is clear that by a sui- table translation in the direction of the x~,- axis we can always choose all the normal c-patches in such a way that Mn-t be represented by the condition x,~ = O; then the transformation of coordinates in a common part of two such c-patches will leave x,~ invariant in the strict sense.
A c-neighborhood of .~/',~_~ will be called normal if it is a union of normal c-patches. Every c-neighborhood contains a normal c-neighborhood.
Let V be a normal c-neighborhood of ,Q'~_I and f a function of class C '~
defined in V. We define the normal derivative of order k, ~ f , at a point P E/I,I,,_ t c~v ~
for k = O, ! , . . . m as follows. Consider any normal c-patch containing P. If the
0 then the normal at P is given by x k = x~ for coordinates of P are x ~ , . . . , x , ,
k - - 1 , . . . , n - - 1 and v = x - - x ~ is a coordinate on the normal independent of
the choice of the c-patch U. This gives a well determined invariant meaning to
c3kf . . . . . . at every point of the normal, in particular at P. it is clear that ~ f (P) - - Ov ~ Ov k
- - O k f (P ) in each normal c-patch containing P. oxk.
The normal derivatives are functions defined on M~_~, the k ~ th derivative
being of class C "-k. These derivatives are mutually independent. More precisely,
for any choice of functions %, q0t, . . . , q0,, s < m, on ~/~_~ such that q~ is of
class C " -k on M~_t, there exists a function f defined and of class C" in V d k
such that f - - % for k = O, 1 , . . . , s. dv k
bIPFERENTIAL OPERATORS ON RIEM&NNIAN MANIFOLDS ~ 8 1
The construction of f is particularly simple if we take m = co and any
finite s. Then all r162 are of class C ~0 on M~-t- Each Q 6 V lies on a unique
normal geodesic passing through a point P of 3?/,,_ t, the projection of Q on
M,,-t. If v denotes the above defined coordinate of Q on the normal through P, we can define
s k ,r
f(Q) = E F., k=O
In the normal c-neighborhood V of ~ - t we define the normal covariant
vector v~, by putt ing in any normal c-patch contained in V, v~ = 0 for
i : I , . . . , n - - 1 and v~ = 1. For any two such c-patches, this definition agrees
with the tensor transformation for covariant vectors, and for an arbitrary c-patch
contained in V we define the components v~ by the tensor transformation from
normal c-patches.
The vector v~ represents the covariant unit vector along the normals to
M=_~. We shall need it especially at the points of M~_I.
Let us recall, finally, a few facts about covariant differentiation. Covariant
derivatives of a tensor a t~ of contravariant rank r and covariant rank s form ~)
a tensor Di,+~ uo)-t~ whose components in each c-patch have the form
" / Dj,+I ah . . . A - - aY, �9 A "~- i" J,+l, ik a j, y~ Oxi,+~ k=~ l ,= l
- - ] , is+ , , - / ' % . i , 1=1 j ' = l
The brackets {i j, k} are the Christoffei's symbols of second kind. _(i) The covariant derivatives of order p, D~'js+~...j~._p ,~j~ are obtained by repeat-
ed differentiation:
D l (,, _(i) DP , i) DPs+l _(0, is+, au~ --- Dis+, u(. h, .~,+~ . . . j~,p ,U~ --= Dj~+p ( , . . . i ,+p-~ uu)l"
In a later section we shall need a formula which is obtained from that for
the divergence of a conlravariant vector u':
282 N. K R O N S Z & . I N , tk. N. ~ t ~ I L G R t M
Z o , " ' = E ' o , , i /~ Ox,
Let do) be a covariant tensor of rank s and b o) a contravariant tensor of
rank (s + 1). In the formula for divergence we put u i - - ~ aA. . .J , d t ...Jsi, J t �9 . . Js
and obtain easily
J l "" " J$ J$ J s + l
- a , ,
'1 " . ' i s J$+l
d " " .i, j,+~ 1/~ axe.., dx"
t=] A .. .Jr
b h . . . is t d x l . . , dx t-I dxt+ ~ . . . axn].
We shall also apply the following relation where c I') is a contravariant
tensor of rank s and r - - l , . . . , s :
Djr( $-r il "' ' ir--lJrJr+l'..Js) E E D)s Js--l'''Jr+l c J'r k Jr+ l J't +z " " ".~
il -.. it-1 Jr Jr+l �9 .. Js DS--r+l Z --is &-l . . .Jr+l Jr C
Jr J,+l"" &
PART 11. DIFFERENTIAL OPERATORS AND DIFFERENTIAL BILINEAR OPE-
RATOR-FORMS.
3. - - Differentiable manifolds. Definitions and statement o f theorems. A li. near differential operator A on a differentiable manifold M, is a mapping of the
space of functions of class C*" on M, into the space of continuous functions
on M, such that for any f the transform g - ~ - A f is given in any c-patch by
a finite sum
Ok, +" " " + k. / (X) (3.1) g (x) , ~ ak~..:k ~ (X) , x _--z (xl, . . ,, x,).
Ox~' . . . Ox~"
(t) See Levi-Civita, loc t i t pp. 153.154, (17') and (17").
DIFFERENTIAl, OpERATI~S ON RIEMANNIAN MANIFOLD~ ~)83
Thus restricted to each c-patch, the operator is assumed to behave like a con- ventional differential operator. The functions aki.. .k,, (x) are assumed independent
of f and continuous in the c-patch. It can be readily seen that the differential operator A provides a linear mapping of the space of functions of C" into the continuous functions. All functions referred to may be real- or complex-valued,
We shall assume always that (3.1) is in reduced form, i .e . that all terms
with the same derivatives of f are collected together. It is then clear that for a given operator A the functions akj.. "k,, (x) are com-
pletely determined in each c-patch. The operator A is ~ 0 if it transforms every function f into 0. This will happen if and only if all the coefficients aki.. ' k,,
in all c-patches vanish identically. If the functions ak,.. .kn (x) are of class C r in every c-patch, we shall say
that A is of class C ". If nt is the greatest number, m ~- kt + . . . + k,, for which in some c-patch is not identically 0, then A will be said to be of some akl . . . k ,~
order m. it is also conceivable that A has no finite order despite the fact that in each individual c-patch only a finite number of functions ak,.." ~,, (xj fail to
be identically 0. However, we shall only be concerned with operators of finite order m and of class C r with r ~ m. If the manifold M,, is compact, then obviously m is finite.
The operators of order m clearly determine a linear mapping of the space of functions of class C" into the continuous functions. In other words, the postulated mapping by A can be extended in an obvious way from the class C" to the larger class C m.
The differential operators form a vector space over the functions of class C ' . In fact, the operators of orders ~ m (m - - 0, 1, . . .) and class C r (r "-- 0, 1 , . . . , co) form a vector space over functions of class C'. This is clear since for functions f and g of class C' and operators A and B of orders < m, we can define the
operator f A + g B by
( fA + z B ) u - - f ( A u ) + g ( B t , )
and the resulting operator is of class C' and order <_ nl. We define the operator A B by the equation (A B) : A (Bu). We see imme-
diately that the product exists whenever the class number r ' of B is ~ the order m of A. The operator A B is then of class number min (r,r" - - m) and of order _<-- ram" where r is the class number of A and m' is the order of B.
284 DIFFERENTIAL OPERATORS ON RIEMANNIAN MANIFOLDS
The set of operators of class C" form a ring under this definition of multipli- cation, since in this case the product is always defined, and the associative and distributive laws are evidently satisfied.
If u and v are functions on the manifold M, and A is a linear differential
operator, Au . v is the simplest case of what we call a bilinear (hermitian) diffe- rential operator on M,. If M, is a Riemannian manifold, then the expression
�9 (Au. v) - - ~ g Au . v d x t . . , dx" is the simplest example of what we call a bilinear differential operator-form of rank n.
Defini t ion 1. A bilinear diff. operator B (u, v) is a mapping which associ- ates a cont inuous function w with each pair of functions u, v of class C" in M~, B (u, v ) - - w , such that in each c-patch, w is given by a finite sum
(3.2) u 0 t~ " v
l~(X) = Z Z bkl...]cn, l l . . . lr t (X) o / l + ' ' ' + k " + " "+ln- -
Here v means the complex conjugate of v. We suppose again that all terms with
the same derivatives of u and v are collected together; therefore the coefficients b are completely determined in each c-patch. The bilinear operator is 0 if it transforms every pair of functions u, v into the function identically 0. This hap- pens if and only if all the coefficients b in all c-patches vanish identically.
The maximal number m --- k~ + . . . -}- k, -1- 1~ ~ - . . . - 3 L- I, for all non-identi- cally vanishing coefficients bk~.., k~, t~. . . t , in all c-patches, is called the total
order of B. Similarly, the maximal numbers m' : k ~ - I - . . . - 1 - k , and m " = li - k . . . - ~ - l, are called the u-order and v-order of B. Clearly, B (u, v) is defined for all u in C" ' and all v in C m".
Defini t ion 2. If to each pair of functions u and v in C", an exterior differential p-form % (u, v) is associated so that in each c-patch the form can be written as
(3.3) ~p (u, v) - - '~. i 1 �9 ip
ate.. "il ' (u, v) dx ~ . . . dx tJ' ,
where the coefficients ac,)(u, v) are expressions of the form (3.2), we shall call % (u, v) a bilinear diff. operator-form (in abbreviation, o-form) of rank p.
Clearly, an o-form of rank 0 is a bilinear diff. operator. We define the
DIFFERENTIAl. OPERATORg ON RIEMANNIAN MANIFOLDS ~ 5
total order of r (u, v) as the maximal total order of all coefficients a/,i (u, v) in all c-patches. Similarly, we define the u-order and v-order of % (u, v).
We can form the differential d (gp (u, v)) which becomes an o-form of rank
p -]- 1. The differential increases the order by at most 1. if M,, is a Riemannian manifold we can also form the . (% (u, v)) which is
an o-form of rank n - p and of the same orders as =p.
The representations for B (u, v ) a n d % (u, v) given in (3.2) and (3.3) are
valid only locally. The question arises if a global representation exists in terms
of global linear differential operators. The affirmative answer is given by the following theorem.
Theorem !. I f ~ (u, v) is a bilinear differential o-form, then gp (u, v) - -
- - - ~ A l u B t v tp, where A t and B i are linear operators on M n and t p are t
p-forms independent o f u and v. This decomposition is not unique and can be
chosen so that total order ~ (u, v ) = max (order of A, q-order of B,). Also l
the u-order and v-order of ~p (u, v) are equal max (order of A,) and 1
max (order of B,) respectively. i
If p = 0, the theorem gives a decomposition of a bilinear diff. operator,
B (u, v) = ~ At u Bi v. l
The most important case for us is the case of rank n. To study this case
we shall choose a fixed exterior differential form g~ independent of u and v and
0 everywhere on M.. Clearly ~0 can be chosen in infinitely many ways (on a
Riemannian manifold we can choose ~0 - - ~ dx~...dxn), and any exterior dif-
ferential form a n can be writted as % : [ ~~ with a function f defined on /14,.
Hence
o~ (u, v ) : {y~,f , A , u B, v~ ~~ : B (u, v)~~ n n I1 P
VT" :
where B is a bilinear diff. operator on the manifold.
If % (u, v) is of u-order O, all At are of order O. Obviously this means that
they are of the form g, u where g, are functions on the manifold. Hence we
get a representation =,, (u, v) - - u. B v o,, where B is a linear diff. operator given
~9 - R e n d . C i rc . M a t e m . P a l e r m o , - - s e r i e I I - t o m o I [ - a n n o I953
2 ~ 6 N . A R O N S Z A J N , A, N . M I L G R . ~ M
by B - - y~ g, f , B,. Similarly, if ~ (u, v) is of v-order 0, we have ~,, (u, v) --- l
- - A u . v ~~
The next theorem can be considered as an extension of a well known for- mula in Euclidean spaces which leads to the notion of adjoint operators.
Theorem II. For any o-form of rank n, ~ (u, v), there exist unique operators
A ' and A " such that % (u,v) -- u A" v O__d (~',,_~ (u, v)) and % (u, v ) - -
-- A " u. v ~: : d (~'~-1 (u, v)). The o-forms ~'~_~ and 15~_~ are not unique. Order
A" - : orderA" ~ total order ~ (u, v). fY can be chosen so that total order of ~' ~ (total order ~) - - 1, u-order ~" <--- (u-order ~) ~ 1, and similarly for ~".
Since the o-forms ~_x and ~:_~ are not unique, there may be other forms,
,, ~ o d (V,,_~ (u, v)) and ~,,(u, v) say Yn-~ and "r,,_l satisfying % (u, v) - - u A' v ,, --- '
- A " u . v : d(Vn_ (u,v)). It fo l lows that is a closed o-form, it
as is also V, ,_~- [3~_~. in general, a closed form is not necessarily an exact
differential. However, for o-forms this is no longer true. indeed, if we call an
o-form ~ (u, v) of rank n closed whenever f . . ~,, (u, v) : O for all z~" and v d M n
vanishing outside of compact sets, and if we regard an o-form of rank 0 as an exact differential if and only if it is identically zero, we can say"
T h e o r e m Ill. Let 0 ~ p <-~ n. I f an o-form of rank p, % (u, v), is closed, then there exists an o-form ~p-~ (u, v) such that % (u, v ) - - d (~p-t (u, v)).
It follows that in Theorem II the o-forms [~,_~ and ,Sn_ ~ are unique except
for an additive exact differential of an o-form. All theorems stated above are proved by first reducing them to the case
of a single c-patch. This is done by using a locally finite covering of A4~ by
c-patches U k, k - - 1 , 2 , . . . , where U~ is a part of a c-patch U~ and corresponds
to an interval lk completely interior (together with its boundary) to the in-
terval I~ corresponding to U~. We consider then a partition of unity 1~0~} cor-
responding to the covering U~ and write, for any o-form
Each term r v) is clearly an o-form vanishing identically outside of U~.
DIFFERENTIAL OPERATORS ON II1EMANNIAN MANIFOLDS 287
it is therefore sufficient to study it only in the c-patch U', and, afort ior i , in
U~. If we prove each of our theorems for such o-forms, then the above decom- position allows us to prove them for general o-forms.
Let us call an o-form or operator restricted (abbreviated r.) if in a given c-patch represented by an interval the operator or o-form vanishes outside a closed subinterval 1' of /.
Theorems I, II, and III remain true if in their hypotheses and conclusions the o-forms and operators referred to are all restricted. We shall prove these theorems in this form in the next section. It is clear from our discussion that the original formulation is an immediate consequence of the restricted versions.
Remark. in Part !11 of this paper we shall need a more general notion of bilinear differential o-forms in systems of functions. This notion treats finite systems of mutually independent functions ut , . . . , uh which form a vector space with the obvious convention that ~ {ut , . . . , uh} a t- ~ { v t , . . . , v h l = {~ u~ q- v/v1 , . . . , ~uh @ "Ovh }. Thus we can consider bilinear diff. operators and also o-forms in two such systems. It is clear that such an o-form can be decomposed as fol lows:
h h
k = l 1=1
where ~ ' ; a r e uniquely determined o-forms in the sense of Definition 2. By this
decomposition the study of o-forms of systems of functions is reduced to the study of o-forms for single functions. It is also immediately seen that Theorems 1, ll, and Ill can be extended to o-forms acting on systems of functions.
4. Proofs of theorems in w 3.
Proof of Theorem I (restricted). Let ~p (u, v) be a r. tions (3. 3) and (3. 2) we have
o-form. By representa-
E il % (u, v) = aq .. ,p (u, v) dx
i I . . . z~
. . . d ; ~
(u, v) ~ a<,)(u, v) = / ~ Qt 1 ip e J e
k I . . . k n I 1 . . . I n
a(,:),k I ... k,~, i t ... l n
k l + . . . + ~ . ~l + . . . + ~,~_
c~ u O v h I k n 11 l n
Ox~ . . . Ox. Ox~ . . . Ox.
All the coefficients a vanish outside of i'. We then introduce a function of
2 8 8 ~q. A R O N S Z ~ J N , A. N. M I L G R A I ~
class C " in I for which there exists an interval 1" such that ]" (2 I " (5 1 "--7 C I and
that ~ - - 0 outs ide of 1", + - - 1 in F . It is clear that if we multiply the coefficients
a_ by de their value is not changed. Hence we can write
(4j)
:l'"IP k~::tk,,"t~
krt l II ... l n
k l ' F "'" "~'-kn i l
Ox I . . .Ox, ]
O :l+'''+tn v~ �9 '~ ~ - , T - - - - ~ t ~ l ( ~ d x t ~ " ' d x t " ) "
O X 1 . . . OX n ]
Ok l + ... + k n Oi l + ... + t n
Clearly 11(~), ~1 ... ~,, ,1 .., ~, k~ k. and d/ ii tn are restricted c~x 1 . . . Ox . Ox~ . . . Ox ,
operators and d e d x t ~ . . . d x 9 defines a restricted p-form. The order assert ions are
n o w obvious.
P r o o f o f T h e o r e m 11. (restricted). A straightforward calculation allows us to verify the fol lowing formula
ok1+'"+'-, (4.2) g, (u, v) ~ akl ... k,: q ... ,, --- ,~-------k~ ~"~---~ ~ ] d X l " ' " dx" :=
kl...k.tl...,. OX, . . . OX. OXl . . ~ OX. _J
,---T - - - E , , , a x ' . . , ax" +
- - k v
Z ',"+"+l+ +" Z "
( - - ( - - 1) ~+ ' - ' 0,~+ + ~ _ ~ + i - , .... ;'~ "~ kV--1 i--I
k l . . . k n i=1 O X ~ 1" " " OXV- - I OXV _ l l . . . in
( _ , , . . . ~,,+, . . . ~ , , . Ox 1 . . . o x ? ] J
a x ~ , . . d x ~ - i t / .X~- i . . . d x n r
DIFFEBENTIAL OPERATOBS ON' RIEMANNIAN MAIqIFOLI)S 2 8 9
where the square bracket under bar represents a r. operator A ' v and the braces
a r. o-form ~,-1 (u, v) which satisfy the requirements of our Theorem. It remains
to be proved that the operator A' is unique.
In fact, if there were another representation cc, (u, v) - - u It" v d x t . . , dx ~ = ~ z
d (~ ~-1 (u, v)) we would have u A v dx t . . . dx ~ = d (~J'~-t (u, v) - - ~J-',-~ (u, v))
where A = A' - - A'. Let 1" be an interval completely interior to I outside of which all the coef-
ficients of A', A' ~'~_t, and ~'~-t vanish identically, if A does not vanish iden-
tically in / there must be a point x ~ (x~. . . x ~ and a function v such that
A v (x ~ =l= 0. Hence Re (A v (x~ # 0 or Im (A v (x~ =~ O. Suppose that the first is true. (We proceed similarly if the second is true.)
The point x ~ must belong to I". Therefore there will exist, by continuity, a small interval 1'" completely interior to I", such that Re (A v (x)) does not vanish and is of constant sign for x E I '" . Consider then a function u of class C ", positive in I" ' and vanishing outside. By Stokes theorem, the integral
j ~ u A v d x l . . . d x ~ : f d (~'~_~ (u, v ) - ~'~_~ (u, v))is equal to the integrals j l tt j ~ s t
of (~J'~-~(u, v ) - ~'~-~ (u, v)) over the boundary submanifolds of I". But the boundary integrals vanish since the o-forms vanish on them identically. Hence
f ~ u A v d x ~ ' " d x ~ = = - f , l , ' . uAv d x ~ ' " d x ~ - - O " This is imp~ h~
since the real part of the last integral --- f u Re (A v (x)) d x t . . , dx ~ ~ O. This d t t t t
contradiction shows that A - - 0 . It is clear that by interchanging u and v and passing to the conjugate, we
obtain similarly that there exists a unique r. operator A" such that c~ (u, v ) -
- - A " u. v dxt . . . dx ~ : d (~J~_~ (u, v)) with A" and ~:_~ having similar proper-
ties to those of A' and ~'~-t, which completes the proof of Theorem If. Remark. in some of the considerations which follow, the coefficients of
c~ (u, v) and also the functions u and v may depend on some parameters besides being dependent on the coordinates x~. . . x~. In the expression of c~ (u, v) there may also enter terms in which there figure derivatives of u and v with respect to the parameters. Formula (4.2) shows that the restricted Theorem ll is still valid for such o-forms c~ (u, v) but then in the operator A' as well as in the o-form ~',_~ the parameters will also appear in the above described manner.
P,oof o f Theorem III. (restricted) The proof proceeds by induction with
2 ~ 0 N. A R O N S Z A J N , A. N. M I L G R A M
respect to the dimension n of the interval / ~ I,. We shall have to consider operators and o-forms which depend on a finite number of parameters t, in addition to the coordinates xk in I,. The functions u and v will also depend on the t,%. In the expression of the operators and o-forms there may also figure derivatives of u and v with respect to the parameters, besides their derivatives with respect to the xk's. in such cases we may consider a finite number of different derivatives of a function u with respect to the parameters as forming a system of functions and then treat the o-forms (or operators) as acting on such systems. Theorem !! (restricted) will remain valid. (See Remark.)
We begin the induction for the case n - - O. Clearly, p - - 0 and % (u, v) = B (u, v)
is a bilinear operator. Since the O-dimensional interval contains only a single point,
x ~ the condition that % (u, v) is closed, means (under our convention) that
v B (u, v) - - B (u (x~ v (x~ - - 0 for every u and v. Thus (u, v) = B (u, v) ~0
is identically 0 and hence by our convention is an exact differential.
Assuming the theorem is proved for n - - n" m 1, we shall then prove it for
n - - n'. Suppose that % (u, v) is defined in l~ and is closed: d o~, (u, v) = 0 for
p < n', or f % (u, v) : 0 for p -~- n'. j i , ,
If p - - n', we have by Theorem 11 (restricted): ~n (u, v) - - u A" v dx ~ . . . dx ~" -~-
d (~',,'-t (u, v)). Again, by Stokes t h e o r e m / i d~',,,-t (u, v) - - 0. Hence by our hy-
pothesis, f u A ; v d x ~ . . . d x ~ ' - - 0 for all u and v. This gives A ' : 0 and j t
(u, v) : d (u, v)). We suppose 0 < p < n'. In the expression of % (u, v) as an exterior dif-
ferential form, the terms which do not contain d x " will be separated from those
which do. The sum of the former will be denoted by ~p (u, v); the latter by
~'p-i (u, v ) d x " . The o-forms ~, (u, v) and d'p-i (u, v) may be considered as
o-forms defined on the variable intervals /,~,_l determined as sections of I,,, by
the hyperplanes x ,~ , - -cons t . ; they are clearly r. o-forms on 1,~,_~. x,,, should
then be considered as a new parameter to be adjoined to the other parameters
t~ which may already figure in % (u, v). When this meaning is given to 0~, (u, v)
and ~ ~',_~ (u, v), it is clear that ~p is the restriction of ~p to /,,,_~. Denoting the
differential on I,,,_~ by d, we write
DIFFEREN'[IAL OPERATORS ON RIEMArqNIAN MANIFOLDS 2(~|
0 : d ~, (u, v) : d [;, (u, v) -~ o~,- (u, v) dx ~'] ---el ~ (u, v)
-{- (-- 1)" % (u, v) dx"" -[- (el "" (u, v)) dx"'.
It follows that
(4.3) ~ ;,, (u, v) : o ,
(4.3') d *" (u, v) + ( - - l y a - =, - , ~ . ; =. ( . , 0 : o.
Since the theorem is assumed valid for n - - n ' - - 1, there exists a r. o-form
which figure i~p_~ (u, v) defined on ./~._~ , depending on all the parameters
in ~p (u, v), in particular on x,. and such that
;, (u, ,,) = ~ l~p-, (u, v).
Since 0__ (,) ~,_t (u, v)) = ~? ~x~-, ~'-' ("' v) , we obtain from (4.3'):
tl [ " ( u , v ) - ~ ( - - l ) ~~ 0 ] ,_, a ~ ~_~ (u, v) = 0.
Again, it follows that for p :> 1, there exists a r. o-form Tp-, (u, v) such that
. 0 ,,'p-i (u, v) + ( - 1), ~ ; ~_~ (u, v) = ~ (~_~ (u, v)).
Hence
(o / a, (u, v) = ,; ~,-, (u, v) + ( - - 1) ' - ' Ox~, 0,-, (u, v) d x ' - ~ [l (~(v-2 (u, v)) dx" =
= a v) + § ].
If p = 1, we obtain
. , O ~ ~, ~_, (u, v) + ( - 1), ~ ~,_., (u, 0 = o,
~ ( ~ N. A R O N S Z A J N , A. N. M I L G R A M
since the left side of this equation is a zero-form which is an exact differential.
Hence
/ ) a, (u, v) --- ~/~,_., (u, v) -~- ( - -1) p - ' ~ ~v-, (u, v) dx ~" - - d ~,..., (u, v).
Finally, when p - - 0, d =p (u, v) - - 0, which means that all the partial derivatives of
=, (u, v) ------ % (u, v) ---- B (u, v) are identically 0 in I,.. Hence B (u, v) is a con-
stant, and since it must vanish outside of an interval completely interior to I /
it is = 0 for every a and v. Thus =0 (u, v ) - - 0 , and therefore by our conven-
tion it is an exact differential. This completes the proof of Theorem Ill.
5. - - Hiemannian manifolds, differential operators and their adjoints. We
assume M~ to be a Riemannian manifold with the metric tensor g,~. Let A be
a linear differential operator. The expression Au. v is then a bilinear differential
operator, i. e. an o-form of rank 0. To this operator we can apply the star
operator and obtain
�9 (A,,. Vg Aa.v d x ' . . . d x n
which is an o-form of rank n.
Theorem !I establishes the existence of a unique operator A* and o-form ~,-t (u, v) such that
of an
(5.1) �9 (A u. 7) = . (u A* v) + d ~._, (u, v).
Here, Theorem II is applied with the n-form =o __ ~/~ d x ' . . . d x " . In view of
Theorem Ill, the o-form ~ - t (u, v) is completely determined except for an ad-
ditive exact differential. The operator A* is of the same order m as A. The
o-form ~,_t (u, v) can be chosen of total order <~ m - - 1 and in all future applica-
tions its choice will be so restricted.
The operator A* is called the adjoint operator of A and plays an impor-
tant role in the theory of differential problems.
The o-form [~_: (u, v ) i s called a concomitant o-form of A. As was seen
it is determined up to an additive exact differential.
D I F F E R E N T I A L OPERATORS ON R I E M A N N I A N ~ I A N I F O L D S 293
By interchanging u and v and taking the conjugate, we deduce immediately
from (5.1) that ( A * ) * - - A . Also, the correspondence between A and A* is anti- linear, i. e. for two operators A and B, and two complex constants a and b, we
have (a A -~- b B)* ---_ a A* + b B*.
Similarly, we verify that if i3._~ (u , v ) is concomitant of A, - - [~ . - i (v, u) is concomitant of A*.
In order to deduce further relations between A and A* we shall establish some invariant representation of the general operator A. To this end we consider in any c-patch the developments of the operator A n :
(5.2) A u = Z u (x). Ah...,~" (x) 0k~+"'+k"
kl...k" OX~I" ' ' OX~ t~
Consider the coefficients of derivatives of u of highest total order k i +... + k,~ == m. i , . . . i m
These coefficients determine a symmetric system of rank m, a , 1 ~< i, ~ n, in the following way: If among the indices it there are ki which are : 1, k2
�9 " k i ~ k ~ which are = 2, . . . , k~ which are ~--- n, we put a q 'ra - - . . . . . A~...~ . We
m
can write
E 0" u (x) A u = ai~...tm (x) 0xi~ " " "Oxl,,,
i I . . . i ra
~- (sum of terms of orders ~ m - 1).
The system a (0 is completely determined by the operator A in each c-patch, and it is easily verified that it forms a contravariant tensor of rank m on the whole
manifold M.~.
Consider now the covariant derivatives of order m, D " t~...ira u (x) and t h e
compound tensor of rank 0, ~ _~0 r, ra u "-'t0 u (x). Clearly this is a differential ope- i l . . . t r a
rator of order m. Since in each c-patch the covariant derivative D m i, ... ,ra u (x) can
be represented by
D m 0 ~ u (x) "-t i,,..im U (X) : Ox,
+ (linear combination of derivatives of u of total orders ~ m - - I),
294 N . A R O N S Z A J N , A . N . M I L G I 1 A M
it is clear that Au -- ~ a <i~ D~) u (x)is a differential operator of order <~ m - 1. it... im
By successively repeating the same procedure we define symmetric contravari-
ant tensors a~ ) of rank k (putting a (i) ~ a~ )) for k - - m, m - - 1 , . . . O, in such a
tn k way that Au -- E E O(i) Dti)u is a differential operator of order < r - - 1.
k=r i 1...i k
Finally, we get the invariant representation of an arbitrary operator A of order
m in the form
m
CA k l J ( t ) II.
k=o i 1... l k
We shall obtain a representation for the adjoint operator A* if we find the ~(i) k adjoint for each operator ~ ~k D(i)ii. To this end w e apply the last formu-
i t ... i h
ias in w 2. For r - - 1, . . . . , k, w e obtain
( - - ' l ) k - - r E Dr..ii it ̀ [l ~ O k-rik ik_l ... ir__l ([t~ l ' ' ' i ' ' l r~ - l~ V ) ~ d X1 � 9 d xn
l I ... l r lr+l.., i k
D,-1 Fik--r+l U ~.~ ~ l k l k _ t ... t r (__ | )k--r§ E it . ' . lr--I
i t ... Jr_ t tr... lit
l , , . ,k V ,tx t . . .
+ d I (--l)k--r ~ ( - |)1--1 Cg i l ~ l -- il ...Eiu D'Iu tu U
x E ik ik--1 ... 1u 1 ir+ 1 ... i k
X dxt ... dx~-i dxt+t." dx" I"
DIFFERENTIAL OPERATORS ON R I E M A N N I A N M A N I F O L D S 295
By adding these equalities for r--: 1 , . . . , k we obtain
(5.5) ( E a~) Do'u" ) ijV'~dx''''dxn=(-1)'u
l l , . . i k i l . . . i k
D• . . . (i) ((l~k t )) V 1/ g E~X ~ dx n +
+ d r- n
k l)k--r l~=l l--1 ~ Z il'"ir--I D r - 1 ~ ( - ( -1 , r-=, i 1 . . i t _ 1
D i i i ( t l ~ l ' ' ' i r - I l i r + l ' " i k ~ ) X Z k--r k k - - l " " r-I-1
ir_t_l ... i k
X d x j . . . dx ~-~ d xt+~...dx"ll
k This formula shows that the adjoint operator of ~ a~ ) D(o u is given by i 1 ... i k
( - l) k F~ Dk (a~" ,). lk. . . i 1
i l ... i h
It follows that the adjoint A* of A represented by (5.4) is given by
m
(5.0) , 4 . , = E ( - 17 ~ D k,~ ,, (a~' .) k=O i, ... ' k
For an operator A of order m, the tensor a~ ) of the decomposition (5.4)
will be called the leading coefficient tensor of A. From (5.0) it follows that the t l . . . i m
leading coefficient tensor of A* is given by (-- 1) m a m
The operator A is called self-adjoint if A - - A*. If A is seif-adjoint, the
leading coefficient tensors for A and A* must be the same; hence
l I ... l m i l . . . I m a m - - ( - - 1 ) m a r e
Therefore if m is even, the leading coefficient tensor must have all real com-
ponents; if m is odd, all components will be purely imaginary. Hence the gener-
2 ~ 0 N. A R O N S Z A J N , A. N. M I L G R A M
al form of the leading coefficient tensor of a self-adjoint operator A is
(-- 1) m/~ b~ "''tin with a symmetric real-valued contravariant tensor b <i~ --rr149
Let ,4. be a self-adjoint operator of order m, and ( - - 1) m/2 b~ "'lm its leading
coefficient tensor. It is obvious that the differential operator
2 O..D,, , .+ . . . , l i I . . . i m 1 1 . . . i m
is seif-adjoint and also has ( - - 1) m/2 ~o b m for leading coefficient tensor. Therefore
subtracting this operator from A, we obtain a self-adjoint operator of order
< m - - 1. Consequently, the general form of a seif-adjoint operator A of
order m is given by
5.7) A . .,i, k Ok =-- ok D(i, u-~- ~ 'k'"" ' k = o 2 i l . . . i k t i . . . i k
where b~ ) are arbitrary symmetric real-valued contravariant tensors of rank k.
If we take the product of two operators A and B, we deduce from (5.1) that
(5.8) (A B)* = B* a*.
In fact, in addition to (5.1) we can write *(Bu.-v) = ,,(u B* v) @ d y,-~ (u, v)
Replacing v by A* v, we get *(Bu A* v) - - *(u B* A* v) - } -d y.-~ (u, A* v). On
the other hand, replacing u by Bu in (5.1) we have
-(A Bu . v) = . (Bu A* v) + d ~._, (B. , v).
Adding the last two formulas, we obtain
. (A Bu . v) - - *(u B* A* v) -+- d ~,,-~ (Bu, v) q- d "r (u, A* v)),
which proves (5.8). (0 For any covariant Let A be of order m with leading coefficient tensor a,,,.
D I F F E R E N T I A L OPERATORS ON R I E M A N N 1 A N M A N I F O L D S 297
vector v, (i. e. a covariant tensor of rank I) we define
i 1 . . i m
(5.9) ['JA (V i ) : E am t'it'"Vira" i! .., ira
Pa (v~) is a homogeneous polynomial of order m in the components v, of
the vector, and it represents a function defined on the whole manifold. For two operators A and B it is easily verified that
(5.10) P.~. (v,) = p ~ (v,) p,, (v,).
We have Pa. (v,) - - ( - - 1) '~ P., (v,). In particular, for a self-adjoint operator A
and a real-valued vector v, P.~ (v,) is real- valued for even order m, and is pu-
rely imaginary for odd order m.
An operator A is said to be elliptic at a point t E M. if at t the coefficient
tensor a (*) is real-valued and for any covariant real-valued vector v, which does
not vanish at t, P., (v~) =i= 0 at t. The operator is parabolic at t if a (') is real-
valued, P~, (v,) is of constant sign or - - 0 at t for any real-valued vector vi and
for at least one such vector non-vanishing at t, P , (v,) = 0 at t. The operator is
elliptic on a subset S of 34. if it is elliptic at each point of S. It is parabolic on S if A is elliptic or parabolic at every point of S and parabolic at some points of S.
An elliptic or parabolic operator on the whole of M. must be of even
order. An operator A is called formally positive if there exists a finite number
of operators such that
(5.11) A - - ~ a] A k k
If mj, is the order of Ak, then tile order of A - - max 2m~. Therefore this k
order is always even and will be denoted by 2m'. if k' runs through the indi-
ces k for which mk = m', then for any real-valued vector v,,
r,_, (v,) : : ~ p,,:~, . , . 0,,) - ( - - 1),,,' ~ t P,,,,, (v,)!~. k" k'
Hence a formally positive operator is always elliptic or parabolic.
2{~8 N. A R O N S Z A J N , A. N. M I L G R A M
If we denote a concomitant o-form of A k by k ' . 13,,_~ (u, v), we obtain, by using
(5.9), the following representation for the positive operator A:
k I This formula is of great importance in the study of differential problems with formally positive operators.
PART 111 -- BOUNDARY OPERATORS AND LINEAR DIFFERENTIAL SYS- TEMS ON RIEMANNIAN MANIFOLDS.
6 . - Boundary operatols on a submanifold M~-t. The developments of the next two sections will form a background for the treatment of linear diffe- rential problems. Ordinarily, the differential problem is stated for a domain D of a manifold (say the Euclidean space) with a boundary B composed of sub- manifolds of different dimensions. The problem asks for a function u in the domaiil satisfying a differential equation in D and boundary conditions on B. The differential equation is of the form A u - - f in D, and the boundary condi- tions B~ u--r where A and B, are linear differential operators, f a function defined in D, and ~i functions defined on the boundary manifolds of D. The boundary conditions may be different on each boundary submanifold.
In this paper we shall treat only certain aspects of the differential prob- lem namely the relations between the operators given in D and the boundary operators on the (n -- 1)-dimensional boundary submanifolds.
We shall be concerned with a boundary submanifoid M~-t and the adja-
cent part of D. We may always assume that the orientation on/1~/~_ t is chosen so
that the adjacent part of D (together with bl,,_t) forms a positive neighbor-
hood of M,,-, (1).
From now on we shall consider a single submanifold M,,-i. The domain D
which is essential for the complete differential problem and for which M,,_~ is assumed to be a boundary submanifold will no longer enter into our develop- ments. We shall consider operators, o-forms, functions and tensors in a c.neighbor-
hood of At,,-1 which may always be assumed a normal neighborhood. There would be no loss of generality if we assume that all the operators, o-forms, etc. are defined on the whole of M,. On the other hand, we may restrict the manifold
(i). In some cases both sides of A~/,~_l may be contained in D. The usual way of handling
his situation is to consider M',~_t in the two opposite orientations as two distinct boundary
DIFFERENTIAL OPERATORS ON RIEMANNIAN MANIFOLDS 2 ~ )
M, to this neighborhood of M,-I. 0). All operators and o-forms will be assu- med to be of class C = (2).
Consider a differential operator B of order m. it can be applied to functions
u of class C" in the neighborhood of h;/,,_~. Bu considered on M, ~ is a well
determined function on the submanifold. The function Bu, so restricted to M',_t,
O~U k _ _ O , will be denoted by /~u. The consecutive normal derivatives of u, 0 v - - -2 '
l , . . . , m form a system of mutually independent functions on ~/,_~ (see w
/~u becomes a linear differential operator defined on M,_~, acting on the system
0~u t
In fact, if we take any normal c-patch U and represent Bu in terms of co- ordinates x~ , . . . , x , _ , x,= ~ v, we have
I n
u= E Z k=O ll' ." I n - 1
O tt+'' '+tn-I O k II
Oxt|, _, t._, 0 ~ ' I , - ' v . . . + 1._~ + # < m. �9 �9 �9 OXn-- z
FL_ Since the tl are mutually independent on /lT/._a, the formula represents a linear O V k
differential operator acting on the system! O'~-u-i.lt can therefore be written as r
g=O
O~ being linear differential operators on .,i4,_ t (considered as a manifold in itself) acting on single functions. The O~. are clearly uniquely determined.
submanifolds of D and for each of these to consider as the adjacent of part D their positive c-neighborhoods contained in D.
(1) In the case considered in the preceding footnote, if an operator or function is given in
D, we shall always assume that they can be continued from one side of 2~/,,_i to the other in
a whole neighborhood of A4,,_t so as to be of the required class C "= or C r in this neighbor- hood. However, it will not be required that this extension from one side to the other coincide on the second side with the definition of the function or operator as given on the second side
(-~) As stated in w the developments are valid even for the case where C | is replaced by C r, when r is sufficiently larger than all orders of the operators and o-forms under conside-
ration.
~ 0 0 N . A R O N S Z A J N , A. N. M I L G R A M
Consider now an o-form % (u, v) of total order m. For any couple of functions
u and v it is an exterior differential form defined in a neighborhood of M,,-l,
and hence we can consider its restriction dp (u, v) to M~_~. In a manner similar
to that employed for operators, we verify that in every normal c-patch the coef-
ficients of the e~ (u, v) are bilinear forms in the two systems ........... o3~u and ..... ,. �9 o3v ~ , , 0 ~ ~ !
Hence ~v (u, v) is an o-form on M.,-I acting on systems of functions and, as was
remarked at the end of w can be written
(6.2) k . l
where the summation indices k,l must satisfy k+l<--m and &,,t are o-forms de- - p
fined on M,,-t of total order <_ m - - k -- l acting on single functions.
A linear differential operator B restricted in the above described manner to
M-- i will be called a linear differential boundary operator f3. It is clear that the
same boundary operator /3 can be obtained from different operators B' defined
in a neighborhood of M--v As shown above, Bu is essentially a linear operator
Nowever, have such defined on ~,,_t and acting on the system to3 v" j"
an operator B {~0~,} defined on M,,-t and acting on systems of functions ~ it
will become a boundary operator only if the ~,, are replaced by ~ u for a fun- o3 v ~
ction u defined in a neighborhood of M,,_t. Such a boundary operator can al-
ways be considered as obtained by restriction to M,,_~ of an operator Bu defi-
ned in a normal c-neighborhood of M,,_~. In fact, in every normal c-patch with
coordinates x : , . . . , x,,_~ on ,~,,_~ and the normal coordinate x,, - - v, we can write
0--~-. Here, the coefficients of Bu do not depend on the variable v.
This extension of B is only defined in a normal c-neighborhood. There are
infinitely many other extensions; in some circumstances there may be extensions
defined on the whole of M~.
Similarly, if ~v {~, +z} is an o-form in the system {~kl and {~?~i defined on
I I I F F E l l E N T I A L O P E I I A T O I S ON R I E M A I C N I A N M A N I F O L D S 3 0 1
M--I we can consider its extension =p (u, v) given in any normal c-patch by
" i ok u 0' 1, I ~p . . . . the coefficients depending only on the coordinates on M~-t. Again 0 v k' 0 r ~'
there are infinitely many other extensions.
From now on we shall use the following notations. As before, Au, B (u, v)
and ~p (u, v) will denote operators, bilinear operators, and o-forms defined in a
neighborhood of M,,-t. If these are defined, then .8,u, 13 (u, v), and ~p (u, v) will
denote their restrictions to /r as explained above. However, the last symbols
may be used without the preceding being defined; they are then defined only
on &/',,_~ and can be written also as a {?~], /~ [~, ~,}, and ~p [q0~, ~} with the 0 k u 0 ~ v
understanding that ?k stays for03v k and ~ for 0v '
It will be convenient to deviate from this general notation for the case of
operators or bilinear operators defined on /~/~-1 and acting on single functions;
these will be denoted by Oreek capitals, e. g. O, 11, etc:O ~, 11 (~, +) and so on.
The symbol 0~u will denote the restriction of the function u to M.,_~; / de- 0 v ~
notes the restriction of the identity operator to A4,,_,/u - - 0~ u. For o-forms de- 0 3 v ~
fined on/~/~_~ and acting on single functions we shall use the notation dp (% +)
as in (6.2).
For any operator A defined in a neighborhood of ~r,~_1 we can form the
product B Au ---- J3 (Au) as long as the function u is of sufficiently high class.
The highest normal derivative of u figuring in /~u is called the normal or.
der of J~. In the representation (0.1) the normal order of 13 may be smaller than
the total order m of B since some of the operators O~ may be identically 0.
/3 is called a normal operator if its normal order is equal to its total order m
and if the operator O,~ (in (6.1))is the identity operator 1.
In the following we shall deal mainly with normal boundary operators.
A finite system of boundary operators 1/~,I is called independent (totally in-
dependent) if for any choice of functions ?, on M,_~ of sufficiently high class
there exists a function u in the neighborhood M._~ such that/~, u "--- 72 for all i. 20 - R e n d . Circ . M a t e m , Pale~ 'mo, - - ~ r i e I I - t o m o I I - a n n o x953
3 0 2 N. A I I O N S Z A J N , A. N. M I L 6 R A M
A system l/~,I is called normal if each B, is a normal boundary operator and the
orders of /~, form a stricty increasing sequence. A normal system l/~l, i = 0
1 , . . . , m -- I, is called a Dirichlet system of order m if the order of 1)4 "--i for
each i. In such a system it is clear that /)0 - - s
A normal system t/~jI can always be considered (in infinitely many ways)
as a part of a Dirichlet system. In fact, let m~ be the order of/~j, j --- l, 2 , . . . , k.
For each non-negative integer m" < mk, which does not figure in {mj}, we take
( "/ an arbitrary normal operator /~'~, of order m" e. g. ~-v,?,] and adjoint all these
operators to the system l/~jl. The resulting system, suitably ordered, will form
a Dirichlet system II2,'~I, of order mk-~ 1.
L e m m a 1. - - Every normal system 1B, I is an independent system.
Proof. In view of the preceding remarks we may limit ourselves to the
case when t/~f is a Dirichlet system of some order m. Given any functions r
on ,4~1~_~, i - - O, 1 , . . . , m - - 1, of sufficiently high class, we must find a function
u in the neighborhood of M~_~, satisfying on /I]._ t the system of equations
l--1 C~ j O' u u (6.3) /~,u - - 0~,' t- ~ 0,,~ OvJ - - ~ ' '
j----o
i = O, l , . . . , m - - 1.
From these equations we determine successively the normal derivatives of u:
0 ~ U - - - ----- % o n /14,,-t,
v ~
c3v - - - - ~ l - - 0 t , o U - - ~ I - - 0 1 , o % ,
c3 ~ u O u ~ s - - O ~ . o u - - 0,~,1 - -
Ov ~ 0 v " - ~o.~ - - 0.~. o ~ o - - (')'~,t (71 - - 0 1 , o % )
and so on.
DIFFERENTIAL OPERATORS ON RIEMANNIAN MANIFOLDS 3 0 3
The normal derivatives - - being so determined for all i - - 0, 1, . . . . m - - 1, <9 r
we can then find a function u possessing these normal derivatives, and it is
easy to verify thai u satisfies /3, u : r
L e m m a 2. --- i f ~ B+i is a Dirichlet system of order m, then every boundary
operator B of normal a unique representation
(6.4)
order ,< m can be expressed in terms of the lJ+ by
m -- I
D : E r, B,, i=0
where I', are operators defined on M.,_~ acting on single functions. In view of (6.1) we can ensure the possibility of (6.4) by proving thai each
c3ku for k < m, can be represented in this way. This pure normal derivative ~ v k ,
is easily achieved by applying the same procedure as in tile proof of Lemma I
with functions epi replaced by /~+ u. The uniqueness of this representation follows from that of (6.1). in fact, if
we express the /J~ in terms of normal derivatives as in (6.3), we can put these expressions in (6.4) and obtain the following expressions for the operators
O~ of the representation (6.1) of /~:
m - - l - - k
(6.5) O~ : I',+ + E r',+s 0~._~,~. j = l
These expressions determine the O / s in terms of the I ' / s , but it is immediately seen that they establish a unique correspondence between the two systems, namely
r , m - 1 " - - ~m.--D,
I",,+_+ - - 0 , . _ m - - I ' . . . . t O , , + - t , ~ - + - - 0+,+_+ - - 0 , . _ i O , , , _ t , , , , _ m
etc, successively for 1'~, with decreasing indices k : m - - 1, m - - 2, . . . . 0. Hence the rk are uniquely determined in terms of 0,.
304 s . A R O N S Z A J N , A. N. M I L G R A M
Remark. In differential problems most commonly encountered, the boundary
conditions can be expressed in terms of normal systems of boundary operators.
The limitation to normal systems is due mainly to two facts:
1 ~ A system which is composed of even one single not normal operator /~
may not be independent in the sense of the definition given above. A simple
example is given by the circumference in the plane which plays the role of M--t.
The arc length s on the circumference can serve as coordinate xl in a normal
c-patch. ~-u- is then a boundary operator, although not a normal one, and the r
r condition -- 1 cannot be satisfied by any function u continuous and uniform
r
in a neighborhood of the circumference. The difficulty may be avoided if the
conditions for independence are weakened. We may replace the notion of inde-
pendence by the notion of local independence which requires that for each c-patch
on M,,_~ there exist a u such that /J~ u -- q0~ is satisfied in the c-patch.
20 In connection with developments in the next section where linear
differential systems and their adjoint systems are introduced, the use of non-
normal boundary operators and systems may lead to the necessity of introdu-
cing adjoint systems in which the boundary operators will not be differential
but integro-differential. Such cases do arise in applications but will not be trea-
ted here since their general theory is much more complicated. In the few instan-
ces when non-normal boundary operators appear in differential problems, they
can be treated by a suitable transformation of the problem, or by specifically
introducing the integro-differentiai boundary operators and making the necessary
changes in the relevant developments.
Oiven two systems of boundary operators /3, I and I/~i~ we say that the
second is weaker than the first if for any function u, sufficiently regular in a
neighborhood of M,,-i, the functions /3~ u completely determine the functions
Bju. In this case we say also that the first system is stroneer than the second.
If the first system is both weaker and stronger than the second we say thai the
two systems are equivalent.
DIFFERENTIAL OPERATORS ON BIEMANNIAN MANIFOLDS 3 0 5
L e m m a 3. In order that the system { BjI be weaker than { t~ 1 it is ne-
cessary and sufficient that for every u sufficiently regular in the neighborhood
of M~-t the conditions B, u - - 0 everywhere on I(I~- t for all i imp.ly B~ - - 0
everywhere on ~[~-~ for all j.
Proof. 1 ~ Necessity. if some ~ u are not 0 on the whole of M,-i we
replace u by c u with an arbitrary constant c. In view of the linearity,
B, (cu) - - 0 - - B, u everywhere on M,~-t whereas /3~. (cu) - - c B} ~ 1~} u in
contradiction to the hypothesis that {/3~] is weaker than [Bi} .
2 ~ Sufficiency. Suppose that for two functions u' and u" we have B, u' - - B~ u"
everywhere on/l~/~_ i for all i. It follows that B, (u' -- u") - 0. Hence/~. (u' - - u") : 0
everywhere on M~_t for all j which gives /3~.u" : Byu." "
T h eorem IV. In order that a system of boundary operators {/~j} be weak-
er than a normal system {/},) it is necessary and sufficient that each operator
13"j be representable in the form
(6. 6) E At, b,.
I f this representation is possible, it is unique and effectively contains only terms
with operators B, of order ~ the normal order of B~.
Proof. It is obvious that the condition is sufficient. In order to prove the
necessity, it is enough to consider the case of the system {/3j.I reduced to a
single operator /~'.
Let m be the highest normal order of all operators B, and of /~'. We can
consider {B, } as a part-system of a Dirichlet system {B~,I of order m-~-l. By
Lemma 2, we have the representation
3 0 6 N. A R O N S Z A J N , A. N. M I L G R A M
m I
k~O
where m' "is the normal order of /~'. A
We shall show that all the operators I'~ corresponding to B~ which do not
occur in the system {/3~1 must vanish identically. This will prove (6. 6) and the second half of the last statement in the theorem.
in fact, we assume that one such P~, say Pk0, does not vanish identi-
cally. Therefore there will exist a sufficiently regular function ~ defined on-/~/,~-i
such that r~o ~0 is not identically 0. By Lemma 1, {/~,} is independent. Therefore
there exists a function u defined in a neighborhood of /l~',, t such t h a t / ~ u - - 0
on Mn-t for all k # k0 and ~ o u = ~0. Hence /~, u = 0 for all i and by (6. 6.1)
B' u = rko n ~ k0 u --- I'~ 0 q~ # o, which is in contradiction to the hypothesis
that /3' is weaker than [ B , I . That the representation (6. 6) is unique follows
from the fact that (6.6. !) is unique (see Lemma 2).
As immediate consequence of Theorem IV we have the following corollaries:
C o r o l l a r y I. I f {/~j} is weaker than the normal system IBm}, each
operator B~ is of the same normal order as some operator B~.
C o r o l l a r y 2. I f the normal system t [~) I is equivalent to a normal
system [ B~}, the two systems contain the some number of operators, which, if numbered by the same index i, are representable by the formulas
(o. 7) Bi = E A,, A,,, = i , j ~ i
In the sequel, if /3, denotes a normal boundary operator, the lower index i
will denote its order. The set {i'} of indices in a normal system [B~,I is the
DIFFERENTIAL OPERATORS ON RIEMANNIAN MANIFOLDS 307
same for all systems equivalent to { B,. }. We shall denote by j ' , k ' , l ' . . . indices
belonging to the set { i" } . Similarly j , k . . , or j", k" . . . or j ' , k' . . . will denote
indices varying in the same set as i or i" or ~7 respectively. If all the operators in a normal system are of orders < m the system will
be said to be of order ~ m (for instance, a Dirichlet system of order m is of order __< m,
�9 We shall treat transformations of normal systems which are best expressed
in terms of matrices of operators. A normal system {/~,, } will be treated as a column vec',or. A matrix {Ar162 of operators, where the row index i' and co- lumn-index i" may vary in different sets of indices {i'} and {i"} will be called an o-matrix if the operators A<.,,, satisfy
Ar is o f order ~ i" ~ i" i f i" > i " .
(6. 8) Ar - - I if i' = i" ,
A,,.,,, - - 0 i f i" < i" .
An o-matrix {A<.<, i transforms a normal system {/)<, } i n t o a system
{ = {At..,..) {B , . . } :
(o. 9) & - E A<,<. &- . i "~i"
B r are of order < i ' but not necessarily normal. However, Here, the operators " ' = A r
if { r 1 c t i" / the transformed system { B <} is normal. For two o-matrices I_A4!!;.I and t-h4,~)e.f we can form their product
t A~,~)i,,,} : IA!,~?;,,t IA~,9.e,,l , where
( 6 . 1 0 ) N 2) - - ~ A0~. A (~) ,
This product is associative, but the product matrix is not in general an o-ma-
trix (A~ z) .... may be :~/*for i' --- i" ') �9 A particularly important class of o-matrices
is the one with row-indices and column-indices varying in the same fixed set I t ' } . Such o-matrices /Ae.~,} will be called triangular; they form a group.
A I - -1 It is immediately verified that { r = {At: r is given by
3 0 8 l'q. AP, O N S Z A J N , A. N. M I L G R A M
(0. 11) A',,..r : ~ - ] ( - - I)" J'[~ 1 i" =i'o > i't >. .. > i 'e= j "
Ar162 Ai.ir ' ... A,,_,r for i '>j', Ar - - I,
where the inner sum is extended over all systems i ' o , . . . , i ' , of indices from {i'l satisfying the marked conditions.
It is dear that the two matrices {&.jl and {A%/ of (6,7) and (6. 7')are reciprocals of each other.
Let {/}~} be a Dirichlet system of order m. Let [i '} denote the set of in-
dices corresponding to a normal system I B'r , i ' < m - l . Denote by { i" l
the complementary set of indices { i} - - { i ' } . The system I/3'r will be said
to be in canonical form with respect to l b~ 1 if for each i" we have
(o. 12) a ' < = & , + Z ai,,,,,B~,, i.e. t ~" - . . + t A,,,,,ItB,,,) j " < i"
T h e o r e m V. Each normal system o[ order ~ m is equivalent to one and only one system in canonical form relative to a fixed Dirichlet system of order m.
Proof. Let i'o be the largest number in the set { i '} . Assuming the theorem
proved for orders less than m, the normal system l B'r 1 , where { f ' } = [ i" } - -
-- l i'o } is equivalent to a canonical system t ~,;,, I relative to the Dirichlet system
l/}il , i < m - - 2 . Since
j < i'o--I
we can set B:~o = /~eo - - ~ A,o.f,, ~;,,, and clearly the adjunction of Bi:o to j " < i " 1
I/~:,,I gives us the required system l h : t in canonical form. The beginning of, the induction, m = 1, is trivial.
The uniqueness of the canonical system follows from the fact that if there
were two such systems l B:;I and t C'::f equivalent to I b~, I they would be equi- valent. Hence
~ '= E r< , , , g := E r, ,<B~,+ E E r,,.,, ~-~,,~,,, j,<e y,<,, y,<e I'<J'
and therefore F,, o, - - ~r i (8,a being the Kronecker symbol) i. e. ~ _: /3~',.
DIFFEI~ENTIAL OPERATORS ON IIIEMANNIAN MANIFOLDS ~ 0 0
7. Linear dif.terential systems, adjoint systems, self-adjoint and positive
systems. An operator A defined in a neighborhood of 2f/~_l and a sequence
l/~, 1 of boundary operators on M~_t form a linear differential system l A; B, I
at 2f/~_~. The boundary operators /3~ will be assumed to be of order smaller
than the order of A. If the /3~ form a normal system, the linear differential system will also be called normal
Consider the homogeneous polynomial Pa introduced in w 5 at the points
of M~__~ for the normal vector v~. Its value will be a function p (t) defined at
every point t of M~-t:
(7. 1) p(t) ~ Pa (t) = PA (v, (t)).
If p( t ) does not vanish at any point t of M~_~ we shall say that A is
non-tangential at M~_,. It is clear that if A is elliptic in a neighborhood of
M,_~ it is a fortiori non-tangential at M~_,, but the converse is not in general true. I I t t
Two o-forms ~ (u, v) and % (u, v) will be called equivalent, written % ~ a" v if a" -- a" is an exact differential.
p P
L e m m a 1. If t~, and C~ are two Dirichlet systems of order m, and ~_~ (u, v)
is an o-form of total order < m on the submanifold M~_~, then there exists one
and only one set of operators O,j on fcI~_~ of orders ~ m - - 1 - - i - - j such that
i+j<,.-1
Proof. By (6. 2) ~_ , (u,v) = k + l < m - - 1 6r
i!l~_, (% 4) has total order =< m-- l - - k - - l . We write
, where
C~ k h
C~V k i ~O
~ h k
E - i . 1=o
. ~ | 0 N. A R O N S Z A J N , A. N. M I L G B A M
We note that in the above relations the order of the operators F,,,, A~, is =< k - i. We have
^ k, 1 ^ k , l The expression ~._, (A~.~/~ u, F~j C;j v) defines an o-form ~_~ (Ak, ~, F~ @) of
total order =< ( m - - l - - k - - l ) + (1r + (I--j) : m - - l - - i - - j , Applying Theo-
rem 11 to this, replacing V and ~ by B~ u and C~ v, and summing over all indices k, l, i, ] gives
, + j < m - i
as asserted in the lemma.
/~, u Oo G v
To demonstrate the uniqueness of the O,j it is sufficient to remark that if
and + are arbitrary functions on 35/,,_, then u and v can be so chosen that
l~ t t ::: ~ , l~.u : 0 for k ~ i , C~v = ~ , C~v : - 0 for l = ~ j . For such u
and v, ~,_, (u, v) ~ ;, (? �9 0~,~). Hence ~,,_~ (u,v) is an o-form in ,~, q~, and it follows from Theorem lI that ~)~# is unique.
Consider a concomitant o-form ~,,~ (it, v) of A. ~,~-~ is always chosen to be of order smaller than the order of A. It is unique up to an equivalence.
By Lemma 1, using B~ : C', : - - , we have 0 v ~
o~' U o~s v c3vJ
where O'~ is uniquely determined and of order < m - l - - i - j . Hence O'
is of order O, i. e. is - - f, ] where f~ is a function defined on ,~,,_, An explicit expression for f, is obtained as follows. We write
A = ~ A j where W - ....~ a0 J is an operator of order j (see (5.4)). j : O '1 ~ Y D(O
Since the concomitants of A j are of total order __< j 1, their restrictions to
M,~_~ cannot contain a term with the product c3*u ~ o~v~ oav,,,_~_ , unless j - - m.
DIFFERENTIAL OPERATORS ON RIEMANNIAN MANIFOLDS 311
Hence, in order to find 0',.,~_,_ t it is enough to consider a concomitant of A '~, as given in (5.5) (where k should be replaced by m). If we consider this formula in a normal c-patch, and consider the restriction of the exact differen-
tial in it to /fL,-,, we obtain for the concomitant
r = l i l ' --1 irq-l".,in:
D m - r / il ""i~--lnZ~'--~-l'"fm---, ] ~a m v)] >( imim_l. . .[rq-1
"/( dx~ . . . dx "-1 "
We use the fact that g - - ,ff in a normal c-patch at a point of M,~_,. From the
above formula it is clear that the product c3ku O"-~-J'v O*u O"-I-kv Ovk Ov "-~-~ ---= Ox~ ox" ,,,-i-~ only oc-
curs in the term corresponding to the indices r - - k-q-l, i t - n for l - - 1 , . . . , r - - 1
and I = rn t -1 / . . . , m. Hence ~ = (-- 1) ''+'-'J' a~ ........ . On the other hand, if we write the expression for v (t) - - l:z (v~) in the same c-patch, we get p(t) :
--- a~ ....... . Hence we obtain
(7. 4) O' - - (--1) ' ' * '*-" p(t) ] h , m - - l - - k - -
T h e o r e m Vl. Let A be a fixed operator defined in a neighborhood of
,~1~_, of order m, and non-tangential at lfL-l . For any Dirichlet system I B, I o]
order m there exists another Dirichlet system I ~" B ~ t of order m, uniquely deter- mined, such that for any concomitant ~,~-~ (u,v) of A we have
(7. 5) ~,:_, (.,~) ~ : p (t) ~ (- ~ . h',,,_,_~, v . k=O
ProoL By Lemma 1, we can write
. :--1 d k
o~v i - - k=O
L-I ( . , v ) ~ ~ ~ B~ u o ~ - fi4-i<m--I
m - - l - - k c~i
where C ~ - - ~ O~i . i=O r
Since the order of 0~ is _< m - - l - - k - - i , it will suffice to show that
0~,,~_~_~ is equal to (--1) m+'-k p(t) I to establish the existence of the Dirichlet
system l / ~ l . From the expression
3 1 ~ N. A R O N , e , Z A J N , A. N. M I L G R A ~,|
/~,u = ~ A~j c~gu A ~ . = / , order Akj < k - - j , L< h c~,~7 ' =
we obtain
j<k ~ f " Or' ~ Alq O~, kq-i~m--1
where A~ is the adjoint of A~ on M,=_, with metric g'~. By formula (7.3), we have
If i "-- m~l - - j , Ayj
m - - l - - i
E Ak] Ok~ j~. k--j
~j . . . . ,_j = 0 ' / . . . . , _ j - - - ( - -1 ) " + " - j p (t) i , by (7.4).
Since Aj~ - - [ , we have Oj .... 1-j : - ( - - 1 ) " + " - / p ( t ) i . Hence B',,,-~-k =
(--1) "+"--* --!__ = L?~ satisfy the condition of the theorem. Uniqueness follows p (t)
o3~ from the remark that /9'k : ( - - l ) ' + " - k 1 ~ Ok, and that O~ are
p (t) i<~-,-k ~ v ' '
unique by Lemma 1. From now on the operator A will be assumed non-tangential at M,,-, .
If the Dirichlet system l/~', I, i -= 0,1 , . . . m-- l , corresponds to the Dirichlet
system 1/~,l (follo,,ing Theorem V) we shall call the linear differential system
(L d. system) I A*;B'~ 1 conjugate to l A;/~ I. Also l/~'~ I will be called conjugate
to l[~l rel. A. Recalling that for A*, the concomitant [3",,_ 1 and the fun-
ction p*(t) are given by ~*~-~(u,v):-~,_,(v,u) and p*(t)= p.~,(v,(t)) :
(-- I)" P,4 (v,(t)) = (-- l)~p(t), we see immediately that l A;I~, 1 is also conjugate
to {
On .,{,/,_, we can consider the adjoint of an operator A (with respect to the
metric tensor g',j). This adjoint is denoted by A';. In connection with the conju- gate systems relative to a non-tangential operator A of order m, we shall need another symbol - - adjoint reL A:
(7. 6) f f _ I x p ~ ) (p (t) A) ; p (t) A ~ (p ( t ) ) ) .
DII*FEIIENTIAL OPERATORS ON" I t IEMANNIAN MANIFOLDS 313
it is clear that [ | = ], A | - - 1 (p (tO A) ~ p (t)
A~ | A
For an o-matrix {At,:,} we introduce
{ A'm_l_i,,,,,_x_,., I :
, (A | 1 7 4 --- A ( A A , ) | =
the adjoint rel. A, IA:.r174 - -
(7.7) A t . _ m-l- i" , ,n-- l - i ' " - ( - - I ) i ' - r ' A.,,,,,| .
It is clear that the adjoint matrix is an o-matrix with row-indices varying in the set {m-- l - - i "} and column-indices in ( m - - i - - i ' } . The adjoint of a triangular matrix is triangular. It is immediately verified that
I f : ' I I~'" I]" | I | ,..,.. ,...,... I : ' l ) : ' - - i e , t t , t zp, l / /
= r {)A.,t ,1 ~ lI . I~ *
Let {i} be the set of indices of a Dirichlet system of order m. The set of
indices { m - 1--i} is now the same as { i}. if {/3", t is conjugate rel. A to t/3, l
and if I C,! = lA,,sl l /3Jl , then I * ' C , I , the system conjugate to IC',I is given by
~.~) le , I - [l~,.,l| -' )~;I.
This formula is obtained by writing, following (7.5)
~._,(.,0~.., ( - i ) ''+" ~ (-1),p(t)8, . "-'-'~i k = U
I m--I m-- 1 (--I) '"+" ~] ( - - l ) ' B , u ~ ( - - l ) ' -" (p (to A~..) ;' C%,-,-k v !
1=0 k=i I
m - - L m - - I
~(-1)'.'+.p(t) X (-~) '&. ~ (-~)"-' A~,, a',.-,-~ ~I I t" i "=0 k~i
Hence I B ,,-1 ~ ~ {| - - C,,-~-,: which proves (7. 8).
314 N. Ano~szAa~r , . ~. ~axLcRAM
Denote by I/~,~I the system conjugate to I ~ I rel. A. Let
(7. 9) i B k - - Ok, ; .
In view of (7. 3) we have
(7.9') 1 Oo , ,+,-i _ _ O ' . = ( - - l ) , j
p (t)
Since l/3,,I is conjugate to I L ~ I , it follows from (7.8)that
As before, let { i} - - {0,1 , . . , m - - I } and let
{ i ' } C { i } , {i"} - - -{ i} - - {i'} .
We shall say that the normal I. d. system lA*;/)',~_i_~,,l is adjoint to the
normal 1. d. system I A; [3r I or that l/~'m-l-i,, I is adjoint to t [~r f rel. A, if there
exists a Dirichlet system t/~r containing l/~i,l such that its conjugate system
rel. A, IL~'~I, contains l/3',,_~_i,,l . From the definition it is clear that the adjoint system is not unique in gen-
eral; it depends on the choice of the operators /~r in the Dirichlet system con-
taining 1 / / r Also, if IA*;/3',,-1-r is adjoint to IA;/~r then lA;l)i,l is ad-
joint to l A*;B'.,~_i,, I and 1/3r I is adjoint to l/)',,-~-i,, I rei. A*. The notion of adjoint systems gains more of an intuitive significance with
the introduction of pseudo-adjoint systems. Consider two arbitrary finite sets
of indices l ? f and t ~ I . Two l. d. systems I A;B? I and I A*;B'~I are pseu-
do-adjoint, or, 1/~'7~ l is pseudo-adjoint to 1/37I rel. A, if for any two functions
u and v which satisfy the conditions /~r u --- 0 and/3',~v - - 0 on M,- i and
where at least one function vanishes together with all its normal derivatives on h / , - i
outside of a compact subset, we have
DIFFEnENTIAL OPERATORS ON RIEMANNIAN MANIFOLDS 315
- o ,
n--I
for any concomitant ~._a(u,v) of A.
Remark 1. By using partitions of unity (see w 3) we prove easily that in the definition of pseudo-adjoint systems we can limit ourselves to functions u and v of which at least one vanishes outside of a closed subinterval of a nor- mal c-patch.
Remark 2. If (7.11) is true for one concomitant ~_x it is also true for any and the fact other concomitant 13',~_~ of A. (This follows from Stokes theorem
that ~'~_~ - - 1~,~-, - - ~/'~-~) �9
T h e o r e m VII. Let A be of order m and non-tangential at A4,_ c and let
tBe t be an arbitrary normal system. Pat { i} - - { 0,1, .... m -- 1 }, {i'} = {i' }{ i},(~/
{ i ' } = {i} - - {i '} . Further, let l[3',~-~-,"t be any system adjoint to
t f l , " I rel. A. In order that a system 11~"~ I be pseudo-adjoint to 1137-I rel. A, it
is necessary and sufficient that l[1"~ I be stronger than l [1",,-1-e'l.
Proof. Sufficiency. Let 111, I and I /3;I be two conjugate systems such that
the first contains 111r I and the second l/~',,_l_W I. Then
(7.12) ~.-i(u,v) = , t ( - -1)"+'~ . , -1 ~, ^ i p( t ) ~=oZ (--1)'L~, u B .... ,_,.v tt ~- d%-2 (u,v).
Since [17u = 0 and /~",~-, v = 0 imply /~r 0 and A, ----- 13 ,,,_ ~_,,, p = 0, we have
for u and v satisfying these conditions, ~ _ , ( u , v ) = d ~ _ ~ ( u , v ) . If one of
the functions u or v vanishes outside of a closed subinterval U of a normal c-patch iwe use Remark 1 above) by Stokes theorem we obtain
~- -1 J -U "~In --1
Necessity. If I /3 ' , , , -1- i - lwere not weaker than 1B"TI there would exist a
(~) I i' I is the intersection of the sets I 7 } and / i } .
316 ,~-. A R O N S Z A J N , A. N. M I L G R A M
^ # function v such that 13"i,;v -- O, and for some i"o, 13 m_,_,,, ~ v # 0 at some
point to E 3~,_~. We choose a neighborhood U of t o such that U is a closed
subinterval of a normal c-patch and choose a function Uo vanishing outside of �9 # l r ^ t and such that 13~ Uo = 0 for i -#: t ,, and /3~,,o tic --~.p.B m-t--,"'oV, where ~0
is an arbitrary non-negative function on M~-t vanishing outside of DM~-, . By
Stokes' theorem it now follows from (7.12) that
j,,., I( . . _ l)"+'~+i% [ P !2 I ~', , , -1- , - V l t tJ �9
--=-0,
i. e. /3',~_t_,,,o v - - 0 inside U M ~_, ; hence a contradiction.
From Theorem Vii we have as immediate consequences:
1 ~ The pseudo-adjointness of the two I. d. systems I A; [3i~- I and 1 A*;B',,, I depends only on the boundary operators B~, and B'e, of orders--< m - 1.
2 ~ An adjoint system is a fortiori pseudo-adjoint. It is minimal with re- spect to the relation of "being stronger,, among all systems pseudo-adjoint to a given system. As a further consequence we have
C o r o l l a r y I . Lel t ]~'m-,-i" l be adjoint to l [3e l rel. A and I C," l be
equivalent to l[3ef. In order that a normal system l C'~-,-i"l be adjoint to
l C~" f rel. A it is necessary and sufficient that it be equivalent to l[l'm-,-,-I.
Proof. in fact if t C"m-~-i,, I is adjoint to l C',' I it is pseudo-adjoint to it also,
hence pseudo-adjoint to 1/~,,l /i) and thus stronger than l/) 'm_,_i-f. The same
argument is also valid in the inverse direction; 1r is weaker than, and
therefore equivalent to I / ) 'm- t - i - I .
If I C"m--~--," I t s equivalent to i l~tm--l--i ,,Iit is necessarily equivalent to any
system I I adjoint to 1 C'~" I. Let l d', f and t C", I be corresponding con- jugate systems. A simple verification shows that if
(9 in the definition of pseudo-adjointness equivalent systems may be interchanged.
D I F F E R E N T I A L OPERATORS ON" R I E M A N N I h N M A N I F O L D S 317
then the conjugate to the Dirichlet system composed of lC'r f and
{ A,~_l_i,,.r~-,-j,, l | {s162 I is composed of l Cttm--l--i" I and { s l .
Since the relation of adjointness assigns a class of equivalent systems
l s a class of equivalent systems t C' , ' l i t is convenient to describe
this relation in terms of systems in canonical form. The formulas are especially
simple if we take two conjugate Dirichlet systems 1/3, I and l/3'i I and consider
the systems l C'i,l and { C"m-l-/" I which are in canonical form with respect to
l/~,t and l/~',t respectively. If we write (see (5.12))
(7.,3) {o,,l = + {A,,,,,I
it is immediately verified thai
(7.13") {C'm--l-i"f = I~J'm--,-i"t - lAi',i"l | tB ' . , - , - i ' l
We turn now to self-adjoint l. d. systems {A;Br }. A is now self-adjoint, A = A*. Moreover, the set of indices It '} must be the same as { m - l - - i " } . It follows that the number m" of indices i" is the same as that of i " ; hence m = 2m' .
From now on, in addition to the assumption that A is non-tangential at /~/,-1, we shall assume that it is self-adjoint and of even order m - - 2 m ' , in the present case the leading coefficient tensor of A is real-valued and p ( t ) - -
p~ (t) - - pa. (t) is real. if ~_~ (u,v) is a concomitant of A, - - ~,~_~ (v,u) is a
concomitant of A* = A; hence 1 ~- [~_i(u ,v) -- [3,~_t(v,u)] : ~'~-t(u,v) is also
a concomitant of A, and it is skew-symmetric: ~',_~(v,u) =- -- ~'~_~(u,v). The investigation of self-adjoint systems will be preceded here by the con-
sideration of self-conjugate Dirichlet systems of order m. Let l / ~ l and 1/3'~t be two conjugate systems tel. A and let ~,~_~(u,v) be a skew-symmetric conco-
mitant of A. We can write ! B ' i l - - l I I i ~ I IBJl with a triangular o-matrix { II~ }. Further
0~-,{U,V) "-' a (--l)"+~p E (--1)' B, U [3".,-t-iV , i = 0
I m--I m--l--i I = ; ( - -1)~+"p ~ ' ( - -1 ) ' / ) , u ~ II,~_,_,.j/~jv
i = 0 j=O t 2 1 . R e n d . C ~ r e . Matem. Palermo, - - s e r i e I I - t o m o I I - a n n o I 9 5 3
318 N . A R O N S Z A f f N , A. N . M I L G R A M
t m--1 - - m--l--j I �9 - ( - - 1 ) ' + " p Y' . ( - -1 ) . / /~ , v 'Y", (--1);--,II | B, u ~ ~,,_,(v,u) .
j = 0 i=0
| | It follows that II,~_i_,,j - - ( - - l ) ; -J+ ~ II~_~_~,~ - - ( _ _ | ) m - - l - - j - - i I l ra - - l - - j , t ,
(7.14) l II ' ' f | : l II''~ I; l II ' ' I is self-adjoint rel. A.
Using this relation it is immediately verified that the Dirichlet system com-
posed of the operators /3~ and ~Tm--~--k for k --- 0, 1 , . . . , m" - - 1, where
m - - l - - k 1 + B, (7.15) C,n-,-. = -~-
i = k - ] - I
is self-conjugate.
Consider now two self-conjugate systems 1/3, I and lC',l �9 If lC',f ~---
1A,,,I l /~ , l , we get, by (7.8),1C-', 1 ----- [IA,.,I | -~ 1/3, I ; hence lA, . , l =
[ IA, , j l | e.
(7.16) A m l J I ,,I --IA,,I | By analogy with terminology in the theory of matrices of numbers, we cal
an o-matrix I A~.jl satisfying (7.16) a unitary o-matrix tel. A. Such o-matrices have the characteristic property that they transform conjugate Dirichlet systems into conjugate systems (even for non-self-adjoint A). The unitary o-matrices form a subgroup in the group of all matrices {A,,~} .
Consider a self-adjoint system l/~,,l . As shown above we have
(7.17) ti" I : l m - - l - - i " t , where l i " l - l i t - - l i ' l .
Consider two conjugate systems I/ t , l and l Z~',l such that each contains
l/3,'I i. e. that /~',. = /~,,. it follows that if l~ ' , l - - t lL.~I t/~Jl we must have
IIr = ~,,.~ ] for all i" and j (~.~ is the Kronecker symbol). If we form
the corresponding self-conjugate system l/3~, E'~-i-~ I , k - - 0, 1, . . . m' - 1, we see that for k" ~ m' N 1, m - - 1 - - k" runs through all the indices i" ~ m',
and by(7.15), C'm-l-k" = /~m-l-e' since IIm-x-v,.; = ~m--~--h",~ 1 . It follows that
1/3,, I t s contained in 1/3~, C'm-l-al . On the other hand, it is obvious that if a
DIFFERENTIAL OPERATORS ON RIEMANNIAN MANIFOLDS 319
normal system l/3,' I with indices satisfying (7.17) is contained in a self-conjugate system, it is self-adjoint. Hence
Theorem Vlll. In order that a normal system 113,, I be self-adjoint reL a it is necessary and sufficient that its indices satisfy (7.17) and that it be con- tained in a self-conjugate system rel. A.
Using formulas (7.13) and (7.13") we can express all self-adjoint systems
I 6",' f which are in canonical form with respect to a given self-conjugate system
l~ , t . We g~tlO,,t = l ~ , , l + IA,,, , t I~,,,l : I ~ , , I - IAi,.,,,I | l~,,,I Thu~
(7.~sl I A,'., " 1 = - I A - ''I |
Clearly, condition (7.1Sl is necessary and sufficient in order that 1 r I = l ~,' I + I A,,,,,II ~,,,I be sel~-adjoint
It follows from condition (7.18) that in a self-adjoint system 16'~, t we can choose the operators of order ~ m' - - 1 arbitrarily, and that the operators of higher order are then completely determined by their parts of normal order ~< m' - - 1. More precisely, if we put
o.~o) 1 7 t = 1 0 , , , . , m ' - - l t , l ~ t = l ~ ' l l T l , 1 7 " l = l r l l T f ,
we have t i , l : l'[,l -~- l n , - l - 7 , , I, l i " l : I7"I + l m - l - t ' l a n d
by (~.~) , l A._~_~,.m_,_~ I = _ I A ~ , , I | l Am_,_~, ~, l = - t ~,,~_,_?, ~,, I |
The last equation characterizes the parts of normal order _< m" -- 1 of (7,,,-1_7,,
in a self-adjoint system l C',,lin canonical form with l cTe I chosen arbitrarily. The following theorem deals with pseudo-self-adjoint systems.
Theorem IX. A normal system t [37 I is pseudo-self-adjoint rel. A if and only if it is stronger than a setf-adjoint system rel. A.
Proof. The sufficiency is evident.
Necessity. To construct a self-adjoint system weaker than I/3z;l, we define
l i ' l - - l iT l l i l as before; extend the system l/~i'l to a Dirichlet system
t/3, 1 and take the conjugate l/3;I. Putting I i" l : I l l - - !i ' I, we have, by
Theorem VII, that l/3',,-i-,,, I is weaker than l/3r I. If l/~;I "-- t lIejt I//Jl it follows that
320 N . s A . N . M I L l ; R A M
(7.20) = E II._,_,,,j, & and H._,_i,,,/, = o. /<m-l-r"
In particular, I m - - l - - i " l C l i" t. Introducing the sets l / ; l and 1/~'I by (7.19)
we see immediately that l i l - [l/~l -t- l m - - l - - i " ' } ] - - l ["I -~- l m - - l - - ~ ' l .
It follows that I i ' t -Jr- I m--1--7"" I is a set of indices of a self-adjoint system.
We now form the self-conjugate system I BT, d',-1-~'I, noting that in view of
(7.15) and (7.20), its self-adjoint subsystem 1Bp, Cm-l-e"l is weaker than 1/3r t"
Besides the one just defined, there may be many other self-adjoint systems
weaker than lBv I. Theorem IX shows that self-adjoint systems are minimal
among pseudo-seif-adjoint systems relative to the relation of "being stronger ,,.
We now introduce semi-concomitants and discuss self-adjoint systems with
respect to them. As before, the operator A will be assumed self-adjoint, of order
m - - 2m', and non-tangential at 34,- i . By the same procedure of partial inte-
gration which served to obtain (5.5), we can transfer at most m' derivations
from u to v in . (Au.v) and obtain an o-form . (A" (u,v)) ~,~ . (Au. v) with a bi-
linear operator A'(u,v) of total order m and of u-and v-orders --- m'. Writing
* CA" Cu, v ) ) ~'. * ( A u .
* (A" (v,u)) ~.~ * ( A v . u) ~'~ * (Au" v)
1 [A ' (u ,v )~-A ' (v ,u ) ] Thebi- we get . ( A " (u,v)) ~ . (An . v) with A" (u,v) - - ~
linear operator A'" (u,v) has the following properties:
(7.21) A'" (u,v) is of u-and v-orders m' and is symmetric: A'" (u,v) - - A'" (v,ui.
By Theorem I! we can write
(7.22) *(Au.v) --- * (A'" (u,v)) H- d~ , - t (u,v),
with an o-form 8~_i (u,v) satisfying
(7.23) ~_~ (u,v) & of total order ~ m- - I and of v-order <= m'--I .
An o-form ~,_~ (u,v) which satisfies (7.23) and (7.22) with a bilinear opera-
tor A'" satisfying (7.21) will be called a semi-concomitant of A relative to A'" (u,v).
DIFFERENTIAL OPERATORS ON RIEMANNIAN b|ANIFOLDS 321
Such a semi-concomitant is uniquely determined except for an additive exact
differential (which may be arbitrarily chosen). From (7.22) and (7.21) it follows
immediately that
* ( u . A * v) - - * ( A v . ~ - - * (A'" (u ,v ) ) -q- d ~ - i (v ,u);
hence ~,~_~ (u,v) - - ~-~ (v,u) is a skew-symmetric concomitant of A. This justifies
our terminology.
^o L e m m a 2. Let ~.-i(u,v) be a semi-concomitant, and let ~n--1 ( / / , V ) be a sym-
o -i (u,v) + (u,v) metric o-form on f4._i of u- and v-orders ~ m ' - - l . The o-form ~o
is the restriction of a semi-concomitant defined in a normal neighborhood of M~_~.
(u,v) Proof. As indicated at the beginning of w we can consider as
a restriction of an o-form ~~ defined in a normal neighborhood V of cY <9'
�9 '~/',,-1. This o-form was obtained in replacing 0v ~ by ~ in the developed expres-
~_l(U,v) the o-form ~_l(u,v) is sion of ~_1 (u,v) in each normal c-patch. Like ^~ ,
symmetric and of u-and v-orders <~ m ' w l . We can now write
. (Au.v-) - - . (A'" (u,v)) -}- d ~ - , (u,v) : [. (A" (u,v)) -- d ~~ (u,v)] --1-
-~- d [ ~-1 (u,v) -}- ~o_~ (u,v) ] which shows that ~_~ (u,v) -~- ~_~ (u,v) is a semi-
concomitant relative to the bilinear operator A"(u,v) - - . d 8~ which is
symmetric and of u-and v-orders <~ m' (hence exactly m').
For the remaining part of the paper it will be more convenient if we put
I i l - - I 0 , 1 , . . . , m ' - I 1.
The set t i ' l will ordinarily be a subset of [ i t . We have
IO, 1 , . . . , m - - 1 1 - - - i l l - ~ - I m - - 1 - - i l .
Let ~,,,~ (u,v) be a fixed semi-concomitant. For the sake of brevity we shall also
call its restriction g,,_l(u,v) a semi-concomitant. Choose an arbitrary Dirichlet
system t]3i, /~,,,-~-I of order m. By Lemma 1 there exist unique operators
32~ N. ARONSZAJN, A. N. MILGRAM
Om-l-i,j of order ~ m-- l - - i - - / and Om-~-l,m-~-j such that
(7.24) ~_~(v,u) ~
(for j :> i) of order j - - i ,
( - - 1 ) re+n+1 p(t) E ( - - 1 ) ' B, U i~0 j=i
.,_1 ]} + F, o._~_i,j ~, v
j=0
Om_l_i,m-l-j Bm--l--j V
~ _ ~ ( u , v ) - %_~(v,u)is a concomitant ~_~(u,v). By using the expression
(7.24) to develop ~,,_~ (u,v), and by transferring all the 0% acting on /3 u's to
the corresponding /~ v's, we arrive at the expression
I _1 . B,n-l-i v -q- ~ (--1) ~-1 -~ B~-x-i u B~ v ( -O~+~p( t ) ~ (-1) ' i3,u "'
i~O i=0
where 1/3;, /~-~-el is the conjugate system to 1/~,, ]~nl--l--tf. A c t u a l computa- i
tion shows that /~ -- ~ ( - - ly-J O~_t_j.,,,_l_~ /3j , hence O,,,-~_t,,,,-1-~ --- i . j=0
Hence, using (7.24) we can write
(7.25) ~_~ (u,v) ~o . (_lim+~+~p(t) mr } e=~o (--1)' (?,,,-1-t u B, v ,
with a normal system t t~m-l-t 1. Lemma 1 tells us that this system is completely
determined by the system l B~ I (for a fixed ~n-l(u,v)). The system I C'm-l-i I
will be called the associate system to l[~ I rel. g,~-i (u,v). The following state-
ments are immediately verified (l C'm-l-if being associate to 1/~,f):
I) The system t [~,, C,,-~-i I is self-conjugate tel. a.
il) For another system l/~, I - - l I',.~ f t/~I the associate system is given by
I C;~-1-,I = [lr , . l+] - ' tC:-~-jl. Now let {i'} C {i} and {i"} = {i} - - {i'}. Consider a normal system 3/3~. I
and extend it arbitrarily to a Dirichlet system 1/3, 1 of order m'. Let 1 t~,,,-1-z I
be the associate system. The system 1/3:, C',,,-~-i,, I will be called self-adjoint
with respect to ~n_~(u,v).
DIffERENTIAl, OPERATORS ON RIEMANNIAN MANIFOLDS 323
Changing 113, I into l/~;I = l P,.y t l/3~ I in such a way that l B;' t be equi-
valent to 1/3i, I, and putting IP}.j I = l Pi.j 1-1, we have l [Ji I - - I Pi.i I I/3~I. Since
l/3i, I and l/)i' I are equivalent, we must have P~,j,, : 0, and /3,, = Y'~ P~,j,/~),. If y'<r ~V' - - we then putt ,,-~-,,,,-*-Jl = lP~.,t | [ lr , jf | -*, we obtain F~_,_,,, ,._,_e : O ,
and hence by statement II),
j">i'"
i. e. l (7;,,-~-i,, I ts equivalent to 1 (,',,,-1-i,, I. We can state the following
II1) Two self-ad;oint systems with respect to ~,_~(u,v), l [3r, Cm-i-r" l and
IBm', C',,-,-r, l are equivalent if and only if ll3rl is equivalent to l/~i, I. More-
over, l[3rl is equivalent to I f3~,f if and only if l~m_x_i,, I is equivalent to
lem-,-i,,l. We define the notion of a system l/3Vl pseudo-self-adjoint with respect to
~,.-1 (u,v) in the same way as for a system pseudo-self-adjoint rel. A, except / -
that condition (7.11), y~_~
(7.26)
~_~ (u,v) --- O, is replaced by
/ ~ ~tl--I (//,V) "--- O.
It is clear that a system pseudo-self-adjoint with respect to ~,,_~ (u,v)is, a fortiori, pseudo-self-adjoint rel. A. By an argument similar to that used for the proof of Theorem VII (but somewhat simpler) we arrive at
T h e O r e m X. Let 113"~ I be a normal system. Put l i'l = l iTl I l l and
It" l : I i l -- It" I. Let l Bi" , 4rn--l--i'" I be self-adjoint with respect to ~,_, (u,v).
Then, l l3"dl is pseudo-self-adjoint with respect to ~,_~ (u,v) if and only if it is
stronger than [ [tr, (:m-l-r' I. Here again we see that systems self-adjoint with respect to ~n-~(u,v) are
the minimal systems among all pseudo-self-adjoint systems. An especially important type of self-adjoint systems are the formally posi-
tive (f. p.) systems. They are defined only for formally positive operators A (see (5.11)).
324 ,~. A R O N S Z h J N , A. N . M I L G R A M
Let A - - Ah Ah be a fixed decomposition of A, (u,v) a concomitant h = l
of Ah and ,.~,, k = l, . . . , s, boundary operators (not necessarily normal) of or-
ders --< m' - - 1. By (5.12) and Lemma 2, the o-form
(7.27) On--i
h = l h=l
is a semi-concomitant. A system self-adjoint with respect to this o-form is said
to be formal ly positive with respect to the decomposit ion A - - ~ A*h Ah and the
system I,S~ I (f. P. [l Ah I, I,S~ I ] ) . The pseudo-formally positive systems are defined similarly.
We mention briefly a simple application of the notions introduced in this section to eigenvalue problems.
Let D C M, be a closed compact domain having a compact submanifold
/~/,,-1 for boundary. We consider the eigenvalue problem
(7.28) Au = F u in D , i37 u - - 0 on /VI,,_1,
where A - - ~A~,Ah is a f.p. operator and I / ~ I is a normal system con.
taining m' operators. For the application of variational methods to this problem, it is essential to have the representation
;o j ;= (7.29) . ( A u . u ) - - , ~ (~ - [ A h u l ' ) + ' ~ ( E [ S ~ a [ ~ )
for all u satisfying [3-~u - - O, ,S~ being some operators of orders < m' - - 1. By using (5.12), Stokes theorem, and (7.27), it is easy to prove that (7.29)
holds if and only if the system IB?I is pseudo-f.p. [I Ahl, l,-q~l]. But if i / ) p i t s pseudo-f, p., it must be f.p. since it contains only m" operators. By means of our developments we may now describe all such systems.
If, instead of the equation Au - - g. u in (7.28), we have Au - - F At u, with a self-adjoint operator Ai of order m~ - - 2m'~ < m, we shall also need in ad- dition to (7.29), a representation
(7.30) fo. (A, (a', = o ,
DIFFERENTIAL OPERATORS ON RIEMANNIAN MANIFOLDS 3 2 5
with a symmetric bilinear operator A' 1 (u,v) of u-and v-orders = m'l. If ~,,_~(u,v)
is a semi-concomitant of A i relative to A~(u,v), then (7.30) will hold if and
only if IB~I is pseudo-self-adjoint with respect to ~'n-l(u,v)" Our results allow
us to determine all systems t B? ] for which (7.29) and (7.30) hold simultaneously.
Systems of this kind will be f.p. [l Ah I, I S~:!] and pseudo-self-adjoint with
respect to ~'n_l(u,v). (Here the system cannot be self-adjoint since it must con-
tain m' > ml operators.) (J).
(i) More details about such applications, especially to approximation methods, can be found in the following papers by N. Aronszajn: The Rayleigh-Ritz and A. Weinsfein methods for approximation of eigenvalues. ! and il, Pror Nat. Acad. ScL vol. 34 (1948) pp. 474-480, 594-601; Studies in eigenvalue problems, loc. cir. Reports il and Ill, Approximation methods for completely continuous symmetric operators, Proceedings of the Sympositium on Spectral Theory and Differential Problems, Oklahoma A. and M. College (1951), pp. 179-203.
