Hypersurfaces of restricted type in Minkowski space

Geometriae Dedicata 62: 319-332, 1996. 319 © 1996 KluwerAcademic Publishers. Printed in the Netherlands.

Hypersurfaces of Restricted Type in Minkowski Space

CHRISTOS BAIKOUSSIS 1 , DAVID BLAIR 2, B A N G -Y EN CHEN 2 and FILIP D E F E V E R 3 . 1Department o f Mathematics, University of loannina, loannina 45110, Greece e-mail: [email protected] 2Michigan State University, Department of Mathematics, East Lansing, Michigan 48824, U.S.A. e-mail: [email protected]; [email protected] 31nstituut voor Theoretische Fysica, Celestijnenlaan 200 D, 3001 Heverlee, Belgium e-mail: [email protected]

(Received: 13 March 1995)

Abstract. A submanifold M~ of Minkowski space ]E~ is said to be of restricted type if its shape operator with respect to the mean curvature vector is the restriction of a fixed linear transformation of ~ to the tangent space of M~ at every point of M~. In this paper we completely classify hypersurfaces of restricted type in ]E~ + i. More precisely, we prove that a hypersurface of ]E~ is of restricted type if and only if it is either a minimal hypersurface, or an open part of one of the following hypersurfaces: S k x ]E~ -k , S~ x ]E "~-k , H k x ]E n-k , S'~, H '~, with 1 < k < n - 1, or an open part of a cylinder on a plane curve of restricted type.

Mathematics Subject Classifications (1991). 53A05, 53A07, 53C40.

1. Introduction

Minimal submanifolds of pseudo-Euclidean spaces are contained in larger classes of submanifolds, e.g. in the class of submanifolds of finite type and also in the class of submanifolds of restricted type. The study of submanifolds of restricted type was initiated in [7]; for a survey of recent results on submanifolds of finite type and various generalizations, see e.g. [6].

Let M ~ be an n-dimensional , connected, nondegenerate pseudo-Riemannian submanifold o f the pseudo-Euclidean space E'~. Denote by x, H , and A respec- t ively the posit ion vector field of M~, the mean curvature vector field of M~, and the Laplace operator on M~ , with respect to the pseudo-Riemannian metric g on M~, induced from the pseudo-Euclidean metric of the ambient space E~ . Then, as is well known (see e.g. [4])

A x = - n i l . (1)

* Aangesteld Navorser N.EW.O., Belgium. This work was done when the first and fourth authors were visiting Michigan State University.

320 CHRISTOS BAIKOUSSIS, DAVID BLAIR, BANG-YEN CHEN, AND FILIP DEFEVER

This shows, in particular, that Mr n is a minimal submanifold of ~ if and only if its coordinate functions are harmonic (i.e. they are eigenfunctions of A with eigenvalue 0).

In this context, Takahashi [17] studied submanifolds in Euclidean space for which Ax = Ax (A E ~), i.e. submanifolds for which all coordinate functions are eigenfunctions of A with the same eigenvalue A. In [5] this study was generalized to pseudo-Euclidean spaces, and a pseudo-Riemannian version of Takahashi's theorem was obtained:

Let x: M~--+IE~ +k be an isometric immersion. Then Ax = Ax, for a real constant A, if and only if one of the following statements holds:

(i) A = 0 and M~ is a minimal submanifold of I~s +k ; (ii) A > 0 and M~ is a minimal submanifold of $2 +k-1 (p) with p = x/'-@A;

(iii)), < 0 and M~ is a minimal submanifold of H2+~ -I (p) with p = x/~/A.

As a generalization of Takahashi's condition, and following an idea of Garay [11], Dillen et al. [9] initiated the study of submanifolds M~ in I ~ such that

Ax = Ax + B (2)

where A is an endomorphism of ~ and B is a constant vector in I ~ . Observe that if A is a diagonal matrix and B = 0, then the latter condition expresses that the coordinate functions of M~ in E~ m , with respect to a certain Cartesian coordinate system are all eigenfunctions of the Laplace operator, but in contrast to Takahashi's condition, belonging to possibly different eigenvalues. In this particular case, we will call submanifolds satisfying condition (2), with A diagonal and B = 0, submanifolds of coordinate finite type. This suggests the following problem:

PROBLEM 1. Classify all submanifolds of pseudo-Euclidean spaces satisfying the condition Ax = Ax + B, in particular those of coordinate finite type.

By taking the covariant derivative of (2), using the well-known equation (1), and applying the Weingarten formula, one may notice that the submanifolds M~ in E~ for which condition (2) holds, have the property that for each tangent vector X at any point p on M~

AH(X) = (AX) T (3)

where A = (1/n)A. ( )T denotes the tangential component and An denotes the shape operator or Weingarten map of M~ at p with respect to the mean curvature vector H. It is well known that for any normal vector ( at a point p E M~, the shape operator A¢ with respect to ( is a linear operator on the tangent space 7~M. Thus, in particular, AH is a natural endomorphism of TpM at each point p C M. Following [7] we will call submanifolds satisfying condition (3), submanifolds of restricted type. The class of submanifolds of restricted type contains the submanifolds satisfying Ax = Ax + B as a subset. Chen et al. [7] also initiated the study of the following problem:

HYPERSURFACES OF RESTRICTED TYPE IN MINKOWSKI SPACE 321

PROBLEM 2. Classify all submanifolds of restricted type of pseudo-Euclidean spaces.

The present paper contributes to the study of Problem 2 by classifying the hypersurfaces M~ of restricted type in the Minkowski space E1 '~+ 1. We prove the following theorem:

THEOREM. A hypersurface M ~ of~11 +1 is o f restricted type if and only i f M~ is one of the following:

(i) a minimal hypersurface;

(ii) an open part o f one o f the following hypersurfaces: S k x ~11 -k , S~ × E n-k , H k × P - k , S ~ , l t ~ , w i t h l < k < n - 1 ;

(iii) an open part o f a cylinder on a plane curve of restricted type.

Our result may be compared with, and adds to, the following known partial results which contribute to the solution of Problem 1 and Problem 2.

Dillen etal. [9] classify locally the surfaces in the 3-dimensional Euclidean space E 3 satisfying (2): an isometric immersion z: M 2 ~ E 3 satisfies Ax = A z + B if and only if it is an open piece of a minimal surface, a sphere or a circular cylinder. Alias et al [2] classify locally the surfaces in the 3-dimensional Minkowski space E~ satisfying (2): an isometric immersion x: M 2 --+ ~ satisfies Ax = Ax + B if and only if M 2 has zero mean curvature everywhere, or M~ 2 is an open piece of one of the following surfaces: L × S l ( r ) , H i ( r ) × ~, S~(r) × ~, H2(r) ,

S l(r). These results have been generalized for hypersurfaces in (pseudo-)Euclidean spaces of any dimension. For hypersurfaces of the Euclidean space E ~+1 we have the following theorem ([8], [13], [7]):

A hypersurface M R in ~-~+1 satisfies A x = Ax + B for a constant matrix A and a constant t~ if and only i f M ~ is a minimal hypersurface, or an open part o f a hypersuu~ace S k × E n-k, with 1 <_ k <_ n. Alias et al. [1] study submanifolds in pseudo-Euclidean spaces satisfying Ax = Ax + 13. In particular for hypersurfaces in Minkowski space it gives the classification: an isometric immersion x: M n --+ ~ + 1 satisfies Ax = Ax + B if and only if M~ is a minimal hypersurface of

E~ +1 , or M~ is an open piece of one of the following hypersurfaces: S k × E~ -k , S~ × E n-k, H k × ~ - k , S~, H n, with 1 _< k _< n - 1. The class of submanit;olds satisfying (2) is contained in the larger class of submanifolds of restricted type. Concerning the latter, we mention the following results: Chen et al. [7] classify hypersurfaces of restricted type in the Euclidean space ~ + ~ : A hypersurface M R of ~ + 1 is of restricted type if and only if M n is a minimal hypersurface, an open part of the hypersurface S ~ × E ~-k, with 2 < k _< n, or an open part of a cylinder on a plane curve of restricted type. In particular, [7] contains the classification of curves of restricted type in the Euclidean plane; the classification of curves of restricted type in the Minkowski plane has been obtained in [10].

322 CHRISTOS BA1KOUSSIS, DAVID BLAIR, BANG-YEN CHEN, AND FILIP DEFEVER

2. Preliminaries

Consider the real (n + D-dimensional vectorspace ~ + l with the standard basis {el, ..., e~+l}. Let <, ) denote the indefinite inner product on ~ + 1 whose matrix with respect to the standard basis is d iag( - 1, + 1, ..., + 1). Then ( , / is called the Lorentz metric on ~ + 1 ; ~ + 1 together with this metric is called the (n + 1)- dimensional Minkowski space and denoted by E~I +1 . A vector X in E~ +1 is called time-like, space-like, or light-like according to whether (X, X / is negative, positive, or zero. A nondegenerate hypersurface M~ of the Minkowski space E~ +1 can itself be endowed with a Riemannian or a Lorentzian structure, according to whether the metric g induced on M~ from the Lorentzian metric on E~ +1 is (positive) definite or indefinite. In the former case a normal vector to M ~ is time- like, in the latter case a normal vector to M~ is space-like. A shape operator of a Riemannian submanifold is always diagonalizable, but this is not always the case for a shape operator of a Lorentzian submanifold. However, since the shape operator Ax at a point x of Mi n is a symmetric endomorphism of the tangent space T~M~ when the metric ( , ) on M~ is of Lorentz type, Az can be put into one of the following four forms with respect to frames for TxM~ whose inner products are given by G (cf. [15], [16]):

A2 +1

A = "'. G = +1 (I)

A,~ +1 (o+1) 0 A +I 0

A : )~3 a = +1 (II) ".. " ,

~ +1

A =

A 0 1 0 A 0 0 1J~

G =

An

0 +1 +1 0

+1

+1

(III)

- v ~ +1 A ---- )~3 G = +1 v ~ 0 . (IV)

' , ' ,

A~ +1

We assume that all manifolds are connected and of class C °°. Let M~ be a hypersurface of the Minkowski space E~ +1 . Denote by V and ~7 the Levi-Civita con-


nections of M~ and ~ + 1 respectively. For any vector fields X, Y tangent to M~, the formula of Gauss is given by

~7xY = V x Y + a(X, Y), (4)

where a is the second fundamental form. The mean curvature vector H = ( 1/n) trace a is a well-defined normal vector field of M~ in E~ +1 . For each normal vector field ~, the formula of Weingarten is given by

~Jx¢ = - A c X + Dx¢, (5)

where A¢ is the shape operator with respect to ~ (A¢ is also called the Weingarten map with respect to ~), and D is the normal connection of M~ in I~1 +1 . For any X, Y tangent to M~ and a normal vector field ~, the second fundamental form a and the shape operator A¢ are related by

(a(X, Y), ¢) = (A(X, Y). (6)

The equation of Codazzi is given by

(~Txa)(Y , Z) = (Vya)(X, Z), (7)

where

(~Txa)(Y, Z) : Dxa(Y, Z) - a(VxY, Z) - a(Y, VxZ). (8)

A hypersurface M~ in ~ + l is said to be of restricted type if the shape operator All is the restriction of a fixed endomorphism A of E] ~+l to the tangent space of M~ at every point of Mr n, i.e.

AHX = (AX) T (9)

for any vector X, tangent to M~, where (AX) T denotes the tangential component of AX. Condition (9) is equivalent to the requirement

(AHX, Y) = (AX, Y) (10)

for all tangent vectors X, Y. On every tangent space the shape operator As is symmetric in the sense that

(AHX, Y) = (ANY, X) for all tangent vectors X, Y. So the restriction of the fixed endomorphism A of I~ +1 to the tangent space Tp M~ at every point p of M reduces to a symmetric operator on TpM~. The fixed endomorphism A ofI~ +1 can always be assumed to be symmetric on ~ + 1 in the sense that (AV, W) = (AW, V) for all vectors V, W of E~ +1 . Indeed, with respect to an orthonormal basis of E~ +~ , a symmetric endomorphism A of E~ +l is represented by a matrix A satisfying ~/A~ = A, where .t denotes the transpose of a matrix, and ~/is the diagonal matrix ~ = diag(-1, + 1, ..., + 1). Taking (A + ~lAt~])/2 instead of A justifies this remark.


3. Hypersurfaces of Restricted Type of the Minkowski Space E~ +1

As a nondegenerate hypersurface M~ of the Minkowski space E~ +1 can be either Riemannian, or Lorentzian, we have to distinguish two cases. Let ~ denote a unit normal vector field; with (~,~/ = e, e = - 1 refers to the Riemannian case, e = + 1 refers to the Lorentzian case. Let S denote the shape operator of ( and 1t = (1 In)trace S the mean curvature. From now on we denote by h the scalar- valued second fundamental form a ( X , Y) = h(X, Y)(. Formulas (4) and (5) read

(~xY = V x Y + h(X,Y){ , (11)

= - s ( x ) , (12)

where (S(X) ,Y) = eh(X,Y). Suppose that M~ is a hypersufface of restricted type. Then for all tangent

vectors X , Y we have

(AX, Y) = H(SX, Y) = Hh(X, Y)e. (13)

We can assume that A is symmetric. Then we can write

AX = HS(X) + ft(X)(, (14)

A~ = Z + f~, (15)

where/3 is a 1-form on M~, Z is a tangent vector field, f is a function and

(Z, X) = eft(X). (16)

As A is a fixed endomorphism ofE~ +1 , the action of A has to commute with the operation of covariant derivation in the ambient space. From

(Tx(AY) = AfTxY, (17)

(Tx(A~) = A(Tx~, (18)

we get, after separating the tangential and normal components,

X(11)S(Y) + 11(VxS)(Y) - f t (Y)S(X) = h(X,Y)Z, (19)

Ha(x , S(Y)) + (Vxf l ) (Y) = h(X, Y) f , (20)

and

d f ( X ) = - 2 f t ( S ( X ) ) , (21)

V x Z - f S ( X ) = - H S ( S ( X ) ) . (22)

CASE 1: M n IS RIEMANNIAN. In this case the metric g induced on M ~ from the Lorentzian metric on ~ + 1 is positive definite and a unit normal vector { will be


C It time-like (e = -1 ) . As M ~ is Riemannian, S is diagonalizable. Let { i}i=l be a local orthonormal* frame in the neighborhood of a point p with respect to which the shape operator is diagonal: Sel = Aiei.

We consider equation (19); after replacing X and Y by basis vectors ei, i = 1 , . . . , n, and considering all possible combinations, we get with i ~ j # k C {1 , . . . , n} the following set E of equations:

H ((V~iS)ei, ei) = 2ciAifl(ei) - eiAiei( H), (23)

H((VeiS)ei, ej) = ciAifl(ej), (24)

H((VeiS)ej , ei) = ~iAifl(ej), (25)

H ( (V~,S)ej, ej ) = -e j Ajei( H), (26)

H((Ve~S)ej, ek) = 0. (27)

However, Codazzi's equation (7) implies that

(V~, S)ej = (V~j S)ei (28)

and hence it follows that for i # j E {1 , . . . , n}

+ = 0.

Here we have to distinguish two subcases, according to whether there are at least two different Ai 7 ~ 0 (rk.qp > 2), or there is only one Ai 7 ~ 0 (rk Sp = 1). In the former case it follows that fl(ej) + ej(H) = 0 for all j C {1, ...n}, in the latter case we have only that fl(ej) + ej(H) = 0 for j > 2 (we can assume that A1 # 0, and A2 . . . . . An = 0).

1. rkSp > 1 andfl(ej) + ej(H) = Oforallj e { 1 , . . . , n } .

Since A is a fixed endomorphism of E~ +1 , the coefficients of the characteristic polynomials of its matrix are constants. In particular, this implies that the trace is constant, and we get

0 = d(trA) = 2 n H X ( H ) + d f ( X ) . (29)

But, together with (21), and for X = ek, k E {1 , . . . , n}, we have

o = + Ak) k(H). (30)

We show now that ek(H) = 0 for all k E {1 , . . . , n} . Indeed, if there were a k for which ek(H) # O, then n H + Ak = 0. On the other hand, from (23)-(27) and dfl + H = 0, we get

= - jAj k(H). (31)

* (el, e i) = 0 for i 7~ j , and (ei, ei) := ei. In the present case ei = +1 for all i E { 1 , . . . , n}. However, in view of later applications of the formulas, we will write them down in a slightly more general form than is needed for the actual purposes.


We may notice that with respect to the frame we have used, ((V~iS)ej , ej) = ej ei ()U). After summing (31) over all j different from k, one gets

H e k ( n H - Ak)= - ( n i l - )~)ek(H). (32)

But if ek(H) ¢ 0, then n H + ),k = 0. With )~k = - n i l , (32) gives ek(H) = 0 which contradicts the assumption.

We conclude that ek(H) = 0 for all k E { 1 , . . . , n } . This implies that both /3 and Z are zero, and from (23)-(27) we see that V S = 0, so M n has parallel fundamental form.

2. rkSp = 1(,~1 ¢ 0,,~2 = . . . . ,~n=O) and~(ej)+ej(H)=O for all j > 1.

From the definition of the mean curvature, we immediately see that nH = ),1. On the other hand, we still have that (ni l + )~k)ek(H) = 0 for k > 1. Together, this immediately gives that ek(~l) = 0 for k > 1.

Expressing (28) explicitly with respect to the choosen basis gives

(33)

After taking the inner product with e j, and choosing j = 1, in view of the fact that e i ( /~ l ) = 0 for i > 1, we find that (ei, Velel} = 0, hence that V~lel = 0.

Taking (33) again, and putting i = 1 leads to S(Vele j ) = --)H~7ejel, from which it follows that V~jel = 0 for j > 1.

Together this means that Vel = 0. Hence el is parallel. A standard argument then shows that a neighborhood o fp is a cylinder on a plane curve. Since M n is of restricted type, this curve is of restricted type.

CASE 2: M~ IS LORENTZIAN. In this case the metric g induced on M~ from the

Lorentzian metric on ~ + 1 is indefinite and a unit normal vector ( will be space- like (e = +1). As M~ is Lorentzian, S is not necessarily diagonalizable. We will have to distinguish four subcases according to whether the shape operator can be brought into one of the four possible forms (I)-(IV) with respect to corresponding suitable local frames.

Subcase 2.1: The shape operator can be brought into the form (I). Let {e,}i=l be a local orthonormal frame in the neighborhood of a point p with respect to which the shape operator is diagonal: Sei = )~ ie i . So (el, e j ) = 0 for i ~ j and (e~, e~) := ci, with c1 = - 1 and q = +1 for i >__ 2.

This case is directly analogous to Case i where M n is Riemannian. Indeed, one can look over the proof of Case 1 in detail (taking advantage of the general form of the formulas) and check that every step holds after replacing el = + 1 --+ el = - 1. At the end, the dependence on the signature of cl drops out. Therefore we can immediately formulate the conclusion:


When the shape operator S of a Lorentzian hypersurface M~ of restricted type

in E~ +1 is diagonalizable in the neighborhood of a point p, then M~ has either parallel second fundamental form, VS = 0, or M~ is in that neighborhood a cylinder on a plane curve of restricted type. Since Mp is Lorentzian, this plane curve of restricted type can be either a curve in the Euclidean or the Minkowski plane.

e n Subcase 2.2: The shape operator can be brought into the form (II). Let { i}i=l be a local pseudo-orthonormal frame in the neighborhood of a point p with respect to which the shape operator takes the form (II): Sel = Ael + e2, Se2 = /Xe2, S e i =

2,~e~ for i > 3. So (e~, e l) = (e2, e2) = 0, (e~, e2) = (e2, e~) = +1 , (e~, ~j) = 0 for i ¢ j _> 3, and (ei, ei) = +1, (el, ei) = (e2, ei) = 0 for i _> 3.

From equation (19), after replacing X and Y by basis vectors ei, i = 1 , . . . , n and considering all possible combinations, we get a set of equations E2 which is the counterpart of the set E(23)-(24) and which we do not list here because of its length. Together with Codazzi's equation (7) it follows that with i E {3 , . . . , n}

~(e~) + e~(H) = 0,

A~(~(el) + e , ( ~ ) ) = 0,

~ ( /~ (e2 ) + e 2 ( U ) ) = 0,

;~(~(e,) + e l ( H ) ) = ~(e2) + e2 (H) ,

a (~ (e2 ) + e l ( H ) ) = 0.

(34)

(35)

(36)

(37)

(38)

From this we see that, as soon as there exist a Ai ~ 0 or A ¢ 0, which is always satisfied since we may suppose that H 7~ 0, then identically

dH + fl = 0. (39)

Since A is a fixed endomorphism of I~1+1, the coefficients of the characteristic polynomials of its matrix are constants. In particular, this implies that the trace is constant, and we get

0 = d(trA) = 2 n H X ( H ) + d r ( X ) . (40)

But, together with (19), and for X = ek, k E {1 , . . . , n} we have

0 ---- ( / z H -t- ~ ) e l ( / - / ) -[- e 2 ( H ) , (41)

0 = (n i l + A)e2(H), (42)

0 = (n i l + Ai)ei(It). (43)

We show now that ek(It) = 0 for all k E {1 , . . . , n}. Indeed, if there were an i _> 3 for which ei(H) 7 ~ O, then n i t + hi = 0. On the other hand, from Y;2 and

328

dE + H = 0, we get

H((v~,s)ej , ej) = -~je~(H),

H((Ve, S)el , e2) = -Ae i (H) ,

R((v~,s)e2, el) = -~e~(H).

CHRISTOS BAIKOUSSIS, DAVID BLAIR, BANG-YEN CHEN, AND FILIP DEFEVER

(44)

(45)

(46)

We may notice that,with respect to the frame we have used, ( (V~S)e j , ej) = ei(Aj), ( ( V ~ S ) e l , e2) = ei(A) and ((V~,S)e2, el) = ei(A). After summing (44) over all j > 3 different from i, and adding (45) and (46), one gets

Hei (nH - Ai) = - ( n i l - Ai)ei(H). (47)

But if e i (H) ¢ 0, then n H + ),i = 0. With Ai = - n i l , (47) gives el(H) = 0 for all i E {3, ...n}. Analogously, one can prove also that ez(H) = 0 and el (H) = O. Together we find that ek(H) = 0 for all k E {1, ...n}. This implies that both/3 and Z are zero, and from ~2 we see that V S = 0, so M r has parallel second fundamental form.

e n Subcase 2.3: The shape operator can be brought into the form (III). Let { i } i= l be a local pseudo-orthonormal frame in the neighborhood of a point p with respect to which the shape operator takes the form (III): Sel = Ael + e3, Se2 = Ae2, Se3 =

e2 q- )~e3, Se i ~- )~iei for i _> 4. So (el , e l) = @2, e2} ---- 0, @1, e2) ----- @2, el) = +1, (ei,ej) = 0 f o r i ¢ j >__ 3, and (ei,ei) = +l , (e l , e i ) = (ez, ei) = 0 f o r i > 3 .

In the same way as in the previous cases we get from equation (19) the basic set of equations E3. In combination with Codazzi's equation (7) they lead to the following minimal system of equations, with i E { 4 , . . . , n}

/3(el) + e l (H) = 0, (48)

/~(e2) + e2(H) = 0, (49)

)~(fl(el) -~ e l ( H ) ) = fl(e3) q- e3 (H) , (50)

t3(ei) + ei(H) -- O. (51)

Hence, automatically, there follows

dH + ~ = o. (52)

Proceeding as before, the counterpart of the system (41)-(43) takes here the form

0 = ( n i l + ) ,)el(H) + e3(H), (53)

0 = ( n i l + A)e2(I-1), (54)

0 = (n~ + ~)e3(H) + e2(H), (55)

0 = ( n i l + Ai)ei(H). (56)


Following the same line of thought as in the previous subcase, one then consecu- tively shows that ei(11) = 0 for i _> 4, e i ( / / ) = 0, e2(H) = 0, and e3(H) = 0. So, in addition, there follows that ek(H) = 0 for all k C {1 , . . . , n}. This implies that both/3 and Z are zero, and from E3 we see that VS = 0, so M~ has parallel second fundamental form.

e n Subcase 2.4: The shape operator can be brought into the form (IV). Let { i}i=1 be a local orthonormal frame in the neighborhood of a point p with respect to which the shape operator takes the form (IV): Sel = Hel -q- t/e2, Se2 = - t / e l + H e 2 , S e i =

Aiel for i _> 3. So (ei,ej) = 0 for i # j , and (ei, ei) := ei, with el = - 1 and ei = +1 for i _> 2.

Here the equations, following from the basic set of equations 24 and Codazzi's equation (7), take the form, with i # j C {3 , . . . , n}

0 = H(/3(e2) + e2(H)) + .(/3(e,) + el(H)),

0 = #(/3(e,) + el(H)) + t/(/3(e2) q- e i (H)) ,

o = +

0 = /~i(/3(el) -1- e l ( H ) ) ,

0 = /\i(/3(e2) q- e2(H)),

o = + eAH)).

From the first two equations (57)-(58) we deduce that

0 = H((/3(e2) -{- e 2 ( H ) ) 2 -{- (/3(e1) q- e l ( H ) ) 2 ) ,

Since u #

0 =

0 =

t / ( ( /3 (e l ) -1- E l ( H ) ) 2 - (/3(e2) -t- E2(H) )2 ) .

0, we always have that

/3(e l ) q- e l ( H ) ,

/3(e2) + e2(H).

(57)

(58)

(59)

(60)

(61)

(62)

(63)

(64)

(65)

(66)

The remaining equations show that we have to distinguish two subcases, according to whether both # = 0 and only one Ai ~ 0 (rkSp = 1), or either # ¢ 0 or both # = 0 and at least 2 differrent Ai ~ 0 (rk S; > 2). In the latter case it follows that/3(ej) + ej( t t ) = 0, for all j E {1 , . . . , n}; in the former case we have only that/3(ej) + ej(H) = 0, for j ~ 3 (we can always assume that A3 # 0, and /~4 . . . . ~--- A n = 0).

1. rkSp > 1 and fl(ej) + ej(H) = Oforall j E { 1 , . . . , n } .

Since A is a fixed endomorphism of ~ + 1 , the trace of its matrix is constant. Together with (19), and the fact that/3 + dH = 0, we have

( (n i l + H) 2 --~ uZ)el(H) --- 0, (67)


(u 2 + (n i l + #)2)e2(//) = 0, (68)

(r~H + Ai)ei(H) = 0. (69)

Proceeding as before, one then shows that ei(H) = 0 for i >_ 3, e l (H) = 0, and e2(H) = 0. This implies that both fl and Z are zero, and from ~4 we see that VS = 0, so M~ has parallel second fundamental form.

2. rkSp = 1(# = 0, N3 7 ~ 0 , ~ 4 -~- . . . . )~n ~-- O)aF td~(e j ) - -~e j (H) = 0

for all j ¢ 3.

From the definition of the mean curvature, we immediately see that nH = A1. On the other hand, we still have that

( n 2 H 2 q- t . , 2 ) e l ( / / ) - - 0 ,

(~2 + n2t/2)e2(n) = 0,

( n n + ~)e¢(~/) = 0

(70)

(71)

(72)

for i > 3. Together, this immediately gives that ek()~3) = 0 for k 7! 3. In view of the fact that still 3(ej) + ej(H) = 0 for all j 7 ~ 3, we also have 3(ek) = 0 for k ¢ 3. The remaining nonzero components of V S are

~<(V~,S)e2, e3) = ~ ( e 3 ) ,

H((Vo2S)e3, el) = .~(e3),

f[((Ve3S)el,e2) = ///~(e3) ,

/-/((Ve3S)e3, e3) = -,~3e3(H) q- 2~3fl(e3).

(73)

(74)

(75)

(76)

From the explicit expressions of (28) with respect to the choosen basis, one can calculate (V~iej, ek) for i , j , k E {1,2, 3} in terms of A3, u, el(u), e2(u), e3(u). From the equations of ~4 other than (73)-(76) it follows that el(U) = 0, ez(u) = 0, e3(u) = 0 and hence hence (V~,ej,ek) = 0 for 1 _< i , j , k _< 3. Equations (73)-(76) now imply that/3(e3) = e3(H) = 0. So VS = 0 and M~ has parallel second fundamental tensor.

Summarizing the partial results of the different cases and subcases, we can formulate the following proposition:

PROPOSITION. A hypersurface M~ of restricted type in E~I+I has either parallel second fundamental form V S = 0 or is a cylinder on a plane curve of restricted type. This plane curve of restricted type can be either a curve of restricted type in the Euclidean or Minkowski plane.


4. Conclusions

Chen et al. [7] classify the curves of restricted type in the Euclidean plane; and the classification of the curves of restricted type in the Minkowski plane has been obtained in [10]. Hence, in order to finish the classification of the hypersurfaces of restricted type in Minkowski space I~ +1 , it suffices to establish which of the parallel hypersurfaces of 1E~1 +1 are indeed of restricted type. To our knowledge, there is no explicit list of parallel hypersurfaces of I~ +1 available in the literature; although partial results are contained in, e.g., [3], [12], [15]. Therefore, we have to devote here some considerations to this intermediate problem. Following the lines of [14], and distinguishing the different subcases which may arise, we see that there exist suitable local frames with respect to which the entries of the shape operator S of a parallel hypersurface of I~ + 1 are the same at all points, and hence constant. In particular, the minimal polynomial of the shape operator is constant. Hence the hypersurface is isoparametric. The Lorentzian isoparametric hypersurfaces have been classified in [16]. A Lorentzian isoparametric hypersurface in E~ +1 is either S k × E~ -k , S~ × I~ -k , or S~, with 1 _< k _< n - 1, if the shape operator is diagonalizable, or falls into one of four categories of generalized cylinders, if the shape operator is not diagonalizable. Along the line of proof in [14], the Riemannian parallel hypersurfaces of E~ +1 are seen to be the hypersurfaces/ /k × i ~ - k and H r~, with 1 < k < n - 1, since the shape operator is always diagonalizable in this case. The hypersurfaces S k x I~1 -k , S~ x Ig m-k, H k × E~. -k , S~, and H n, with 1 _< k ___ n - 1, are all parallel hypersurfaces of I~1+1 ; it can easily be checked that they are in fact of restricted type. In order to complete the classification of the hypersurfaces of restricted type of IE~ +1 , we still need to decide which hypersurfaces of the four types of generalized cylinders of [16] are indeed of restricted type. We distinguish eight subcases, according to whether the minimal polynomial of the Lorentzian isoparametric hypersurface is z2( z - a ), z 2, ( z - a )2:c , (z - a) 2, z3(z - a), z 3, ( z - a)3z, ( z - a) 3, with a a nonzero constant. We remark that the hypersurfaces with minimal polynomial z 2 and z 3 are in fact minimal. Hence they are automatically of restricted type, as included in the set of minimal hypersurfaces. Following the line of proof of Lemma 1 of [14], we see that hypersurfaces with minimal polynomial z 2 ( z - a) or z 3 ( z - a) cannot be parallel; hence they cannot be of restricted type. For the remaining cases, where the minimal polynomial takes one of the forms (x - a)2x, (x - a) 2, ( x - a)3x, or (x - a) 3, a long, although straightforward, calculation shows that they are not parallel, and consequently not of restricted type.

Summarizing, we can formulate the following theorem which classifies the hypersurfaces of restricted type of the Minkowski space I~+1:

THEOREM. A hypersurface M n o fE~ +1 is o f restricted type i f and only i f M ~ is one o f the fo l lowing:

(i) a min imal hypersurface;


(ii) an open part o f one o f the following hypersurfaces: S k X IE~ -k , Ski × E ~-k , H k × E ~ - k , S ~ , I - I n , w i t h l < _ k < n - 1 ;

(iii) an open part o f a cylinder on a plane curve o f restricted type.

Acknowledgement

The fourth author (ED.) would like to thank M. Magid for helpful discussions.

References

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Hypersurfaces of restricted type in Minkowski space