Normal Coordinates in the Geometry of Paths

Normal Coordinates in the Geometry of PathsAuthor(s): Harry LevySource: Proceedings of the National Academy of Sciences of the United States of America,Vol. 16, No. 7 (Jul. 15, 1930), pp. 492-496Published by: National Academy of SciencesStable URL: http://www.jstor.org/stable/85687 .

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492 MATHEMATICS: H. LEVY PROC. N. A. S.

NORMAL COORDINATES IN THE GEOMETRY OF PATHS

BY HARRY LEVY

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF ILLINOIS

Communicated May 28, 1930

1. Riemannl first proved that in a metric space characterized by the linear element2

ds2 = gjdx'dxj (1.1)

it is always possible to choose a co6rdinate system first so that the first derivatives of the g's are zero at any preassigned point and second so that the geodesics through that point have linear equations. Fermi3 extended Riemann's results, establishing the existence of a co6rdinate system in which the g's are stationary along any preassigned curve, but in such a co6rdinate system the geodesics will be given by power series in the arc s in which the coefficients of only the quadratic terms are zero. Veblen4 and Eisenhart5 extended Riemann's and Fermi's results, respectively, to non-metric characterized by the paths

d2x' . dx3 dxk d + rk d d =0. (1.2) ds2 ds ds

Eisenhart's work has been carried somewhat further by Whitehead and Williams.6 But aside from Veblen no one has found any new coordinate systems in which is retained the vital property of Riemann's results, namely, that geodesics have linear equations. We propose to do this. The main purpose of this paper is to prove the following theorem:

Let Vm be an arbitrary m-dimensional manifold in the space characterized by equations (1.2) and let X(,)i (i = 1, 2, ... n) be the components of n - m (ao = m + 1, m + 2, . . . n) arbitrary directions defined at points of Vm but none of which lie in Vm; then there exist (infinitely many) coordinate systems such that the paths through the points of Vm in directions linearly dependent on the n - m preassigned directions X(/) have linear

equations. If the space (1.2) is Riemannian, the following somewhat special form

of the theorem seems most useful: In any Riemannian space there exist coordinate systems in which the

geodesics through the points of a preassigned subspace in the directions orthogonal to that subspace have linear equations.

We observe that this theorem may, for m = 0 be regarded as identical with Riemann's.

2. The proof is direct. Let Vm be given parametrically by the equations



VoL. 16, 1930 MA THEMA TICS: H. LEVY 493

x- = xi(ul, u2, ... u. ) i = , 2, ... n, (2.1)

and let Xi be any direction linearly dependent on the X(\)s2

Xi = AaX(,)i (2.2)

The path through a point x in a direction X is given by4

Xi = x + XiS - 1/2s xks2 - 1/3! rjx X - .. (2.3)

where the r's are evaluated at the point x. We write

us = A s f = m +- 1, ... n, (2.4)

so that2 is = UaX(,)

and

Xi= x + uz(X( - '/2 rjk(/X ) ku,u - (2.5)

If we regard the x's in these equations as the functions of u1, u2, ... um

given by (2.1) the X's are then defined as independent functions of the n u's with single valued inverses in the neighborhood of Vm and hence the u's may be regarded as a new set of co6rdinates.

The path (2.3) determined by the initial conditions

xi i Xz/.l1 2 " = x'(, (U , U ... O

X' = A(uo) X(,/)(uo)

that is, by a point (ul, u2, ... u) of Vm and a direction there, must sat- isfy the conditions

u~ = ka

u = k"s (2.6)

where the k's are constants. We shall speak of co6rdinates of this type as normal with respect to Vm.

3. Let the co6rdinates (ul, u2, ... un) be normal with respect to the Vm given by

u= 0 = + 1, ... n, (3.1)

and let us introduce new co6rdinates ii defined by the equations

2 = fR(ul, u2, ... um) (3.2)

ia = a u7 (3.3)

where the a's are constants and the u's are functionally independent. It follows that the f's are likewise normal with respect to Vm. With the aid of (3.2) and (2.5) we note that when we make a general trans-



494 MATHEMATICS: H. LEVY PROC. N. A. S.

formation of the parameters ul... um of Vm the normal coordinates will undergo the transformations given by (3.2) and

u = U a = + 1, ... n,

while if we make a general transformation of the space co6rdinates or if we replace the n - m directions X\()' associated with the normal co6rdinates by a new set of X's the transformation of the normal co6rdinates will be given by (3.3) and i" = u, (a = 1, 2, . . ., m).

4. Let L,k be the coefficients of the linear connection determined by (1.2) in terms of co6rdinates normal with respect to a given Vm. Then (2.6) must be a solution of

d2U duj duk + Ljk - - = 0, (4.1)

ds2 ds ds whence it follows that

L'tTkk' = 0. (4.2)

Multiplying by s2 we find by means (2.6) and the equations immediately preceding that the relations

L,TUuT = = 1, 2, ...,n (4.3)

hold identically throughout the space. Likewise from (4.2), since the k's are arbitrary, it follows that

Lr _ i = ,2, ...,n, (4.4) O T,T = m + 1, ..., n,

for all points of Vm. If we differentiate (4.3) partially with regard to u' we obtain

uau + 2L,u = 0,

whence it follows that the relations

aLaT uauvu" = 0 (4.5) 6u,

hold throughout the space; again, since the k's in (2.6) are arbitrary that the relations

P = 0, (4.6) bu,

where P stands for the sum obtained by permuting the indices i7, a, r cyclically, hold for all points of Vm. We can continue this process and obtain a sequency of identities of the types of (4.3) and (4.5) which for m = 0 reduce to those obtained by Veblen.4



Vol. 16, 1930 MA THEMA TICS: H. LEVY 495

5. Let us suppose our space is Riemannian with linear element (1.1), that in it we are given a non-minimal subspace Vm defined by (3.1) and that the linear element of Vm is given by2

ds2 = -gadudu ,

where a bar, as over the g, denotes that the function barred is evaluated on Vm, i.e., for u' = 0 (o- = m + 1, ... n). We assume, moreover, that the co6rdinates u' are normal with respect to Vm and that the X's associated with normal co6rdinates have been taken as a set of independent mutually orthogonal normals. Since the curves of parameters u" and uh, respectively, through a point of Vm are orthogonal, it follows that

g((h = ?m - 1, .... n) .

Moreover we may, by a proper choice of u', make

g = 1 == ,. (5.2)

If we define the quantities (4,).a

(<)a2 = - y (5.3)

2 bu'

it follows by a direct computation using (5.1), (5.2) and (4.4) that these 2's are the coefficients of the second fundamental forms of Vm. A similar

computation shows that the functions /(T), defined by

ar t)a = =2 b ) (5.4)

are the coefficients of the linear forms associated by Voss with a subspace.7 This interpretation of the 02's is identical with Bianchi's form = n - ,8 and one could by following his method for that case obtain the Gauss- Codazzi equations for a general subspace directly by means of (5.3) and (5.4).

6. Let m = 1 so that Vm is a curve C. We assume that space is Rie- mannian referred to co6rdinates normal with respect to C, and that the X's of ?1 are the n - 1 principal normals to C.9 The Frenet equations for C"

du j a- 1 X(a) ,i d = - N(- \( -l) + 1-?+ X(T+,),' (6.1)

ds Pa - 1 P

where the 1/p's are the curvatures of C, reduce in this case by means of (4.4), (5.1) and (5.2) to



496 PHYSICS: W.A. MARRISON PROC. N. A. S.

L+I= _ 1'+, L_ ( - __ Cl-1 (6.2)

Pa Pa -

so that

1 l 1 (bg 1 -+l _ 6g4U p, 2 agl,+ ugl (6.4) Po- 2 au' 8bu +

or

-= (+la).1) (6.3) Pa

that is, the curvatures of a curve correspond to the ,u's of a subspace. The analogues of the 2's are, of course, zero.

1 Gesammelte Werke, 1876, p. 261. 2 The Latin indices h, i, j, ... range through the values 1, 2, ... n, the Greek a, j, , ...

through 1, 2, ... m, and the Greek 7r, a, r, ... through m + 1, m + 2, . .. n. Repeti- tion of an index indicates the sum obtained by allowing that index to take on all values of its range.

3 Rend. Lincei, Rome, 311, 21, 51 (1922). 4 These PROCEEDINGS, 8, 192-197 (1922). 5 "Non-Riemannian Geometry," Amer. Math. Soc. Colloq. Publ. (1927), p. 64. 6 Ann. Math., 312, 151 (1930). 7 L. P. Eisenhart, Riemannian Geometry, Princeton, pp. 159-163. 8 "Lezioni di Geometria Differenziale," Bologna, 2, 450-455 (1924). 9 Riemannian Geometry, pp. 106-107.

TIIE CRYSTAL CLOCK

BY W. A. MARRISON

BELL TELEPHONE LABORATORIES, NEW YORK CITY

Read before the Academy, April 29, 1930

The crystal clock is a relatively new device for keeping accurate time. It consists essentially of a generator of constant frequency controlled by a resonator made of quartz crystal, with suitable means for producing continuous rotation controlled by it to operate time indicating and related mechanisms.

A crystal clock of this sort has been set up in the Bell Telephone Lab- oratories and has been operating over a considerable period. The appara- tus was designed especially as a reference standard of frequency for the Bell System, but it was recognized that it might also serve as a reference standard of time, if developed for that purpose. As a matter of fact, since time interval and frequency are so closely related, it would be very

496 PHYSICS: W.A. MARRISON PROC. N. A. S.

L+I= _ 1'+, L_ ( - __ Cl-1 (6.2)

Pa Pa -

so that

1 l 1 (bg 1 -+l _ 6g4U p, 2 agl,+ ugl (6.4) Po- 2 au' 8bu +

or

-= (+la).1) (6.3) Pa

that is, the curvatures of a curve correspond to the ,u's of a subspace. The analogues of the 2's are, of course, zero.

1 Gesammelte Werke, 1876, p. 261. 2 The Latin indices h, i, j, ... range through the values 1, 2, ... n, the Greek a, j, , ...

through 1, 2, ... m, and the Greek 7r, a, r, ... through m + 1, m + 2, . .. n. Repeti- tion of an index indicates the sum obtained by allowing that index to take on all values of its range.

3 Rend. Lincei, Rome, 311, 21, 51 (1922). 4 These PROCEEDINGS, 8, 192-197 (1922). 5 "Non-Riemannian Geometry," Amer. Math. Soc. Colloq. Publ. (1927), p. 64. 6 Ann. Math., 312, 151 (1930). 7 L. P. Eisenhart, Riemannian Geometry, Princeton, pp. 159-163. 8 "Lezioni di Geometria Differenziale," Bologna, 2, 450-455 (1924). 9 Riemannian Geometry, pp. 106-107.

TIIE CRYSTAL CLOCK

BY W. A. MARRISON

BELL TELEPHONE LABORATORIES, NEW YORK CITY

Read before the Academy, April 29, 1930

The crystal clock is a relatively new device for keeping accurate time. It consists essentially of a generator of constant frequency controlled by a resonator made of quartz crystal, with suitable means for producing continuous rotation controlled by it to operate time indicating and related mechanisms.

A crystal clock of this sort has been set up in the Bell Telephone Lab- oratories and has been operating over a considerable period. The appara- tus was designed especially as a reference standard of frequency for the Bell System, but it was recognized that it might also serve as a reference standard of time, if developed for that purpose. As a matter of fact, since time interval and frequency are so closely related, it would be very



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Normal Coordinates in the Geometry of Paths