On Riemann Spaces with Proper Affine Collineations

Revista Brasileira de Flsica, Vol. 18, n9 1, 1988

On Riemann Spaces with Proper Affine Collineations B. LESCHE and M.L. BEDRAN

lmtituto de Flsica, Universidade Fedral do Rio de Janeiro, Caixa Postal 68528, Rio de Janeiro,

21944, RJ, Brasil

Recebido em 14 de outubro de 1987

Ab-t I t i s shown t h a t a Riemannian m a n i f o l d (wi t h p o s i t i v e d e f i n i t e m e t r i c ) w h i c h c a r r i e s an a f f i n e c o l l i n e a t i o n can l o c a l l y be w r i t t e n as a C a r t e s i a n p r o d u c t o f m a n i f o l d s i n each one o f which t h e a f f i n e c o l l i n e a t i o n a c t s as a h o m o t h e t i c motion. 1. INTRODUCTION

A f f i n e c o l l i n e a t i o n s a r e symmetries o f a f f i n e spaces d e f i n e d by t h e v a n i s h i n g L i e d e r i v a t i v e o f t h e a f f i n e connections l

a

where ; denotes c o v a r i a n t d e r i v a t i v e and R

i s t h e Riemann tensor. I n

B Y ~

Riemannian and pseudo Riemannian spaces eq.

(1.1)

i s equivalent t o

where t h e ( ) i n d i c a t e s y m m e t r i z a t i o n i n t h e i n d i c e s a,B.

S p e c i a l cases

are o f a f f i n e c o l l i n e a t i o n s i n Riemannian spaces w i t h r n e t r i c g aB t h e t i c motions

h;B) = c o n s t .

homo-

%B

and K i l l i n g v e c t o r

A f f i n e c011 i n e a t i o n s a r e t r a n s f o r m a t i o n s t h a t keep t h e geodes i c s unchanged, a l t h o u g h t h e y may change t h e m e t r i c . Hojman

set

of

et

a12

showed t h a t a f f i n e c o l l i n e a t i o n s a r e non- Noetherian symmetries and cons t r u c t e d new c o n s t a n t s o f rnotion a s s o c i a t e d t o them. An example o f a f L f i n e c o l l i n e a t i o n was g i v e n by K a t z i n and ' l e v i n e 3 i n a two

dirnensional

a f f i n e space wi t h o u t m e t r i c . Bedran and ~ e s c h egave ~ an example o f

af-

Revista Brasileira de Ffsica, Vol. 18, n? 1, 1988

f i n e c01 1 i n e a t i o n i n t h e Robertson- Walker m e t r i c (a pseudo

Riemannian

space) w h i c h t u r n e d o u t t o be a h m o t h e t i c m o t i o n . I n t h i s paper we s h a l l d e r i v e c o n d i t i o n s on t h e geometry o f Riemannian space i n o r d e r t o a d m i t a p r o p e r a f f i n e c o l l i n e a t i o n , an a f f i n e c o l l i n e a t i o n t h a t i s n o t a h o m o t h e t i c motion. The a f f i n e c o l l i n e a t i o n s i n Riemannian spaces i s o f i n t e r e s t

for

i.e,

study

of

dynamic

systems. The o r b i t s o f a system o f p a r t i c l e s under s c l e r o n o r n i c s t r a i n t s w i t h o u t f o r c e s a r e g i v e n by t h e

a

con-

g e o d e s i c s i n conf i g u r a t i o n

space, where t h e m e t r i c i s g i v e n by t h e k i n e t i c energy.

Aff ine

col-

l i n e a t i o n s define sygmetries i n the s e t o f o r b i t s .

2. RESTRICTIONS ON ~ H GEOMETRY E IMPOSED BY THE EXISTENCE OF PROPER AFFINE COLLINEATIONS

L e t M be a f i n i t e dimensional Riemannian m a n i f o l d w i t h p o s i t i v e definite metric g

aB'

a c c o r d i n g t o eq. (1.2)

I f t h e r e e x i s t s an a f f i n e c o l l i n e a t i o n on M, then t h e r e must e x i s t a symmetric t e n s o r SolBsuch t h a t

'ol~;y

= O

I f t h e a f f i n e c o l l i n e a t i o n i s a p r o p e r one we

(2.1) h a v e w i t h eq.

(1.3) saB

Z const.

As SaB i s symmetric, L?B

x gaB

d e f i n e s a 1i n e a r mapping o f t h e

(2.2) tan-

g e n t spaces i n t o t h e t a n g e n t spaces w h i c h i s s e l f - a d j o i n t w i t h r e s p e c t t o t h e s c a l a r p r o d u c t d e f i n e d by t h e m e t r i c t e n s o r . Thus, a p p l y i n g t h e s p e c t r a l theorem, we can w r i t e

k

wi t h

and

Revista Brasileira de Flsica, Vol. 18, n? 1, 1988

and

and S R E R . The f a c t t h a t SaB i s c o v a r i a n t l y constant means t h a t s a B ( x ) can be obtained from S o l B ( g ) by a p p l y i n g a p a r a l l e l t r a n s p o r t o p e r a t o r which corresponds t o a p a t h lead i n g from x t o y

SOB( y ) =

I n s e r t i n g eq. (2.3)

I'

Sq (z)

;'2

i n t o eq. ( 2 . 7 ) g i v e s

c

II

(ylphB ( Y ) =

v

R

lJ

Sp, ( x )Ta

p R V ~( x )

,? ( y ) S T V ~ l J ((s) ~ RcrB E B EV~J f u l f i 11s eqs. (2.4) -(2.6). Then the uniqueness

of

the

spectral dec

p o s i t i o n implies

S R ( y ) ' s (z) z S R R

(2

and

Paa6(y) = T~' T 6lJ P RW(") By def i n i t i o n o f p a r a l l e l t r a n s p o r t eq.(2.10)

(2.

gives

T h i s r e s u l t means t h a t the tangent spaces Tx can be w r i t t e n as a d i r e c t sum o f orthogonal subspaces

i n such a way t h a t a p a r ' a l l e l t r a n s p o r t .c f rom x t o y maps V&

into

Y

v

along

any

curve y leading

RY T

y

v

=

(2.13)

Revista Brasileira de Flsica, Vol. 18, nP 1, 1988

Consequently the Riemann tensor

such t h a t i t maps

V&

fl

BlJv

d e f i n e s a rnap

into

From eq. (2.15) and

R u o r ~+~RuyaB + RV B Y ~=

O

and from the o r t h o g o n a l i t y o f the s'paces Vhone

f u r t h e r concludes f o r

I n the f o l l o w i n g we w i l l denote c o v a r i a n t d e r i v a t i v e s by D. Lema: Let x ( c , X ) be a one-parameter fami 1y o f geodesics, where 5 i s t h e curve parameter ( p r o p o r t i o n a l t o the p a t h l e n g t h s) and X i s the f a m i l y

axCL/a T h e n v t h e Lemma g i v e s t h e d e s i r e d r e s u l t a/au (y) E

c o n n e c t i o n . W i t h eq.(2.21) ementof V

Eq.(2.22)

RP'

element o f V

EP

we have t h a t Dt/DA a t X = 0 ,

ensures t h a t D a / D g a t A = O ,

'

vky. L e t us now w r i t e t h e m e t r i c t e n s o r i n t h e s e a l i g n e d c o o r d i n a t e s . As t h e spaces Vlluca r e o r t h o g o n a l i t f o l l o w s t h a t g t a k e s b l o c k d i a g o n a l form

F u r t h e r , f rom eq. (2.13)

f o r t h e C h r i s t o f f e l symbols we conclude

we g e t f rom eq. (2.24)

,

', ' ifn f m

+

r cn, k x m , i

ixp,s> (2.25)

Revista Brasileira de Flsica, Vol. 18,

So (a) gji

um ' s,i

n9 1, 1988

depends o n l y o n t h e c o o r d i n a t e s

The s e t MQ C

= O f o r m # Q i s l o c a l l y a submanifold o f M

.

(i)

M

with

can be thought

5

o f as a m e t r i c t e n s o r f o r MR. On MQ t h e a f f i n e c011 i n e a t i o n

acts

as

a homothetic m o t i o n because t h e tangent spaces o f MQ a r e t h e VQ and c(