Cartanian symmetrical property of Riemannian projective-symmetric spaces - tensor geometr

Reynolds, R. F.

Thompson, A. H.

English

NASA Technical Reports Server

f 
SPACE RESEARCH COORDINATION CENTER 
./ PROJECTIVE-SYMMETRIC SPACES 1 
BY 
R. F. REYNOLDS AND A. H. THOMPSON 
DEPARTMENT OF MATHEMATICS 
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SRCC REPORT NO. 16 
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ff 653 July65 
UNIVERSITY OF PITTSBURGH 
PITTSBURGH, PENNSYLVANIA 
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.A"-.*-** 
22 OCTOBER 1965 
https://ntrs.nasa.gov/search.jsp?R=19660003680 2020-03-16T22:10:05+00:00Z
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i 
PFUIJECTIVE-SYMMETRIC SPACES* 
R. F. Reynolds and A. H. Thampson 
m c t i q .  
Gy. Soor [ l ]  and B. Gupta [2 ]  have dircussed the properties 
of Riemannian spaces Vn ( ~ 2 )  in which the first covariant deri-  
vat ive of Weyl's project ive curvature tensor is everywhere zero; such 
spaces they c a l l  Pro.lective-Smnnetric spaces. In t h i s  paper we wish 
t o  point out t ha t  a l l  Riemannian spaces w i t h  thin property are 
symmetric in the sense of Cartan [3] ;  tha t  is t h e  first covariant 
der ivat ive of the Riemann curvature tensor of the space vanisher. 
Further sections are  devoted t o  a discussion of projective-symmetric 
a f f ine  spaces A,, w i t h  oymmetric a f f ine  connexion. Throughout, the 
geometrical quant i t ies  discuared w i l l  be as defined by Eisenhart [4] 
and [ 5 ] .  
1. g r o l e c t i  ve-SvrPnretric Riemannirn Spaces. 
For a vn , weyl'r projective curvature tensor Wabcd i s  
a a 2 a 
bcd bcd ox [d %]b} @ 
*Supported in par t  by tha National Aeronautics and Space Adminiatration 
under grant Nr G-416; U n i V 8 r S i t Y  of Pittsburgh. 
Awtraliaa Journal - of Mathematicr. 
Accepted for publication i n  
4 
d-- 
2. 
tensor, of the space. The Vn Is a projective-symetrlc space i f  
and only If 
a - 0. bcd;e 
a We define the tensor U by 
a 
where 
proj ective-synnnetrlc, then 
R = R a , and from (1.1) It follows tha t  I f  the space I s  
a 
= 0 .  d;e 
For n>2, equation (1.2) and the twlca-contracted Blunchl Ident i ty  
and thus w e  have R a b t e  = 0. With (1.1) t h i s  gives the r e su l t  
wabcd;e< > R a  bcd;e - 0 9  
from which follows:- 
Theorem 1. v
- A Riemannian space Vn ( ~ 2 )  l e  a projective-symmetric apace 
i f  and only if it l a  eyrmnetric i n  the eense of Cartan [3].  --
a 
For n 2,  w bcd I s  ident ical ly  zero and (1.1) l e  a degene- 
rate condition in  a V2. We ramark however tha t  a V2 l e  a eymmetrlc 
. 
r' 3. 
space if snd only if it has constant scalar curvature R. 
The results of Gupta [2] follow immediately since they are 
trivially true for synrnetric spaces. The paper of Soos [2] contains 
theorems for projective-symmetric spaces which are generalisations of 
results found by Sinjukaw [6] for symmetric spaces. 
Affine SDacea With m x i o n .  2. -
For the remainder of this paper we consider the application 
of the preceding theorem tn 8n Affine space with rymetric connexion. 
Such a space we will denote by 
covariant dif ferentiation with respect to this connexion by 
a An, its connexion by I' bc, and 
" ;" . 
The curvature tensor of A is defined n 
a h a 
bcd 2rab[d,c] + 2r b[d I' c]h ' 
for which the identities 
and Bianchi's identity 
hold. The analogue of the Ricci tensor for an An is Bbc = Babca, 
but in this care it ir not necessarily rymmetric; it follows from 
(2.2) that 
h e r e  scd = Baaed. From (2.3) we have also 
- 0 ,  [cd;eI S 
( 2 . 5 )  
a 
bcd;a 2Bb[c;d] ' 
Weyl's projective curvature tensor for an An is 
(2.6) Wabcd Ba - -  1 0 -  2 B  da  -- s 6a 
bed n+l *ab 'cd n-1 b[c  d] *2-1 b[c dl 
This tensor is invariant for  projective transformations of the space 
and i t s  vanishing implies tha t  the A has the same pathe as f l a t  
space [ 5 ] .  By sproiective-eynnnetric a f f ine  space we w i l l  mean an 
A, ( ~ 2 )  such tha t  
n 
Wa = 0 ,  bcd;e (2.7) 
throughout; an A is # m e t r i c  [3] i f  and only i f  n 
(2.8) Ba bcd;e - 0 ,  
at  ail polsitc. 
1 
Equation (2.8) implies that every symmetric An is a projec- 
t ive-symetr ic  A . Such projective-symmetric spaces we w i l l  c a l l  
degenerate, and from Beortag 1 we see tha t  a l l  projective-syrmaetric 
Riemannian epacss V (n72) are  degenerate i n  t h i s  sense. We w i l l  
show tha t  t h i s  is not t rue  for  a general An and w i l l  coneider i t s  
va l id i ty  in re la t ion  t o  cer ta in  sub-claores of Affine spaces. 
n 
n 
3. Noa-Doneaerate Prolective-Synnnetric 4. 
Conrider the 41 with connexion coeff ic ients  
5 .  
(3.1) 
a 
i n  a coordinate system {x 1 such that  
The latter condition i s  expressed covariantly as 
The 41 
t u r e  tenror  vanishes and therefore it  is a projective-symmetric 
space. From (2.1) we have f o r  t h i s  An 
i s  project ively re la ted  t o  f l a t  space; i t s  project ive curva- 
and using (3.2) 
For 
have the r e s u l t  :- 
$, # 0, the  curvature tensor of the space i e  non-zero and we 
Theorem 2. -
-- There exirt  prolrctive-symmetric & ' a  which are non-degenerate. 
3. The Decomposable %. -
If two spaces and &lp are  given with coordinate8 
a A 
x : (a,@,y = 1,2,...,m) and x : (A,B,C = Wl,...,n) and the connexlons 
then the & with coordinates x : (a,b,c - l,Z,...,n) a and A BC, 
a 
A 1, Is ca l led  the  product of Am a By' BC and connexion r bc z U 
and %-. An An t ha t  ir a product space is sa id  t o  be decomposable. 
A geometric object in a decomposable 
if i t s  components with respect t o  the spec ia l  coordinates are always 
zero when they have indices from both ranges, and the  components 
belonging t o  the rubspace A, (h-) are functions of x (x ) only. 
I n  a decomposable %, B bcdp BbC and t h e i r  covariant derivatives 
are decomposable; w bed and w bcdie are not i n  general decomposable. 
An is decomposable i f  and only 
a A  
a 
a a 
Theorem 3. 
1
- A projective-symmetric which is decompoeable is neces- 
s a r i  ly degenerate . 
We (USUPLO t h a t  5 {hX 4-1 
relate t o  pt,, and A,B,C - u+l,....n relate t o  From the  
def in i t ion  of the  projective-curvature tensor we have f o r  
posable An. 
where indices a ,$ ,y  - l....m 
the decom- 
and 
(3.2) 
The assumption tha t  % is a projective-symmetric space gives with (3.1) 
and therefore in (3.2) 
7. 
Similarly w e  have 
and s in@ Bbd;e is a decomposable tensor of t he  % it  follows tha t  
%d;e = 0 .  
With the  above, the  d i f fe ren t ia t ion  of (2.5) gives 
a a * = bcd;e bcd;e 
and t h e  decomposable 4 is a aymet r i c  space. 
4. 111 The proSectIve-Smmetric gn. 
An A, i n  which there  ex is t s  a rynnnetric two index tensor 
gat, of rank n such tha t  
f o r  same covariant vector oc is cal led a Wn and was f i r s t  discussed 
ab ab a 
by Weyl [7].  Define the  contravariant tensor g by g gbc = 6 c' 
then from (4.1) 
(4.1) 
ab ab 
g ; c  = 20c 8 
ab 
We can uae gab (g ) t o  define 8 correspondence between covariant 
and contravariant quant i t ies  i n  A,; i n  f ac t  i f  Oc is a gradient 
and 
bc 
(4.3) Qad Babcd 
.. 
vector  0 the  Wn is a Riemannian space Vn w i t h  metric tensor  
*c - 20 
= e gab' 
e we def ine  With 'abed 8ae "bed and Babcd Iae bed* 
(4.2) 
From the Ricci I d e n t i t y  applied t o  gab* and the use of (4.1) and 
(4.la) w e  have 
(ab) cd = -2 gab $[c;d] * 
which y i e lds  a f t e r  cont rac t ion  
Qad - Bad O4 4[a;d] * 
and 
'2n4[c;d] 
We e x t r a c t  the symmetric and anti-symmetric pa r t8  of theoe equations 
t o  obta in  
(4.4) B[adl * "+[a;d] * 
Ql.4 (n-4) + [a; d] 
With equation (4.2), t h e  de f in i t i on  of the p ro jec t ive  curvature tensor  
. 
9. 
gives 
bc bc 
where B = g Bbc - 8 Qbco Frequent use of the relations 
(2.4) and (4.4) give the decomposition( 
n2-4 
B[ab] .) T[abl n(n-1) 
Lemma 1. -
-- In a grolective-symmetric Wn ( m 2 )  Tab;= - 0 .  
Proof . -
ef d d ef 
Tab ; c gad g efb;c 4- 8ad;c b + 8 ;c 'aefb a 
From (4.1) and (4.la) the sun sf the second and third terms on the 
right hand side of the above equation is zero. Hence 
ef d 
Tab ; c gad 8 efb;c B 
which vanirhar if Wn is projective-symmetric. Q.E.D. 
7 L e m a  1 applied to the second equation of (4.5) gives 
and with (4.6) in the first equation of (4.5) 
10 
n3rcmf* 
We have 
2 
Bb[c;d] 0 -  
n-2 -- a a * bcd;a bcd;a n2-1 Scd;b n-l 
From (2.3) end (4.6) S,d;e = 0, and we see t h a t  
a 2 
Bb[c;d] bcd;a n-l I -  
However from the  contracted Biaachi Ident f ty  (2,5) we have 
a 
bcd;a 2Bb[c;d] * 
and t h e  r e su l t  of the lema follows. 
From lemma 2 and (4.7) we have 
and a f t e r  contraction 
Referring t o  equation (4.7) we deduce t h a t  Bab;c 0 ,  and there'fore 
f o r  a Wn 
0 I) Wabcd;e < - 0. 
11. 
Theoreq 4, 
Every poiactive-symmetric Wn & degenerate. 
We also remark that i f  B # 0 i n  ( 4 . 3 )  then $c is neces- 
sarily a gradient:. 
Theorem 5, 
P 
-- The "scalar curvature" B of a prod ective-symmetric Wn which 
is  not a Riemannian space i s  neceraarily zero. --- 
i 
V 
I 
REFERENCES 
Soos, Gy., Ueber d i e  Geodaetiochem Abbildung vm Riemannrche 
Raeumen auf Projectiv-Symmctrische Riemanusche Raeme. 
Math. Acad. Sci. Hung., 9, (1958) 349-361. 
Gupta, B., On Projective-Spanettic Spacer. 
SOC. IV, (1964). 113-121. 
Acta. 
J. Austr. Math. 
Schouten, J. A*,  Ricci Calculus, 2nd Ed., p. 163. Springer- 
Verlag, Berlin 1954. 
Eirenhart ,  L. P., Riemannian Geometry, Princeton U. Presr ,  
Princeton. (1926). 
Eisenhart, L. P., Non-Riemannian Geometry, h e r .  Math. SOC. 
New York (1927). 
Sinjulcw, N. S., On t he  Geodesic Correspondence of a Riemannian 
Space with a Synnnetric Space (Russian) D. A. N. (U.S.S.R*).  
98 (1954) 21-23. 
Weyl, H., Reine Inf ini tes imal  Gearnettie. Math. Ze i techr i f t ,  
2, (1918) 384-411. 
Department of Mathematics 
University of Pit tsburgh 
Pittsburgh, Pennsylvania 
U.S.A. 


Projective-symmetric spaces

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