THE THEOREM OF K. NOMIZU ON FINSLER MANIFOLDS. MATC by. SORIN DRAGOMIR. §1. INTROICTION. In a preceeding, paper, [2] , we have given a des-.
f
THE TI.IEORII'i OF
K.
asc.
2
NOi.{IZU OI{ FINSLER IIAi'IIFOLDS
by SORlI.I IJRAGO}{IR
§
1"'
INTR0l-''cTr0h'
1n a Preceeding Paper , lz7 , wc have qiven a dessubcription of Finsler space forms by means.-of -reqular form' It' is :[anj_folc]s wtrich nave a vanishing sécond frlnda:nental ,,..,ci1. xnowrr, i+ ] , that tirese are totally geodesic s;ui:rrarnifolds'
aswellasintheRiemanniangeoinetly.Butgeneral]l.Lheconverse fails for the.rinslerian case. the cletermination of the-Lola'lil,gecocsicsubinanifo}dsofaglvenfinslermanifo}disadifficuIL probl.em. In 1-his note we shall prove that the connected given conponents of the fixed point set of an izometry of a FinslermanifoldMaretotal}ygeodesicsubmanifoldsoffi. §2.
PRELI]''INARIES AND NOTATIONS
give a brief account of notations and conventions' real For detaj.Is we refeare t" [:] ' Let M be a n-dimenslonal c' - differentiable *.rìitorO' we denote by: T(lrr)*.r lI the the subbundle of tangent.bundle over I'1 , ancl by n : il-->ll non-nul tangent vectors on li' Let P : n-}ru--+Il be the bundle ring naruraliy induced by T (M) and n . we denote by 5P rrtt tne ofrealvalue 0 so tnai we have a !rel1 defined exponentlal map BY theore,fi 2'L',
exP x i B, -*_>
,
by exp *(x) = o*(1) L
-c
-
M
where
x
M with the origin Let i 'U" a fixed non-nul tangent vector on at x. Hence we can define a natural lift: t B.-r!'!
expof the exponential
x tTtdPr bY:
exp-x=exp. >:
-ìI
ìlX
: -l - {;ì --t't . where :', = i"j x BL , ani ? r-iTli* second factor: of the Prodrlct ---T (ì4) E t-he prclection on the of X to be L(X) manifold ii, v' "(Fl) ' ff we
r
and:
T -(I{ ll
r*
II
r(n
l
-
{}1)
-1
I I
I
l:
I I I
s
I
/!, \
P**
x
§4. izol'{trRriis
F*x
tirt
0N FTNSLER Ì,lANrFoLDS
on M. Then tire dlife-
I' : i,i--*I"1 be a transformation
Let renLial
TY
F i.ndu.ces naturally
l'..- of
il -l-Tl'l -6-t (L)F)- : :-tlt -r:: XX.^)'
a
map
Di: : r -r-Ti'l defincri by
-1 Ttr1 ,
,*-'
'(DP)-.x = (r.x,
Then F is called an i.zctneiry O{ the Fittsier rnattif oi.i
r-';x;' (}4, Il ,
c
(Drr) ,: , (Dr)- Y) = c^ (J: , T; >ix>i for any I: , .- e :-it't: , ani an1' x e tc ' rf there is no danoc:: ):of confusir:n wcr wri i:e simply: g ( (Df') x' (Df ) Y) = 9(x . Ì) .Ile i' l:
{
recariì ltre ri:rrpiirsr;:
:
i t r (iti ---r i:-rTM
)
\, -)
ì -
.1
-j,t A.:L.i '-rr
i7 i.re the lned by: t,txl = i*, :,. x) , lor x € ?-x (11) ' i'et As §Z is r:eguiar anci unlque Flnsler connectlon associated to 9' is an izo;{èr ttl = T}t , Ehe restriction e irrn : Tù"*- '-i':lt ' v - )-1 it's inverse' rnorphlsm. lle denote by B = t i'ltmr, :ief
The I'lnsler connection V
= i rg (l , Zt I -
2q(V-Z,T)
(4.r)
it
+
z (g
1s glven by:
(x, Y)') + s(t[i
r g(t[Y, xi,
, i],
i'tg-(7, T)
Y) + g (l[i,
i7
+
,xt
Z)
for anY IÌ-- vector f lel
