9 Springer-Verlag 1989. Metrics with only f'mitely many isometry invariant geodesics. Hans-Bert Rademacher. Forschungsinstitut fiir Mathematik der ETH Zfirich, ...
Ilalhemt he AnWm
Math. Ann. 284, 391-407 (1989)
9 Springer-Verlag 1989
Metrics with only f'mitely many isometry invariant geodesics Hans-Bert R a d e m a c h e r Forschungsinstitut fiir Mathematik der ETH Zfirich, ETH-Zentrum, CH-8092 Ziirich, Switzerland
Dedicated to Professor Wilhelm Klingenberg with best wishes on his 65th birthday Introduction In this paper we generalize results for closed geodesics by the author [10] to the case of isometry invariant geodesics. A non-constant geodesic c : I R ~ M on a c o m p a c t manifold M with a Riem a n n i a n metric g with isometry f of finite order s is f-invariant if c(t + 1) = f(e(t)) for all t 9 IR. The theory of f-invariant geodesics was first developed by G r o v e [2, 3]. These geodesics can be described as critical points of the energy functional E: on the Hilbert manifold A ( M , f ) of f-invariant curves. A ( M , f ) carries a canonical SZ-action. The index ind (c, f ) of an f-invariant geodesic c is defined as the index of the critical orbit S ~ 9c. Two f-invariant geodesics c 1 , c2 : I R ~ M are geometrically equivalent if c l ( N ) = e z ( I R ). F o r an equivalence class of f-invariant geodesics (geometric f-invariant geodesic) there are numbers mo, so, 1 < m 0 < s o, s o divides s, such that S ~ .c a+ms~176 m 9 mso+mor is the set of geometric equivalent f-invariant geodesics for an f-invariant geodesic with c " : R ~ M , c"(t)=c(ut) for u 9 We say c is regular if there exists m ' > 0 such that for all m>m' ind (c 1 +,,so/mo, f ) > 1. Then the generalized average index ec > 0 exists with c~- 1 = lira A* (N) N~o N and A* (N) = ~ {m > 0lind (c I +'s~176 f ) < N}. In Sect. 1 we show using results of G r o v e and T a n a k a [5] that there is a n u m b e r tic>0 such that for all N > 0
A*(N) - N
O, mso(k ) +mo(k ) %0} and if the quotient spaces Bk/S 1 are simply-connected. Then any f-invariant geodesic c is regular and its generalized average index ~c is welldefined. For each k = 1 ..... r we have the following metric invariants: ~k=~c, ~)k=Tc for c E B k with y,s {0, -I-1/2, + 1}, controlling the parity of the sequence (ind (c1§ m~o/mo(k),f ) ) modulo 2 and Xk being the Euler characteristic of Bk/S 1. Then we obtain in 2.1 the following Theorem 1. Let M be a simply-connected compact Riemannian manifold with isometry f of finite order s and with finitely many connected non-degenerate manifolds of geometric f-invariant geodesics. Let ~tk, 7k, "Zk,k = 1 ..... r be the above defined metric invariants. Then the topological invaraint
B(M,f):=
lim --1 ~ ( _ l ) i b , ( A ( M , f ) / S 1 ; f f ~ ) r~o~
m
i=o
(where b i is the i-th Betti number) can be expressed as B(M,f)= ~ k=l
7k)~k O~k
Since all critical submanifolds are non-degenerate it is an immediate consequence of (1) that the sequence (bi(A(M, f ) ; Q))i>__ois bounded. In [4] topological conditions for M resp. f are given for such manifolds. In [5] Grove and Tanaka prove, that on a compact simply-connected Riemannian manifold M with isometry f of finite order and with unbounded sequence b~(A(M, f ) ; F ) of Betti numbers for a field F there are infinitely many geometric f-invariant geodesics. In [12] Tanaka shows that this result remains true for any isometry. For H* (M; Q) isomorphic to the truncated polynomial algebra Td,n§ 1(x) with generator x of degree d and height n + 1 with an isometry fixing x we compute in 3.2
B(M, f ) =A (d, n) - z(Fix F ) where A (d, n) = (n + 1 ) ( d - 2)/(2 d(n + I) - 4) for even d and A (d, 1) = ( d + 1)/(2 d - 2) for odd dand where x(Fix f ) is the Euler characteristic of the fixed point set Fix f of f on M. In Sect. 4 we study f-invariant geodesics on the d-dimensional sphere S a with the standard metric and isometrics of finite order as examples of metrics with finitely many non-degenerate connected manifolds of geometric f-invariant geodesics. In 4.1.1 we give examples of rotations f o f S a with only [(d+ 1)/2] geometric f-invariant geodesics. As an application of theorem 1 we study the stability properties of f-invariant geodesics of metrics with only finitely many geometric f-invariant geodesics. A closed geodesic c is hyperbolic if none of the eigenvalues of the linearized Poincar6 map Pc has norm 1. We say an f-invariant geodesic c : l R ~ M is hyperbolic if the corresponding closed geodesic c ~: I R / Z + M is hyperbolic. In Sect. 5 we show Theorem 2. Let M be a simply-connected compact Riemannian manifoM with isometry f of finite order s. Let f * fix the generator x of H* (M; ~ ) ~ Td,n+ 1(x) and
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let H* (Fix f; Q) ~- Td,l+1(x), 1 =o with the following properties: a) AZ(oa) = AZ(aS) b) I f there is an o ~ S 1 with A~(oa)>0 then z~J~~ c) There are at most 2(dim (M) - l ) points on S ~ where A ~ is not locally constant and the splittin9 numbers S~ (oa):= lim A ~ ( e % O - A ~ ( o a ) t~ +0
are non-negative. d) ind (c' + m~o/,.o,f ) =
X
E
X
A z(oa) .
~a/So= 1 O)mSo+mo =~0 Zm~o ~-ko=~ - I
Let Vc be the complexified vector space of vector fields along c urn~orthogonal to 3. Define for oaES t S,(m, oaf) : = {Xe V~]X solves (2) with 2 < 0, X ( t + m) = oaf ( X ( t ) ) V t e ~1,} . Then ind (c x +"~~176 f ) = dim Sc (ms o + m o, f ) and 1.1 d) follows from S~(mso+mo,f)=
9 QB/SO=1 r
9
9
S~(1,oaf "~
Q ~m~to+ko=~-I
c~ker (f~o - z id) with A~(oa) : = d i m So(1 , cof'o)c~ker ( f . o - z id) . In particular if s = s o then A~(oa)v~O only if z = 1 i.e. ind (c 1 +"~~176 f ) =
~ ores +rag= l
A 1 (oa) .
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F r o m the index t h e o r e m of Bott [1 ] it follows that for a closed geodesic c with iterates c m the average index a~= lim (ind(cm)/m) exists and that for ~c=0 i n d ( c ' ) = 0 V m > 0 and ,n~o lind (c") -merci < dim M - 1 for all m > 0 [10], 1.4. In general the corresponding limit lim (ind (c 1 +,,so/,,o, f ) / m ) m~oo
for an f-invariant geodesic will not exist. We get the following 1.2. Proposition. Let 0 < l < s. Then f o r all m =- l (rood s) we have either ind ( c m'/m~ f ) = 0 where m ' : = ms o + m o or there exists a number S l > 0 such that lind (e m'/m~ f ) -r c(l)m'l < St with
1 Z
Y
I
~*S/XO=1 grtl! +rlo~ q - 1 0
Proof. F o r m = l (mod s)
lind (c m'/m~ f ) - m%(c)]
z'%+ks
= qs/~=l
z #S/SO= Y1 where S Z = ~
(eJ~'=oaz((D)-m'iAZ(e2nit)dt)
E glno+ko=
:st 0 - 1
max(S~. (co), S ~_ (co)) with the splitting numbers S~.(o9) defined
r162 1
in 1.2c).
[]
If at(l) > 0 we get from 1.2 for A* (l, N) : = # {m > 0lind (c "'/m~ f ) < N, m = l (mod s)) the following estimate: There exists a number/3,(l) > 0 such that for all N > 0 : .4* (l, N) - N ~
~AOSSo
= 0. F o r a regular f-invariant geodesic c we define the n u m b e r A* (N) : = @ {m > Olind (c m'/m~ f ) < N) .
Then there exists a tic' > 0 with A*(N) -N
0 sufficiently small with A~(M, f ) : = {c E A(M, f)lE:(c) < x} and ~ ( M , f ) : = A~(M, f ) / S x : H , (A~ (M, f ) , A~-~(M, f ) ; Q) ~ H , (D (c~'/m~
S(cra'/m~
; 11~)
Let k = ind (c~'/~~ f ) then we have for the Betti numbers
b,(D(c"/~~ =J'l
; ;
, i=k
S(c"/'o)/z~; r
and otherwise
b(c"/'~
(mod2)
In order to estimate the number
y'
At(N) : =
( - 1)ibi(D(e"'/"~
S(c"'/"o)/71~; if).)
(cm'/mo, f) < N mEZ, m'#O
ind
the invariant ycE{0,-I-l/2,___l}: Let bi,k~{O, 1}, i = 0 , 1 with b i - b (c"':'~ k i - ind (c"'/m~ f ) for m = i (mod 2). Then 7~= 0 iffeither b o = b I = 1 or we define
b o = b I = 0 and k o + k I = 1. lycl= 1 iff b o = b 1 = 0 and k o = k 1 . If yr :/: 0 then 7c > 0 if there is an i~ {0, 1} with k i = 0 and h i = 0 . Then we obtain the 1.4. Proposition. For a regular f-invariant 9eodesie c there is an tic > 0 such that for
all N> 0: A ~ ( N ) - Z N Y~ 2 are included, see 4.1.2. A connected submanifold B (without boundary) o f f-invariant geodesics in A ( M , f ) is a non-degenerate critical submanifold of constant multiplicity, if the energy, the index, the nullity and the multiplicity are constant on B (hence we can write E:(B), ind (B, f ) , null (B, f ) and mul (B)) and if null (B, f ) = d i m ( B ) - 1 . We say a Riemannian metric 9 on a simply-connected c o m p a c t manifold M with isometry f o f finite order s has finitely many connected non-
398
H.-B. Rademacher
degenerate manifolds of geometric f-invariant 9eodesics if the set o f f-invariant geodesics as a subset o f A ( M , f ) is the union o f disjoint non-degenerate critical submanifolds B where all quotient s p a c e s / 1 = B/S 1 are simply-connected and if there are only finitely m a n y critical submanifolds B with mul ( B ) < s. In the case f = id~t this definition is the definition of an "admissible" Riemannian metric given in [10, Chap. 2]. Hence there is a n u m b e r r ~ Z + such that for any k = 1 .... , r there are numbers mo(k), so(k ) with l < m o ( k ) < s o ( k ) , so(k ) divides s such that there are nondegenerate critical submanifolds Bk, k = l ..... r with mul(Bk)=smo(k)/so(k)o, k = 1 ..... r such that c = Co ~+mso~k)/,,o~k)for some co ~ Ok. F o r each k = 1 ..... r and 0 < l < s the invariants ~k (l) = Ctc (l), ~k = 7c for any c e B k are well-defined (cf. 1.2, 1.3). If for some l:~k(l ) = 0, then there are infinitely m a n y critical submanifolds of index 0 and hence it follows from the Morse-inequalities, that there are infinitely m a n y critical submanifolds o f index 1, since A (M, f ) is connected. This is not possible in our case, since 4t: {m e Z[ind (c1+mso(k)/mo(k),f ) = 1} is finite (cf. 1.2). Hence all f-invariant geodesics are regular and for any k = 1,..., r the generalized average index ~k = ~c > O, C~ B k is defined as well as the invariant 7k = Yc, C6Bk, 7c~{0, + 1 / 2 , ___1}" Hence a Riemannian metric g with isometry f of finite order s determines the following set o f metric invariants: ~k, Yk, Zk, k = 1 ..... r with Zk being the Euler characteristic Z (Bk/$1) o f the quotient space Bk/S I. Given an Sl-space X we denote b y ) ? t h e quotient space X / S 1 and by b i(X, A; Q ) for a space pair (x, A) the i-th Betti number, i.e. b~(X, A ; ~ ) = dim H~(X, A ;Q). We show the following generalization o f T h e o r e m 3.1a) in [10]: 2.1. Theorem. Let M be a simply-connected Riemannian manifoM with isometry of finite order s and with only finitely many connected non-degenerate manifolds of geometric f-invariant geodesics. Let ~k, Yk, Xk, k = 1 ..... r be the metr& invariants defined above. Then the topological invariant
B ( M , f ) : = lira 1 ra--*~ m
~ (-1)'b,(A(M,f);Q) i=o
is finite and can be expressed in terms of the metric invariants ~k, 7k, Xk by ~kZk
B(M, f ) = k=l
~k
Proof. Let (cz)l_o be the sequence o f critical values o f the energy functional E y with q < ct+ 1 and (at)t ~ o be a sequence o f regular values with a o = 0, a I < c t < a t + 1 and define for a E R A ~ ( M , f ) : = { t r e A ( M , f ) l E Y ( t r ) < a } . Let w , = ~. b , ( A ~ ' §
Q)
/=0
and
bi :=bi(A(M, f ) , A~
f ) ; Q)
then we can write d o w n the Morse inequalities in the following form (cf. [10, 2.3]):
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There exists a sequence (ql)~_o of non-negative integers ql such that
wi=b~+qi+qi_l resp. (-
=
(- 1)%1=0
( - 1)'b, .
(3)
i=0
From 1.2 it follows that (wi)i>=ois bounded hence also (bi)i~_o and (qi)i~_o are bounded sequences. If the metric g is ]:bumpy, then r=2r', r'eZ + and the manifolds B k of f-invariant geodesics with mul(Bk)__ois bounded we get from (3) that there is a number fl' > 0 such that for all N > 0
i~=o( - l ) ' b i - N k~=t 7~'Z~lO m--~are constant. We have for O
