8
D. BAO, S.S. CHERN AND Z. SHEN
comparison theorem
([BCl], [BCSl])
and the Bishop-Gromov volume comparison
(Shen
[Sl])
for Finsler manifolds. However, there is still room for improvement in
estimating Jacobi fields by curvature bounds.
We digress to echo the fact (noted in
[Sl])
that infinitesimal invariants defined
by the flag curvature are not able to completely control some global invariants
such as volume (see §7). This is so because the flag curvature does not control
the geometry within each Minkowskian tangent space
(TxM, Fx)·
The problem can
perhaps be remedied by carefully exploiting
P/kt'
which might also shed some light
on how to improve the Jacobi field estimates.
Estimates on Jacobi fields can be integrated to give distance estimates; but in
Finsler geometry the latter are inevitably
([El])
of a coarse nature. A treatment
of such technique on Riemannian manifolds can be found in Buser and Karcher
[BK];
see also Eberlein
[Eb].
Egloff
[El]
has used this tool rather effectively in the
study of complete simply-connected uniform Finsler manifolds with negative flag
curvature, proving that these are uniformly visible and hence 8-hyperbolic in the
sense of Gromov.
Jacobi fields can be interpreted as invariant vector fields of geodesic flows,
thereby accentuating the latter's importance. For compact Finsler manifolds with
negative flag curvature, it is known (Foulon
[F2])
that their geodesic flows are of
Anosov type. This fact allows one to bring in machinery intended for hyperbolic
dynamical systems. See Egloff
[E2]
for discussions and applications.
We now describe the ingredients that center around the study of complete
simply-connected n-dimensional Finsler manifolds which are strictly i-pinched,
that is,
i
K :::; 1. The goal is to prove that such spaces are homeomorphic
to
sn.
In Riemannian geometry, the Sphere Theorem and its variants [for example,
strengthening the homeomorphism to a diffeomorphism, or weakening it to say
(n-
I)-connectedness] have a long history and a distinguished list of geometers
who contributed to it. We refer the reader to the excellent accounts in
[CE], [DC],
[Ka], [Ko],
and references therein.
For Finsler manifolds, Dazord
([Dl], [D2])
sketched a proof for the Homotopy
Sphere Theorem in dimensions
n
~
3, and Kern
([Kl], [K2])
contributed to the
differentiable version. A systematic exposition of Dazord's approach can be found
in
[BCSl].
Central to all works on the Sphere Theorem is Klingenberg's estimate of the
injectivity radius, together with a decomposition of the manifold into two balls
joined along their common boundary. For Riemannian manifolds, both tasks can be
accomplished by involving Toponogov's comparison theorem at some stage. Since
the latter holds only on Riemannian spaces, a different strategy is needed in the
Finsler setting. It is well known that the cut locus estimate can be achieved with
Morse theory (see
[DC]);
this was how Dazord obtained his Homotopy Sphere
Theorem for the Finsler case. As for the decomposition, even though it can be
proved using Rauch's theorem (see Tsukamoto
[T])
instead of Toponogov's, there
are still doubts (raised by angle related issues) regarding its successful generalization
to Finsler spaces.
Finally, as most geometers would most certainly concur, two items have dis-
tinguished themselves among the extensive list of tools and concepts we just enu-
merated. They are the conjugate loci and the cut loci; see Kobayashi
[Ko]
for
a beautiful treatment. Their relevance in Finsler geometry has been secured by
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