FINSLER GEOMETRY OVER THE REALS
7
scalar Ric. It is a scalar function on SM. Alternatively, one can express Ric as
f)
Rjikl
f
1.
See §9 for more discussions about this scalar.
In case the flag curvature depends on neither the transverse edge nor the flag-
pole, and
M
has dimension at least 3, then there is no dependence on the location
x either. That is, K is constant in such settings. For further discussions of this
nature, see [Ml]
or
[AIM].
Finsler spaces with constant flag curvature K constitute an important class.
The cases
K
0 and
K =
0 were systematically studied by Akbar-Zadeh [AZl]. In
both cases, the completeness of M was hypothesized and certain growth conditions
were imposed on the Cartan tensor A (these are all automatic if M is compact).
Specifically, A must grow slower than exponentially for the negative case, and slower
than linearly for the flat case. Under such assumptions, those Finsler manifolds
with K 0 must in fact be Riemannian, and those with K
=
0 are actually locally
Minkowskian. Bryant [Br2] has counterexamples which show that none of the
above hypotheses is removable.
Unlike the
K =
0 and
K
0 cases, the structure of Finsler manifolds with
constant positive flag curvature K has not been completely understood yet. Akbar-
Zadeh [AZl] showed that the universal cover of
M
must be diffeomorphic to the
Euclidean n-sphere. He also proved that Landsberg spaces with constant K 0
are necessarily Riemannian. Results of more recent vintage were obtained by Shen
[82] as well as Bryant [Brl]; both articles appear in this proceedings volume. In
[Brl], explicit examples of non-Riemannian Finsler structures with constant K
=
1
were given on the 2-sphere.
This last statement, together with what we said about the K
=
0 case, makes
it quite clear that the definition of a space form in Finsler geometry cannot just
involve the flag curvature. Rather, new and possibly subtle invariants derived from
the Cartan tensor are needed to efficiently categorize the numerous isometry classes
of constant curvature models.
6. Comparison theorems
The past decades have seen great developments in global Riemannian geometry;
for excellent surveys, see Berger [Ber] and Sakai [Sa]. It is a natural and important
project to try to generalize these to the Finsler setting. As in the Riemannian case,
the tools that have demonstrated their utility fall roughly into several camps which
we next describe.
On the one hand, we have the index form which results from the second varia-
tion of arc length. It is somewhat remarkable that the Jacobi equation, the second
variation, and the index form for Finsler spaces look exactly like their counterparts
in Riemannian geometry (see [Ch2],
or
[BCl] for an exposition). As a result,
various arguments involving the comparison of index forms (as in Cheeger-Ebin
[CE]
and Kobayashi
[Ko])
all generalize straightforwardly to the Finsler setting.
Immediate consequences include the Finslerian versions of the Cartan-Hadamard
theorem, the Bonnet-Myers theorem, and Synge's theorem. These were first estab-
lished by Auslander
[Al],
but one of his hypotheses is unnecessary in light of an
observation of Shen's (see [BCl], pp.l77-178).
Not surprisingly, the Morse Index theorem also generalizes. That was due to
Lehmann [L]; see Matsumoto [Ml] for an exposition, and Milnor [Mi] for back-
ground. The method of Riccati inequalities (Karcher
[Ka])
unifies all the compari-
son procedures involved up to this point. This technique then effects the first Rauch
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