FINSLER GEOMETRY OVER THE REALS 9
the successful (albeit expected) generalization of the Hopf-Rinow theorem. The
ability to visualize the cut locus and the conjugate locus of strategic points is tan-
tamount to an understanding of the global behaviour of geodesics. Indeed, most
comparisons theorems are formulated and proved with insights of this nature. Per-
haps super-computers can play a role in helping Finsler geometers cultivate such
insights.
7. Finster manifolds as metric spaces
Many geometrical aspects of Finsler manifolds can be explored strictly from
a metric space point of view. Several works of Busemann's ([Bul], [Bu2], [Bu3],
[Bu4]) testify to the viability of this approach. For example, in [Bu2] he introduced
the notion of a volume form on Finsler manifolds and proved that the volume of a
subset equals its Hausdorff measure with respect to the induced distance.
There are many concepts of importance here. For example the filling radius
(see Gromov [Gl]) and the notion of topological entropy (see Egloff [E2] for a
discussion in the context of Finsler manifolds). But a most remarkable feature of
this school of thought is the resourceful use of geometrical objects to make sense of
curvature bounds.
In Alexandrov's works ([Al], [A2]), this is done by comparing small geodesic
triangles with their analogues in an appropriately specified Riemannian space form.
However, spaces whose curvature is bounded above in this sense are automatically
Riemannian. By modifying Alexandrov's construction (but still using geodesic
triangles), Busemann [Bul] was able to define the notion of non-positive curvature
for a large class of metric spaces which include Minkowskian spaces. For arbitrary
metric spaces, Gromov
[G2]
has recently proposed the use of metric balls in the
definition of non-positive and non-negative curvatures.
8. Geometric analysis on Finsler manifolds
On Riemannian manifolds, many deep relationships between curvature and
topology were established through the use of hard analysis (see the book by Schoen
and Yau
[SY]
and references therein) involving the Laplacian. We believe that
Finsler manifolds
(M, F)
shall pose no exceptions.
As a first step in this direction, Bao and Lackey [BL2] have recently con-
structed Laplace-Beltrami operators (and deduced a Hodge theorem) directly on
the smooth differential forms of
M.
These linear elliptic operators are manifestly
non-Riemannian. Nevertheless, the resulting Laplacian on functions is again ex-
pressible in a divergence form, thereby leading to a divergence theorem for vector
fields. Many analytic issues can now be addressed. These include potential theory
for complete Finsler manifolds, eigenvalue estimates on compact Finsler manifolds,
and Bochner type vanishing theorems.
Other approaches to the Laplacian (on functions) are also being pursued. For
example, using the dual Finsler structure
F*
on
T* M
and the Busemann vol-
ume form (§7) on
M,
Shen has investigated the nonlinear eigenvalue problem un-
derlying the Rayleigh quotient Q(f)
:=
fuj:*j'!fll
2
on functions. The resulting
Euler-Lagrange equation is non-linear and elliptic. A divergence theorem is again
obtained.
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