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 (ยง7) 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.