Some remarks on Finsler manifolds with constant flag curvature,
Houston Journal of Mathematics, vol. 28 no. 2
(2002),
pp. 221–262 [MR2003h:53102], [math.DG/0107228]
(last updated on 2003/12/08)
Abstract:
This article is an exposition of four loosely related
remarks
on the geometry of Finsler manifolds with constant
positive
flag curvature.
The first remark is that there is a canonical Kahler
structure on the space of geodesics of such a manifold.
The second remark is that there is a natural way
to
construct a (not necessarily complete) Finsler
n-manifold of
constant positive flag curvature out of a hypersurface in
suitably general position in complex projective n-space.
The third remark is that there is a description of
the
Finsler metrics of constant curvature on the 2-sphere in
terms of a Riemannian metric and 1-form on the space
of its
geodesics. In particular, this allows one to use any
(Riemannian) Zoll metric of positive Gauss curvature
on the
2-sphere to construct a global Finsler metric of
constant
positive curvature on the 2-sphere.
The fourth remark concerns the generality of the
space
of (local) Finsler metrics of constant positive flag
curvature in dimension n+1>2 . It is shown that such
metrics
depend on n(n+1) arbitrary functions of n+1 variables
and
that such metrics naturally correspond to certain
torsion-
free S^1 x GL(n,R)-structures on 2n-manifolds. As a by-
product, it is found that these groups do occur as the
holonomy of torsion-free affine connections in
dimension 2n,
a hitherto unsuspected phenomenon.
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