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Publications [#243380] of Robert Bryant
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 Bryant, RL, Some remarks on Finsler manifolds with constant flag curvature,
Houston Journal of Mathematics, vol. 28 no. 2
(January, 2002),
pp. 221262, UNIV HOUSTON [MR2003h:53102], [math.DG/0107228]
(last updated on 2019/08/21)
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
nmanifold of
constant positive flag curvature out of a hypersurface in
suitably general position in complex projective nspace.
The third remark is that there is a description of
the
Finsler metrics of constant curvature on the 2sphere in
terms of a Riemannian metric and 1form on the space
of its
geodesics. In particular, this allows one to use any
(Riemannian) Zoll metric of positive Gauss curvature
on the
2sphere to construct a global Finsler metric of
constant
positive curvature on the 2sphere.
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 2nmanifolds. As a by
product, it is found that these groups do occur as the
holonomy of torsionfree affine connections in
dimension 2n,
a hitherto unsuspected phenomenon.


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