Projectively Flat Finsler Metrics of Constant Curvature
Abstract
It is the Hilbert's Fourth Problem to characterize the (notnecessarilyreversible) distance functions on a bounded convex domain in R^n such that straight lines are shortest paths. Distance functions induced by a Finsler metric are regarded as smooth ones. Finsler metrics with straight geodesics said to be projective. It is known that the flag curvature of any projective Finsler metric is a scalar function of tangent vectors (the flag curvature must be a constant if it is Riemannian). Thus it is a natural problem to study those of constant curvature. In this paper, we study the Hilbert Fourth Problem in the smooth case. We first give a formula for xanalytic projective Finsler metrics F(x,y) of constant curvature K=c using a power series with coefficients expressed in terms of f(y):=F(0, y), h(y):=(1/2)F(x,y)^{1}F_{x^k}(0, y)y^k and c. Then, for any given pair {f(y), h(y)} and any constant c, we give an algebraic formula for smooth projective Finsler metrics with constant curvature K=c and F(0, y)=f(y) and F_{x^k}(0, y)y^k=2f(y)h(y). By these formulas, we obtain several special projective Finsler metrics of constant curvature which can be used as models in Finsler geometry.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 September 2001
 arXiv:
 arXiv:math/0109178
 Bibcode:
 2001math......9178S
 Keywords:

 Differential Geometry;
 Metric Geometry;
 53C60;
 53B40
 EPrint:
 26 pages, revised in September, 2001