Projectively Flat Finsler Metrics of Constant Curvature
| dc.creator | Shen, Zhongmin | |
| dc.date | 2001-09-23 | |
| dc.date.accessioned | 2026-07-07T04:43:30Z | |
| dc.date.available | 2026-07-07T04:43:30Z | |
| dc.description | It is the Hilbert's Fourth Problem to characterize the (not-necessarily-reversible) 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 x-analytic 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. | |
| dc.description | 26 pages, revised in September, 2001 | |
| dc.identifier | https://arxiv.org/abs/math/0109178 | |
| dc.identifier | http://arxiv.org/abs/math/0109178 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/62252 | |
| dc.subject | Differential Geometry | |
| dc.subject | Metric Geometry | |
| dc.subject | 53C60; 53B40 | |
| dc.title | Projectively Flat Finsler Metrics of Constant Curvature | |
| dc.type | text |