J. Aust. Math. Soc.  82 (2007), 249-262

A system of PDEs for Riemannian spaces

Padma Senarath   Mathematics   Institute of Fundamental Sciences   Massey University   Private Bag 11222   Palmerston North   New Zealand   P.Senerath@massey.ac.nz Gillian Thornley   Mathematics   Institute of Fundamental Sciences   Massey University   Private Bag 11222   Palmerston North   New Zealand   G.Thornley@massey.ac.nz and Bruce van Brunt   Mathematics   Institute of Fundamental Sciences   Massey University   Private Bag 11222   Palmerston North   New Zealand   B.vanBrunt@massey.ac.nz

Abstract

Matsumoto [10] remarked that some locally projectively flat Finsler spaces of non-zero constant curvature may be Riemannian spaces of non-zero constant curvature. The Riemannian connection, however, must be metric compatible, and this requirement places restrictions on the geodesic coefficients for the Finsler space in the form of a system of partial differential equations. In this paper, we derive this system of equations for the case where the geodesic coefficients are quadratic in the tangent space variables yi, and determine the solutions. We recover two standard Riemannian metrics of non-zero constant curvature from this class of solutions.

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