J. Aust. Math. Soc.
82 (2007), 249262

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



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 nonzero
constant curvature may be Riemannian spaces of
nonzero 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 y^{i}, and determine the solutions. We recover two
standard Riemannian metrics of nonzero constant
curvature from this class of solutions.

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