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A system of PDEs for Riemannian spaces

Part of: Partial differential equations on manifolds; differential operators Local differential geometry

Published online by Cambridge University Press:  09 April 2009

Padma Senarath ,
Gillian Thornley  and
Bruce van Brunt
Bruce van Brunt
Affiliation:
Mathematics Institute of Fundamental Sciences Massey Univeristy Private Bag11222 Palmerston NorthNew Zealand e-mail: P.Senerath@massey.ac.nzG.Thornly@massey.ac.nzB.vanBrunt@massey.ac.nz
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Abstract

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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 Remannian metrics of non-zero constant curvature from this class of solutions.

MSC classification

Secondary: 53B40: Finsler spaces and generalizations (areal metrics) 58J60: Relations with special manifold structures (Riemannian, Finsler, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 82 , Issue 2 , April 2007 , pp. 249 - 262
Copyright
Copyright © Australian Mathematical Society 2007

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