| Summary: | This thesis presents a study of Einstein Randers metrics. Initially introduced
within the context of relativity, Randers metrics have a strong
presence in both the theory and applications of Finsler geometry. The starting
point is a new characterization of Einstein metrics of Randers type by
three conditions. The conditions form a coupled, highly non-linear (due to
the presence of a Riemannian Ricci tensor), second order system of partial
differential equations. The equations are polynomial in the unknowns; a
Riemannian metric ã and differential 1-form b.
Recently Z. Shen has generalized Zermelo's problem of navigation on the
plane to arbitrary Riemannian manifolds. (The goal is to identify the paths
of shortest time on a Riemannian manifold (M, ă) under the influence of an
external force W = Wi∂xi.) In this context, Randers metrics may be viewed
as solutions to Zermelo's problem. The navigation structure yields the main
result of the thesis, a succinct geometric description of Einstein metrics
of Randers type. Explicitly, the Randers metric arising as the solution to
Zermelo's problem on (ă, W) is Einstein if and only if the Riemannian metric
ă is Einstein itself, and W is an infinitesimal homothety of ă.
The navigation description quickly yields a Schur lemma for the Ricci
curvature of Randers metrics. It is a testament to the navigation description
that this result, the first Schur lemma for Ricci curvature in (non-
Riemannian) Finsler geometry, is obtained with relative ease. An extension
of Matsumoto's Identity for Randers metrics of constant flag curvature to
the Einstein setting then follows.
Having established these general results, I then explore three scenarios:
Einstein metrics on surfaces of revolution, constant flag curvature metrics,
and Einstein metrics on closed manifolds. The thesis closes with a collection
of open questions. === Science, Faculty of === Mathematics, Department of === Graduate
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