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Einstein metrics of Randers type Robles, Colleen

Abstract

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.

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