Isometry group and geodesics of the Wagner lift of a Riemannian metric on two-dimensional manifold

Abstract

In this paper we construct a functor from the category of two-dimensional Riemannian manifolds to the category of three-dimensional manifolds with generalized metric tensors: for each two-dimensional oriented Riemannian manifold, on the total space of the principal bundle of the positively oriented orthonormal frames of the manifold we construct a metric tensor (in general, with singularities), called the Wagner lift. We study the relation between the isometry groups of the metric and its Wagner lift, in particular find the structure of the Lie algebra of projectable infinitesimal isometries of the Wagner lift. We prove that the projections of the geodesics of the Wagner lift onto the base are solutions of a second order ordinary differential equation which is expressed in terms of the curvature of the metric on the base, and study properties of the solutions.