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 $(M,g)$ we construct a metric tensor $\hat g$ (in general, with singularities) on the total space $SO(M,g)$ of the principal bundle of the positively oriented orthonormal frames on $M$. We call the metric $\hat g$ the Wagner lift of $g$. We study the relation between the isometry groups of $(M,g)$ and $(SO(M,g),\hat g)$. We prove that the projections of the geodesics of $(SO(M,g),\hat g)$ onto $M$ are the curves which satisfy the equation \begin{equation*} \nabla_{\frac{d\gamma}{dt}}\frac{d\gamma}{dt} = C K J (\dot\gamma) - C^2 K grad K, \end{equation*} where $K$ is the curvature of $(M,g)$, $J$ is the operator of the complex structure associated with $g$, and $C$ is a constant. We find the properties of the solutions of this equation, in particular, for the case when $(M,g)$ is a surface of revolution.