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

TL;DR

This paper introduces the Wagner lift, a metric on the orthonormal frame bundle of a 2D Riemannian manifold, and analyzes its geodesics and isometry groups, revealing new geometric relationships and properties.

Contribution

It constructs the Wagner lift metric on the frame bundle and studies its geodesics and symmetries, linking 2D Riemannian geometry with 3D generalized metrics.

Findings

Proves the relation between geodesics of the lifted metric and a specific differential equation.

Analyzes the isometry groups of the original and lifted manifolds.

Examines properties of geodesics on surfaces of revolution.

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.