Wagner Lift of Riemannian metric to Orthogonal Frame Bundle

TL;DR

This paper introduces the Wagner lift, a method to extend a Riemannian metric from a 2D manifold to its orthonormal frame bundle, linking the geometries of the base and total space.

Contribution

It applies Wagner's technique to lift metrics via the Levi-Civita connection, establishing relations between the geometries of the manifold and its frame bundle.

Findings

Wagner lift constructed for 2D Riemannian manifolds

Relations established between base and total space geometries

Extension of metric via Levi-Civita connection

Abstract

In the present work we construct a lift of a metric $g$ on a 2-dimensional oriented Riemannian manifold $M$ to a metric $\hat{g}$ on the total space $P$ of the orthonormal frame bundle of $M$. We call this lift the \textit {Wagner lift}. Viktor Vladimirovich Wagner (1908 -1981) proposed a technique to extend a metric defined on a non-holonomic distribution to its prolongation via the Lie brackets. We apply the Wagner construction to the specific case when the distribution is the infinitesimal connection in the orthonormal frame bundle which corresponds to a Levi-Civita connection. We find relations between the geometry of the Riemannian manifold $(M,g)$ and of the total space $(P,G)$ of the orthonormal frame bundle endowed with the lifted metric.