Submanifolds and the Sasaki Metric

TL;DR

This paper discusses the Sasaki metric on tangent bundles of Riemannian manifolds, showing it coincides with the induced metric on submanifolds only when they are totally geodesic, using Dieudonné's connection.

Contribution

It provides a natural expression of the Sasaki metric via Dieudonné's connection and characterizes submanifolds with induced Sasaki metrics as totally geodesic.

Findings

The Sasaki metric can be expressed naturally using Dieudonné's connection.

The induced metric on a submanifold's tangent bundle is Sasaki if and only if the submanifold is totally geodesic.

The paper clarifies the relationship between submanifold geometry and tangent bundle metrics.

Abstract

This is the content of a talk given by the author at the 2009 Lehigh University Geometry/Topology Conference. Using the definition of connection given by Dieudonn\'e, the Sasaki metric on the tangent bundle to a Riemannian manifold is expressed in a natural way. Also, the following property is established. The induced metric on the tangent bundle of an isometrically embedded submanifold is the Sasaki metric if and only if the submanifold is totally geodesic.