Non-Natural Metrics on the Tangent Bundle

TL;DR

This paper explores a broader class of metrics on tangent bundles that incorporate interactions between vertical and horizontal components, extending the traditional Sasaki metric, with applications demonstrated on SO(3).

Contribution

It introduces a new class of non-natural metrics on tangent bundles that include interactions between components, expanding beyond the classical Sasaki metric.

Findings

Defined a general class of metrics with vertical-horizontal interactions

Clarified application methods for non-constant vertical components

Provided an example with the SO(3) group

Abstract

Natural metrics provide a way to induce a metric on the tangent bundle from the metric on its base manifold. The most studied type is the Sasaki metric, which applies the base metric separately to the vertical and horizontal components. We study a more general class of metrics which introduces interactions between the vertical and horizontal components, with scalar weights. Additionally, we explicitly clarify how to apply our and other induced metrics on the tangent bundle to vector fields where the vertical component is not constant along the fibers. We give application to the Special Orthogonal Group SO(3) as an example.