Weighted metrics on tangent sphere bundles

TL;DR

This paper investigates natural metric structures on tangent and tangent sphere bundles of Riemannian manifolds, focusing on contact structures and $G_2$-twistor spaces, with implications for geometric analysis.

Contribution

It derives equations for metric connections with torsion on these bundles and explores their reducibility to almost Hermitian structures, advancing understanding of geometric structures on tangent sphere bundles.

Findings

Derived equations for metric connections with torsion.

Analyzed conditions for reducibility to almost Hermitian structures.

Studied natural contact structures and $G_2$-twistor spaces.

Abstract

Natural metric structures on the tangent bundle and tangent sphere bundles $S_rM$ of a Riemannian manifold $M$ with radius function $r$ enclose many important unsolved problems. Admitting metric connections on $M$ with torsion, we deduce the equations of induced metric connections on those bundles. Then the equations of reducibility of $TM$ to the almost Hermitian category. Our purpose is the study of the natural contact structure on $S_rM$ and the $G_2$-twistor space of any oriented Riemannian 4-manifold.