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