Isometries and curvatures of tangent sphere bundles

TL;DR

This paper investigates the geometric properties of tangent sphere bundles, including their characteristic classes, curvature, and homothety conditions, with applications to contact structures and G_2-structures.

Contribution

It provides new results on when tangent sphere bundles are homothetic and determines their curvature properties under various conditions.

Findings

Characterizes when two tangent sphere bundles are homothetic.

Calculates Riemannian, Ricci, scalar, and mean curvatures in specific cases.

Constructs positive scalar curvature metrics on tangent sphere bundles.

Abstract

Natural metric structures on tangent bundles and tangent sphere bundles enclose many important problems, from the topology of the base to the determination of their holonomy. We make here a brief study of the topic. We find the characteristic classes of some of those structures. We solve the question of when two given tangent sphere bundles S_rM of a Riemannian manifold M,g are homothetic, assuming different variable radius functions r and weighted metrics induced only by the conformal class of g. We determine their Riemannian, Ricci, scalar and mean curvatures in some cases. We find a family of positive scalar curvature metrics on S_rM when M has positive scalar curvature or when it has bounded sectional curvature and index of nullity 0. Our objective is the study of contact structures and gwistor spaces, a recently found natural G_2-structure on S_1M.