Homotheties and topology of tangent sphere bundles

TL;DR

This paper explores the geometry and topology of tangent sphere bundles over Riemannian manifolds, providing new examples of manifolds with constant scalar curvature that are not Einstein, and analyzing associated complex and symplectic structures.

Contribution

It introduces homothety results for tangent sphere bundles with variable radii and weighted Sasaki metrics, and computes characteristic classes for these structures.

Findings

New examples of manifolds with constant scalar curvature that are not Einstein

Homothety theorems for tangent sphere bundles with variable radii

Calculations of Chern and Stiefel-Whitney classes for tangent bundles

Abstract

We prove a Theorem on homotheties between two given tangent sphere bundles $S_rM$ of a Riemannian manifold $M,g$ of $\dim\geq 3$, assuming different variable radius functions $r$ and weighted Sasaki metrics induced by the conformal class of $g$. New examples are shown of manifolds with constant positive or with constant negative scalar curvature, which are not Einstein. Recalling results on the associated almost complex structure $I^G$ and symplectic structure ${\omega}^G$ on the manifold $TM$, generalizing the well-known structure of Sasaki by admitting weights and connections with torsion, we compute the Chern and the Stiefel-Whitney characteristic classes of the manifolds $TM$ and $S_rM$.