Sasakian structures on tangent sphere bundles of compact rank-one symmetric spaces

TL;DR

This paper proves the existence of Sasakian structures on tangent sphere bundles of certain symmetric spaces, even when their sectional curvature varies, expanding understanding of geometric structures on these manifolds.

Contribution

It demonstrates that tangent sphere bundles of compact rank-one symmetric spaces admit unique Sasakian structures induced from almost Hermitian structures, regardless of curvature constancy.

Findings

Existence of Sasakian structures on tangent sphere bundles of symmetric spaces.

Uniqueness of the K-contact and Sasakian structures in this context.

Construction of these structures from almost Hermitian structures on punctured tangent bundles.

Abstract

A positive answer is given to the existence of Sasakian structures on the tangent sphere bundle of some Riemannian manifold whose sectional curvature is not constant. Among other results, it is proved that the tangent sphere bundle Tr(G/K), for any r > 0, of a compact rank-one symmetric space G/K, not necessarily of constant sectional curvature, admits a unique K-contact structure whose characteristic vector field is the standard field of T(G/K). Such a structure is in fact Sasakian and it can be expressed as an induced structure from an almost Hermitian structure on the punctured tangent bundle T(G/K)\{zero section}.