Curvature Properties of 3-$(\alpha,\delta)$-Sasaki Manifolds

TL;DR

This paper explores the curvature characteristics of 3-$(oldsymbol{ extalpha,oldsymbol{ extdelta}})$-Sasaki manifolds, focusing on their Riemannian and canonical connection curvatures, spectra, and conditions for strongly definite curvature.

Contribution

It provides a detailed analysis of curvature operators, spectra, and eigenforms of 3-$(oldsymbol{ extalpha,oldsymbol{ extdelta}})$-Sasaki manifolds, extending understanding of their geometric properties.

Findings

Spectrum of curvature operators characterized

Eigenforms identified for specific curvature conditions

Conditions for strongly definite curvature established

Abstract

We investigate curvature properties of 3-$(\alpha,\delta)$-Sasaki manifolds, a special class of almost 3-contact metric manifolds generalizing 3-Sasaki manifolds (corresponding to $\alpha = \delta = 1$) that admit a canonical metric connection with skew torsion and define a Riemannian submersion over a quaternionic K\"ahler manifold with vanishing, positive or negative scalar curvature, according to $\delta = 0$, $\alpha\delta > 0$ or $\alpha\delta < 0$. We shall investigate both the Riemannian curvature and the curvature of the canonical connection, with particular focus on their curvature operators, regarded as symmetric endomorphisms of the space of 2-forms. We describe their spectrum, find distinguished eigenforms, and study the conditions of strongly definite curvature in the sense of Thorpe.