Spectral geometry of $eta$-Einstein Sasakian manifolds

TL;DR

This paper demonstrates that certain geometric properties of $ ext{eta}$-Einstein Sasakian manifolds and Sasakian space forms with constant $ ext{phi}$-sectional curvature are uniquely determined by their spectral data, extending previous results to contact manifolds.

Contribution

It proves that being an $ ext{eta}$-Einstein Sasakian manifold or having constant $ ext{phi}$-sectional curvature in Sasakian space forms is spectrally determined, extending spectral geometry results.

Findings

Spectral data uniquely determines $ ext{eta}$-Einstein Sasakian structure.

Spectral data determines constant $ ext{phi}$-sectional curvature in Sasakian space forms.

Extension of Patodi's result to contact manifolds.

Abstract

We extend a result of Patodi for closed Riemannian manifolds to the context of closed contact manifolds by showing the condition that a manifold is an $\eta$-Einstein Sasakian manifold is spectrally determined. We also prove that the condition that a Sasakian space form has constant $\phi$-sectional curvature $c$ is spectrally determined.