A note on generalized Dirac eigenvalues for split holonomy and torsion

TL;DR

This paper investigates the Dirac spectrum on special Riemannian spin manifolds with split holonomy and torsion, establishing an optimal lower bound for the first eigenvalue that extends classical estimates.

Contribution

It introduces a generalized lower bound for the Dirac operator with torsion on manifolds with split holonomy, broadening Friedrich's classical Riemannian estimate.

Findings

Established an optimal lower bound for the first Dirac eigenvalue with torsion.

Extended Friedrich's estimate to manifolds with split holonomy and torsion.

Analyzed the Dirac spectrum in the context of split holonomy and skew torsion.

Abstract

We study the Dirac spectrum on compact Riemannian spin manifolds $M$ equipped with a metric connection $\nabla$ with skew torsion $T\in\Lambda^3 M$ in the situation where the tangent bundle splits under the holonomy of $\nabla$ and the torsion of $\nabla$ is of `split' type. We prove an optimal lower bound for the first eigenvalue of the Dirac operator with torsion that generalizes Friedrich's classical Riemannian estimate.