Survey on eigenvalues of the Dirac operator and geometric structures

TL;DR

This survey reviews how the holonomy of Riemannian spin manifolds influences the eigenvalues of the Dirac operator, including a new proof of Kirchberg's result.

Contribution

It provides a comprehensive overview of existing results and introduces a new proof connecting holonomy and Dirac eigenvalues.

Findings

Holonomy constrains Dirac eigenvalues in Riemannian spin manifolds

New proof of Kirchberg's theorem on Dirac spectrum

Summarizes key relationships between geometry and spectral properties

Abstract

We give a survey of results relating the restricted holonomy of a Riemannian spin manifold with lower bounds on the spectrum of its Dirac operator, giving a new proof of a result originally due to Kirchberg.