The Space of Dirac-Minimal Metrics is Connected in Dimensions 2 and 4

TL;DR

This paper proves that on closed connected spin manifolds of dimension 2 or 4, the set of Riemannian metrics minimizing the Dirac operator's kernel dimension forms a connected space, revealing topological structure of these metrics.

Contribution

It establishes the connectedness of the space of Dirac-minimal metrics specifically in dimensions 2 and 4, a new topological insight into spin geometry.

Findings

The space of Dirac-minimal metrics is connected in dimension 2.

The space of Dirac-minimal metrics is connected in dimension 4.

Provides new understanding of metric spaces related to Dirac operators.

Abstract

Let $M$ be a closed connected spin manifold. Index theory provides a topological lower bound on the dimension of the kernel of the Dirac operator which depends on the choice of Riemannian metric. Riemannian metrics for which this bound is attained are called Dirac-minimal. We show that the space of Dirac-minimal metrics on $M$ is connected if $M$ is of dimension 2 or 4.