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

Bernd Ammann^a, Mattias Dahl<sup>b

a</sup> Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany
^b Institutionen för Matematik, Kungliga Tekniska Högskolan, 100 44 Stockholm, Sweden

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 $4$.

Keywords: Dirac operator, Atiyah–Singer index theorem, generic Riemannian metrics, minimal kernel.

MSC: 53C27, 19K56, 58C40, 58J5

Received: August 11, 2025; in final form November 25, 2025; Published online December 6, 2025

Language: English

DOI: 10.3842/SIGMA.2025.102

ArXiv: 2508.01420