Holonomy restrictions from the curvature operator of the second kind

TL;DR

This paper establishes that certain curvature conditions on Riemannian manifolds impose strong restrictions on their holonomy groups, leading to flatness or constant curvature in symmetric cases, with improvements under Einstein or Kähler assumptions.

Contribution

It proves new holonomy restrictions based on the curvature operator of the second kind, extending classical results without requiring completeness and refining them for special geometries.

Findings

Manifolds with n-nonnegative or n-nonpositive curvature operator of the second kind have restricted holonomy SO(n) or are flat.

Locally symmetric spaces under these curvature conditions have constant curvature.

Refined bounds are provided for irreducible locally symmetric spaces, depending on dimension.

Abstract

We show that an $n$-dimensional Riemannian manifold with $n$-nonnegative or $n$-nonpositive curvature operator of the second kind has restricted holonomy $SO(n)$ or is flat. The result does not depend on completeness and can be improved provided the space is Einstein or K\"ahler. In particular, if a locally symmetric space has $n$-nonnegative or $n$-nonpositive curvature operator of the second kind, then it has constant curvature. When the locally symmetric space is irreducible this can be improved to $\frac{3n}{2}\frac{n+2}{n+4}$-nonnegative or $\frac{3n}{2}\frac{n+2}{n+4}$-nonpositive curvature operator of the second kind.