Holonomy restrictions from the curvature operator of the second kind

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 Kähler. 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.

10 pages

References in corpus (1)

• Manifolds with nonnegative curvature operator of the second kind

Cited by in corpus (3)

• Betti numbers and the curvature operator of the second kind
• Product manifolds and the curvature operator of the second kind
• Einstein manifolds of negative lower bounds on curvature operator of the second Kind