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 3n2n+2n+4-nonnegative or 3n2n+2n+4-nonpositive curvature operator of the second kind.