  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.
