Product manifolds and the curvature operator of the second kind

TL;DR

This paper establishes optimal rigidity results for product Riemannian manifolds with specific curvature operator bounds, characterizing their universal covers as standard product spaces like spheres, hyperbolic spaces, or complex projective spaces.

Contribution

It provides new pointwise algebraic rigidity theorems for product manifolds with curvature operator bounds, extending classical results to a broader class of spaces.

Findings

Universal cover of certain non-flat product manifolds is isometric to standard product spaces.

Rigidity results for product manifolds with curvature operator bounds.

Characterization of product Kähler manifolds with curvature conditions.

Abstract

We investigate the curvature operator of the second kind on product Riemannian manifolds and obtain some optimal rigidity results. For instance, we prove that the universal cover of an $n$-dimensional non-flat complete locally reducible Riemannian manifold with $(n+\frac{n-2}{n})$-nonnegative (respectively, $(n+\frac{n-2}{n})$-nonpositive) curvature operator of the second kind must be isometric to $\mathbb{S}^{n-1}\times \mathbb{R}$ (respectively, $\mathbb{H}^{n-1}\times \mathbb{R}$) up to scaling. We also prove analogous optimal rigidity results for $\mathbb{S}^{n_1}\times \mathbb{S}^{n_2}$ and $\mathbb{H}^{n_1}\times \mathbb{H}^{n_2}$, $n_1,n_2 \geq 2$, among product Riemannian manifolds, as well as for $\mathbb{CP}^{m_1}\times \mathbb{CP}^{m_2}$ and $\mathbb{CH}^{m_1}\times \mathbb{CH}^{m_2}$, $m_1,m_2\geq 1$, among product K\"ahler manifolds. Our approach is pointwise and algebraic.