Product manifolds and the curvature operator of the second kind

Xiaolong Li

Vol. 332 (2024), No. 1, 167–193

DOI: 10.2140/pjm.2024.332.167

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 nonflat complete locally reducible Riemannian manifold with $\left(n+\frac{n-2}{n}\right)$-nonnegative (respectively, $\left(n+\frac{n-2}{n}\right)$-nonpositive) curvature operator of the second kind must be isometric to ${\mathbb{𝕊}}^{n-1}\times ℝ$ (respectively, ${ℍ}^{n-1}\times ℝ$) up to scaling. We also prove analogous optimal rigidity results for ${\mathbb{𝕊}}^{n_1}\times {\mathbb{𝕊}}^{n_2}$ and ${ℍ}^{n_1}\times {ℍ}^{n_2}$, $n_1,n_2\ge 2$, among product Riemannian manifolds, as well as for ${ℂ\mathbb{P}}^{m_1}\times {ℂ\mathbb{P}}^{m_2}$ and ${ℂℍ}^{m_1}\times {ℂℍ}^{m_2}$, $m_1,m_2\ge 1$, among product Kähler manifolds. The approach is pointwise and algebraic.

Keywords

curvature operator of the second kind, product manifolds, rigidity theorems

Mathematical Subject Classification

Primary: 53C20

Secondary: 53C24

Milestones

Received: 15 February 2024

Revised: 24 June 2024

Accepted: 4 October 2024

Published: 20 November 2024