Product manifolds and the curvature operator of the second kind

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 ${\mathbb{H}}^{n_1}\times {\mathbb{H}}^{n_2}$, $n_1,n_2\ge 2$, among product Riemannian manifolds, as well as for ${\mathbb{CP}}^{m_1}\times {\mathbb{CP}}^{m_2}$ and ${\mathbb{CH}}^{m_1}\times {\mathbb{CH}}^{m_2}$, $m_1,m_2\ge 1$, among product Kähler manifolds. Our approach is pointwise and algebraic.

23 pages, comments are welcome. arXiv admin note: text overlap with arXiv:2208.14505

References in corpus (3)

• Betti numbers and the curvature operator of the second kind
• Manifolds with nonnegative curvature operator of the second kind
• Holonomy restrictions from the curvature operator of the second kind

Cited by in corpus (1)

• Einstein manifolds of negative lower bounds on curvature operator of the second Kind