New Sphere Theorems under Curvature Operator of the Second Kind

TL;DR

This paper establishes new sphere theorems and curvature characterizations for Riemannian manifolds under specific eigenvalue conditions of the curvature operator of the second kind, extending previous results and demonstrating sharpness with examples.

Contribution

It introduces optimal curvature conditions involving eigenvalues of the second kind curvature operator, leading to new sphere theorems and characterizations in various dimensions.

Findings

Proves differentiable sphere theorems in dimensions three and four.

Establishes a homological sphere theorem in higher dimensions.

Provides curvature characterizations of Kähler space forms.

Abstract

We investigate Riemannian manifolds $(M^n,g)$ whose curvature operator of the second kind $\mathring{R}$ satisfies the condition \begin{equation*} \alpha^{-1} (\lambda_1 +\cdots +\lambda_{\alpha}) > - \theta \bar{\lambda}, \end{equation*} where $\lambda_1 \leq \cdots \leq \lambda_{(n-1)(n+2)/2}$ are the eigenvalues of $\mathring{R}$, $\bar{\lambda}$ is their average, and $\theta > -1$. Under such conditions with optimal $\theta$ depending on $n$ and $\alpha$, we prove two differentiable sphere theorems in dimensions three and four, a homological sphere theorem in higher dimensions, and a curvature characterization of K\"ahler space forms. These results generalize recent works corresponding to $\theta =0$ of Cao-Gursky-Tran, Nienhaus-Petersen-Wink, and the author. Moreover, examples are provided to demonstrate the sharpness of all results.