Product manifolds and the curvature operator of the second kind (Q6644090)

For a Riemannian \(n\)-manifold \((M,g)\) the curvature operator of the second kind refers to the endomorphism defined by the projection of the restriction of the curvature operator, viewed as an operator on symmetric two-forms, to trace-free symmetric two-forms \(S^2_0(T_p M)\) (see the start of the review [\textit{J. Nienhaus} et al., J. Lond. Math. Soc., II. Ser. 108, No. 4, 1642--1668 (2023; Zbl 1530.53048)]) or its equivalent bilinear form \(\mathring{R}:S^2_0(T_p M) \times S^2_0(T_p M) \rightarrow \mathbb{R}\). Recall that a Riemannian \(n\)-manifold \((M,g)\) is said to have \(\alpha\)-positive (resp. nonnegative) curvature operator of the second kind if on any orthonormal basis \(\{\varphi_i:i=1,\dots, (n-1)(n+2)/2\}\) for \(S^2_0(T_p M)\) one has\N\[\N\left(\sum_{i=1}^{\lfloor \alpha \rfloor} \mathring{R}(\varphi_i,\varphi_i)\right) + (\alpha -\lfloor \alpha \rfloor)\mathring{R}(\varphi_{\lfloor \alpha \rfloor+1},\varphi_{\lfloor \alpha \rfloor+1}) > 0 \quad (\text{resp. }\geq 0 ).\N\]\NSimilarly, a Riemannian \(n\)-manifold \((M,g)\) is said to have \(\alpha\)-negative (resp. nonpositive) curvature operator of the second kind if on any orthonormal basis \(\{\varphi_i:i=1,\dots, (n-1)(n+2)/2\}\) for \(S^2_0(T_p M)\) one has\N\[\N\left(\sum_{i=1}^{\lfloor \alpha \rfloor} \mathring{R}(\varphi_i,\varphi_i)\right) + (\alpha -\lfloor \alpha \rfloor)\mathring{R}(\varphi_{\lfloor \alpha \rfloor+1},\varphi_{\lfloor \alpha \rfloor+1}) < 0 \quad (\text{resp. }\leq 0 ).\N\]\NThe author shows, under the assumption that \(n\geq 4\), \(M\) is complete, locally irreducible, and not flat, that\N\begin{enumerate}\N\item[(1)] if \(M\) has \((n+(n-2)/n)\)-nonnegative curvature of the second kind, then is is covered by \(S^{n-1} \times \mathbb{R}\) up to homothety,\N\item[(2)] if \(M\) has \((n+(n-2)/n)\)-nonpositive curvature of the second kind, then it is covered by \(\mathbb{H}^{n-1} \times \mathbb{R}\) up to homothety,\N\end{enumerate}\Nwhere \(S^{n-1}\) denotes the \((n-1)\)-sphere of curvature \(1\) and \(\mathbb{H}^{n-1}\) denotes the \((n-1)\)-hyperbolic space of curvature \(-1\).\N\NHe also shows that for a Riemannian \(m\)-manifold \((M,g)\) and a Riemannian \(n\)-manifold \((N.h)\) where \(m, n \geq 2\) that the Cartesian product \((M \times N, g \oplus h)\) has\N\begin{enumerate}\N\item[(1)] if \(M \times N\) has \(\alpha\)-nonnegative (resp. \(\alpha\)-nonpositive) curvature operator of the second kind for some \(\alpha < 1+mn+(m(n-1)+n(m-1))/(m+n)\), then \(M\) is flat,\N\item[(2)] if \(M \times N\) has \((1+mn+(m(n-1)+n(m-1))/(m+n))\)-nonnegative curvature operator of the second kind then \(M\) and \(N\) both have constant sectional curvature \(c >0\), namely \(M \times N\) is covered by \(S^m \times S^n\) up to homothety,\N\item[(3)] if \(M \times N\) has \((1+mn+(m(n-1)+n(m-1))/(m+n))\)-nonpositive curvature operator of the second kind then \(M\) and \(N\) both have constant sectional curvature \(c <0\), namely \(M \times N\) is covered by \(\mathbb{H}^m \times \mathbb{H}^n\) up to homothety.\N\end{enumerate}\N\NThe approach displayed in the proofs of the two main results effectively builds on earlier work of the author in [J. Geom. Anal. 32, No. 11, Paper No. 281, 14 p. (2022; Zbl 1501.53046)] and [Proc. Am. Math. Soc. 151, No. 11, 4909--4922 (2023; Zbl 07735836)] respectively that deal with low dimensional cases.