Related papers: Manifolds with harmonic curvature and curvature op…

We prove that a compact Einstein manifold of dimension $n\geq 4$ with nonnegative curvature operator of the second kind is a constant curvature space by Bochner technique. Moreover, we obtain that compact Einstein manifolds of dimension…

We prove that a compact Riemannian manifold of dimension $n\ge 8$ with harmonic Weyl curvature and $\frac{3(n-1)(n+2)}{4(3n-1)}$-nonnegative curvature operator of the second kind is either globally conformally equivalent to a space of…

Using Bochner techniques, we prove that a compact Einstein manifold of dimension $n \ge 4$ has constant curvature provided that the curvature operator of the second kind satisfies a cone condition that is strictly weaker than nonnegativity.…

We show that an $n$-dimensional Riemannian manifold with $n$-nonnegative or $n$-nonpositive curvature operator of the second kind has restricted holonomy $SO(n)$ or is flat. The result does not depend on completeness and can be improved…

It is established in [6, 14, 23] that any closed Einstein manifold with two-nonnegative curvature operator of the second kind is either flat or a round sphere. In this paper, we refine this result by relaxing the curvature condition to a…

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…

We investigate the curvature operator of the second kind on Riemannian manifolds and prove several classification results. The first one asserts that a closed Riemannian manifold with three-positive curvature operator of the second kind is…

The harmonicity condition of the curvature 2-form of a pseudo- Riemannian manifold is formulated on the basis of annulment of this form by the de Rham-Lichnerowicz Laplacian. The following theorem is proved: The curvature 2-form of any…

In this note, we study Einstein manifolds whose curvature operator of the second kind $\mathring{R}$ satisfies the cone condition \[ \alpha^{-1}\big(\sum_{i=1}^{[\alpha]} \lambda_i+ (\alpha - [\alpha] ) \lambda_{[\alpha] + 1} \big) \ge…

In this short note, as a simple application of the strong result proved recently by B\"ohm and Wilking, we give a classification on closed manifolds with 2-nonnegative curvature operator. Moreover, by the new invariant cone constructions of…

We show that compact, $n$-dimensional Riemannian manifolds with $\frac{n+2}{2}$-nonnegative curvature operators of the second kind are either rational homology spheres or flat. More generally, we obtain vanishing of the $p$-th Betti number…

We prove that a compact Riemannian manifold of dimension $m \geq 3$ with harmonic curvature and $\lfloor\frac{m-1}{2}\rfloor$-positive curvature operator has constant sectional curvature, extending the classical Tachibana theorem for…

We show that a closed four-manifold with $4\frac{1}{2}$-positive curvature operator of the second kind is diffeomorphic to a spherical space form. The curvature assumption is sharp as both $\mathbb{CP}^2$ and $\mathbb{S}^3 \times…

Let $(M,g)$ be a noncompact complete $n$-manifold with harmonic curvature and positive Sobolev constant. Assume that $L_2$ norms of Weyl curvature and traceless Ricci curvature are finite. We prove that $(M,g)$ is Einstein if $n \ge 5$ and…

We demonstrate that $n$-dimension closed Einstein manifolds, whose smallest eigenvalue of the curvature operator of the second kind of $\mathring{R}$ satisfies $\lambda_1 \ge -\theta(n) \bar\lambda$, are either flat or round spheres, where…

We classify those curvature-homogeneous Einstein four-manifolds, of all metric signatures, which have a complex-diagonalizable curvature operator. They all turn out to be locally homogeneous. More precisely, any such manifold must be either…

This article aims to investigate the curvature operator of the second kind on K\"ahler manifolds. The first result states that an $m$-dimensional K\"ahler manifold with $\frac{3}{2}(m^2-1)$-nonnegative (respectively,…

This paper is devoted to the first systematic investigation of manifolds that are Einstein for a connection with skew symmetric torsion. We derive the Einstein equation from a variational principle and prove that, for parallel torsion, any…

In this paper, we investigate manifolds for which the curvature of the second kind (following the terminology of Nishikawa) satisfies certain positivity conditions. Our main result settles Nishikawa's conjecture that manifolds for which the…

We prove that any closed, convex hypersurface in an $(n+1)$-dimensional Riemannian manifold with $\lceil \frac{n}{2} \rceil$-positive curvature operator is a rational homology sphere with finite fundamental group. The same conclusion holds…