相关论文: Einstein manifolds under cone conditions for the c…

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…

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…

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 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 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…

In the following work we investigate the structure of Einstein manifolds with positive scalar curvature whose curvature operator is sufficiently close to the identity operator in dimensions below 12. An Einstein manifold with positive…

We prove that complete Riemannian manifolds of dimension $n\ge3$ with harmonic curvature and $\frac{n(n+2)}{2(n+1)}$-nonnegative curvature operator of the second kind must be Einstein. In particular, We show that complete Einstein manifolds…

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…

In this paper, we show that a closed $n$-dimensional generalized $(\lambda, n+m)$-Einstein manifold of constant scalar curvature with weakly radially zero Ricci curvature is isometric to either a sphere ${\Bbb S}^n$, or a product ${\Bbb…

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*}…

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…

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…

Given an Einstein structure with positive scalar curvature on a four-dimensional Riemannian manifolds, that is $Ric=\lambda g$ for some positive constant $\lambda$. For convenience, the Ricci curvature is always normalized to $Ric=1$. A…

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…

On a given closed connected manifold of dimension two, or greater, we consider the squared $L^2$-norm of the scalar curvature functional over the space of constant volume Riemannian metrics. We prove that its critical points have constant…

Let the warped product $M^n=L^m\times_\varphi F^{n-m}$, $n\geq m+3\geq 8$, of Riemannian manifolds be an Einstein manifold with Ricci curvature $\rho$ that admits an isometric immersion into Euclidean space with codimension two. Under the…

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…

Throughout the history of Einstein manifolds, differential geometers have shown great interest in finding the relationships between curvature and the topology of Einstein manifolds. In the paper, first, we prove that a compact Einstein…

We simplify V\'etois' Obata-type argument and use it to identify a closed interval $I_n$, $n \geq 3$, containing zero such that if $a \in I_n$ and $(M^n,g)$ is a closed conformally Einstein manifold with nonnegative scalar curvature and…

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…