Hessian Tensor and Standard Static Space-times

Vacuum static spaces with vanishing of complete divergence of Bach tensor and Weyl tensor

In this paper, we study vacuum static spaces with the complete divergence of the Bach tensor and Weyl tensor. First, we prove that the vanishing of complete divergence of the Bach tensor and Weyl tensor implies the harmonicity of the metric, and we present examples in which these conditions do not imply Bach flatness. As an application, we prove the non-existence of multiple black holes in vacuum static spaces with zero scalar curvature. On the other hand, we prove the Besse conjecture under these conditions, which are weaker than harmonicity or Bach flatness. Moreover, we show a rigidity resu…

On the stability of L^p-norms of Curvature Tensor at Rank one symmetrics spaces

We study stability and local minimizing properties of $L^p$- norms of Riemannian curvature tensor denoted by $\mathcal{R}_p$ by variational methods. We compute the Hessian of $\mathcal{R}_p$ at compact rank 1 symmetric spaces and prove that they are stable for $\mathcal{R}_p$ for certain values of p > 2. A similar result also holds for compact quotients of rank 1 symmetric spaces of non-compact type. Consequently, we obtain stability of L^{n\2}- norm of Weyl curvature at these metrics.

On Static Manifolds and Related Critical Spaces with cyclic parallel Ricci tensor

The aim of this paper is to classify three dimensional compact Riemannian manifolds $(M^{3},g)$ that admits a non-constant solution to the equation $$-Δf g+Hess f-fRic=μRic+λg,$$ for some special constants $(μ, λ)$, under assumption that the manifold has cyclic parallel Ricci tensor. Namely, the structures that we will study here will be: positive static triples, critical metrics of the volume functional, and critical metrics of the total scalar curvature functional. We shall also classify $n$-dimensional critical metrics of the volume functional with non-positive scalar curvature and satisfyi…

具有循环平行里奇张量的空间的直接证明

Direct proofs for spaces with cyclic parallel Ricci tensor

该研究为梯度里奇孤子、带边界紧致拟爱因斯坦流形及具循环平行里奇张量的临界空间的刚性结果提供直接证明，通过推导张量恒等式等方法得到相关刚性结论与里奇平行性结果。

Stationary or static space-times and Young tableaux

Algebraic curvature tensors possess generators which can be formed from symmetric or alternating tensors S, A or tensors θwith an irreducible (2,1)-symmetry. In differential geometry examples of curvature formulas are known which contain generators on the basis of S or A realized by differentiable tensor fields in a natural way. We show that certain curvature formulas for stationary or static space-times contain such differentiable realizations of generators based on θ. The tensor θis connected with the timelike Killing vector field of the space-time. θlies in a special symmetry class from the…

Vanishing conditions on Weyl tensor for Einstein-type manifolds

In this paper we consider an Einstein-type equation which generalizes important geometric equations, like static and critical point equations. We prove that a complete Einstein-type manifold with fourth-order divergence-free Weyl tensor and zero radial Weyl curvature is locally a warped product with $(n-1)$-dimensional Einstein fibers, provided that the potential function is proper. As a consequence, we prove a result about the nonexistence of multiple black holes in static spacetimes.

Riemannian Metrics on Locally Projectively Flat Manifolds

The expression (-1/u) times the Hessian of u transforms as a symmetric (0,2) tensor under projective coordinate transformations, so long as u transforms as a section of a certain line bundle. On a locally projectively flat manifold M, the section u can be regarded as a metric potential analogous to the local potential in Kahler geometry. If M is compact and u is a negative section of the dual of the tautological bundle whose Hessian is positive definite, then M is projectively equivalent to a quotient of a bounded convex domain in R^n. The same is true if M has a boundary on which u=0. This th…

正曲率真空静态空间的刚性

Rigidity of positively curved vacuum Static spaces

本文研究维数≥4且具非负截面曲率的紧致真空静态空间，证明正截面曲率且Weyl曲率张量完全散度非负的这类空间等距于标准球面，还推广了临界点方程相关的刚性结果。