Conformal metrics and Ricci tensors on the sphere

We consider tensors on the unit sphere , where , is the standard metric and is a differentiable function on . For such tensors, we consider the problems of existence of a Riemannian metric , conformal to , such that , and the existence of such a metric that satisfies , where is the scalar curvature of . We find the restrictions on the Ricci candidate for solvability, and we construct the solutions when they exist. We show that these metrics are unique up to homothety, and we characterize those defined on the whole sphere. As a consequence of these results, we determine the tensors that are rotationally symmetric. Moreover, we obtain the well-known result that a tensor , 0 $">, has no solution on if and only metrics homothetic to admit as a Ricci tensor. We also show that if , then equation has no solution , conformal to on , and only metrics homothetic to are solutions to this equation when . Infinitely many solutions, globally defined on , are obtained for the equation

where . The geometric interpretation of these solutions is given in terms of existence of complete metrics, globally defined on and conformal to the Euclidean metric, for certain bounded scalar curvature functions that vanish at infinity.

     

Nonparallelism of the Ricci Tensor of Pseudo-cylindric Metrics

We describe examples of metrics in the conformal class [g] on some conformally flat Riemannian manifolds (M,g]. These metrics have a constant scalar curvature and an harmonic curvature with nonparallel Ricci tensor.     

General spherically symmetric Finsler metrics with constant Ricci and flag curvature

In this paper, we investigate the flag curvature of a special class of Finsler metrics called general spherically symmetric Finsler metrics, which are defined by a Euclidean metric and two related 1-forms. We find equations to characterize the class of metrics with constant Ricci curvature (tensor) and constant flag curvature. Moreover, we study general spherically symmetric Finsler metrics with the vanishing non-Riemannian quantity χ-curvature. In particular, we construct some new projectively flat Finsler metrics of constant flag curvature.     

Asymptotically Euclidean Ends of Ricci Flat Manifolds,and Conformal Inversions

We prove that a Ricci flat end of a Riemannian manifold is asymptotically Euclidean if it is obtained from a smooth metric by a conformal inversion. A number of consequences are discussed.     

Ridigity of Ricci solitons with weakly harmonic Weyl tensors

In this paper, we prove rigidity results on gradient shrinking or steady Ricci solitons with weakly harmonic Weyl curvature tensors. Let be a compact gradient shrinking Ricci soliton satisfying with constant. We show that if satisfies , then is Einstein. Here denotes the Weyl curvature tensor. In the case of noncompact, if M is complete and satisfies the same condition, then M is rigid in the sense that M is given by a quotient of product of an Einstein manifold with Euclidean space. These are generalizations of the previous known results in 10 , 14 and 19 . Finally, we prove that if is a complete noncompact gradient steady Ricci soliton satisfying , and if the scalar curvature attains its maximum at some point in the interior of M, then either is flat or isometric to a Bryant Ricci soliton. The final result can be considered as a generalization of main result in 3 .     

A Comparison Theorem on the Ricci Curvature in Projective Geometry

We show that if two Riemannian metrics and g are pointwiseprojectively equivalent and the Ricci curvatures satisfy Ric, then the projective equivalence is trivialprovided that g is complete. In this case, is parallel with respect to g and the Riemann curvatures of g and are equal.The Ricci curvature condition can be weakened when the manifold iscompact. This rigidity theorem actually holds for more general geometricstructures, such as Finsler metrics and sprays. In this paper, we willalso discuss several examples and show that the completeness of g cannot be dropped.     

Group-invariant solutions for the Ricci curvature equation and the Einstein equation

We consider the pseudo-Euclidean space $({\mathbb{R}}^n,g)$, $n\ge 3$, with coordinates $x=(x_1,\dots ,x_n)$ and metric $g_{ij}={\delta }_{ij}{?}_i$, ${?}_i=\pm 1$, where at least one ${?}_i$ is positive, and also tensors of the form $A={\sum }_{i,j}{A}_{ij}d{x}_{i}d{x}_j$, such that $A_{ij}$ are differentiable functions of x. For such tensors, we use Lie point symmetries to find metrics $\stackrel{￣}{g}=\frac{1}{u^2}g$ that solve the Ricci curvature and the Einstein equations. We provide a large class of group-invariant solutions and examples of complete metrics $\stackrel{￣}{g}$ defined globally in ${\mathbb{R}}^n$. As consequences, for certain functions $\stackrel{￣}{K}$, we show complete metrics $\stackrel{￣}{g}$, conformal to the pseudo-Euclidean metric g, whose scalar curvature is $\stackrel{￣}{K}$.     

Smoothing Riemannian metrics with bounded Ricci curvatures in dimension four

We obtain a local smoothing result for Riemannian manifolds with bounded Ricci curvatures in dimension four. More precisely, given a Riemannian metric with bounded Ricci curvature and small L²-norm of curvature on a metric ball, we can find a smooth metric with bounded curvature which is C1,α-close to the original metric on a smaller ball but still of definite size.     

On Einstein Matsumoto metrics

Einstein metrics are solutions to Einstein field equation in General Relativity containing the Ricci-flat metrics. Einstein Finsler metrics which represent a non-Riemannian stage for the extensions of metric gravity, provide an interesting source of geometric issues and the (α,β)-metric is an important class of Finsler metrics appearing iteratively in physical studies. It is proved that every n-dimensional (n≥3) Einstein Matsumoto metric is a Ricci-flat metric with vanishing S-curvature. The main result can be regarded as a second Schur type Lemma for Matsumoto metrics.     

On the Ricci curvature of steady gradient Ricci solitons

Assume (Mn,g) is a complete steady gradient Ricci soliton with positive Ricci curvature. If the scalar curvature approaches 0 towards infinity, we prove that , where O is the point where R obtains its maximum and γ(s) is a minimal normal geodesic emanating from O. Some other results on the Ricci curvature are also obtained.     

A priori Estimates and Existence for a Class of Fully Nonlinear Elliptic Equations in Conformal Geometry

In this paper we prove the interior gradient and second derivative estimates for a class of fully nonlinear elliptic equations determined by symmetric functions of eigenvalues of the Ricci or Schouten tensors. As an application we prove the existence of solutions to the equations when the manifold is locally conformally flat or the Ricci curvature is positive.     

On a Kaehler hypersurface with the cyclic Ricci semi-symmetric tensor

The purpose of the present paper is to prove that a Kaehler hypersurface with the cyclic Ricci semi-symmetric tensor is locally symmetric.     

Singular Ricci Solitons and Their Stability under the Ricci Flow

We introduce certain spherically symmetric singular Ricci solitons and study their stability under the Ricci flow from a dynamical PDE point of view. The solitons in question exist for all dimensions n + 1 ≥ 3, and all have a point singularity where the curvature blows up; their evolution under the Ricci flow is in sharp contrast to the evolution of their smooth counterparts. In particular, the family of diffeomorphisms associated with the Ricci flow “pushes away” from the singularity causing the evolving soliton to open up immediately becoming an incomplete (but non-singular) metric. The main objective of this paper is to study the local-in time stability of this dynamical evolution, under spherically symmetric perturbations of the singular initial metric. We prove a local well-posedness result for the Ricci flow in suitably weighted Sobolev spaces, which in particular implies that the “opening up” of the singularity persists for the perturbations as well.     

Curvature in Special Base Conformal Warped Products

We introduce the concept of a base conformal warped product of two pseudo-Riemannian manifolds. We also define a subclass of this structure called as a special base conformal warped product. After, we explicitly mention many of the relevant fields where metrics of these forms and also considerations about their curvature related properties play important rolls. Among others, we cite general relativity, extra-dimension, string and super-gravity theories as physical subjects and also the study of the spectrum of Laplace-Beltrami operators on p-forms in global analysis. Then, we give expressions for the Ricci tensor and scalar curvature of a base conformal warped product in terms of Ricci tensors and scalar curvatures of its base and fiber, respectively. Furthermore, we introduce specific identities verified by particular families of, either scalar or tensorial, nonlinear differential operators on pseudo-Riemannian manifolds. The latter allow us to obtain new interesting expressions for the Ricci tensor and scalar curvature of a special base conformal warped product and it turns out that not only the expressions but also the analytical approach used are interesting from the physical, geometrical and analytical point of view. Finally, we analyze, investigate and characterize possible solutions for the conformal and warping factors of a special base conformal warped product, which guarantee that the corresponding product is Einstein. Besides all, we apply these results to a generalization of the Schwarzschild metric.      

李奇曲率平行的黎曼流形的孤立现象

本文研究李奇曲率平行的封闭黎曼流形，证明了黎曼曲率平方的一个拚挤定理。