相关论文: Ricci Tensors with Rotational Symmetry on R^n

The Ricci tensor (Ric) is fundamental to Einstein's geometric theory of gravitation. The 3-dimensional Ric of a spacelike surface vanishes at the moment of time symmetry for vacuum spacetimes. The 4-dimensional Ric is the Einstein tensor…

Efficient formulae of Ricci tensor for an arbitrary diagonal metric are presented.

Suppose $M$ is a manifold with boundary. Choose a point $o\in\partial M$. We investigate the prescribed Ricci curvature equation $\Ric(G)=T$ in a neighborhood of $o$ under natural boundary conditions. The unknown $G$ here is a Riemannian…

Let G be a three-dimensional unimodular Lie group, and let T be a left-invariant symmetric (0, 2)-tensor field on G. We provide the necessary and sufficient conditions on T for the existence of a pair (g, c) consisting of a left-invariant…

In this article, we study $c$-spherical Ricci metrics, that is, Riemannian metrics whose Gaussian curvature $K$ satisfies \begin{equation*} (K - c)\Delta K - |\nabla K|^2 - 4K(K - c)^2 = 0, \end{equation*} for some $c>0$. We explicitly…

A Ricci soliton is a natural generalization of an Einstein metric. On a pseudo-Riemannian manifold (M, g), it is defined by : $LX g + \r{ho} = {\lambda} g, where X is a smooth vector field on M , LX denotes the Lie derivative in the…

A. Derdzinki [D] gave examples of Riemannian metrics with harmonic curvature and non parallel Ricci tensor on some compact manifolds $(M,g]$ . We examine their existence as well as their number wich naturally depends on the geometry of the…

In this paper, using the Milnor-type theorem technique, we provide on each nilpotent five dimensional Lie group, some global existence result of a pair (g, c) consisting of a left-invariant Riemannian metric g and a positive constant c such…

The prescribed Ricci curvature problem consists in finding a Riemannian metric $g$ on a manifold $M$ such that the Ricci curvature of $g$ equals a given $(0,2)$-tensor field $T$. We survey the recent progress on this problem in the case…

It is known that on $\mathrm{RCD}$ spaces one can define a distributional Ricci tensor ${\bf Ric}$. Here we give a fine description of this object by showing that it admits the polar decomposition $${\bf Ric}=\omega\,|{\bf Ric}|$$ for a…

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

In Riemannian geometry the prescribed Ricci curvature problem is as follows: given a smooth manifold $M$ and a symmetric 2-tensor $r$, construct a metric on $M$ whose Ricci tensor equals $r$. In particular, DeTurck and Koiso proved the…

The aim of this note is to study the measure-valued Ricci tensor on smooth metric measure space with boundary, which is a generalization of Bakry-Emery's modified Ricci tensor on weighted Riemannian manifold. As an application, we offer a…

We consider Riemannian manifolds $(M^n,g_0)$, $(M^n,h)$, where $(M^n,h)$ is smooth, complete, with curvature bounded in absolute value by $K_0 < \infty$, and $(1-\varepsilon_0(n)) h \leq g_0 \leq (1+\varepsilon_0(n)) h$ for some small…

Let $M$ be a domain enclosed between two principal orbits on a cohomogeneity one manifold $M_1$. Suppose $T$ and $R$ are symmetric invariant (0,2)-tensor fields on $M$ and $\partial M$, respectively. The paper studies the prescribed Ricci…

We determine an explicit expression for the Ricci tensor of a K-manifold, that is of a compact Kaehler manifold M with vanishing first Betti number, on which a semisimple group G of biholomorphic isometries acts with an orbit of codimension…

Let (M,g) be a steady gradient Ricci soliton of dimension n \geq 4 which has positive sectional curvature and is asymptotically cylindrical. Under these assumptions, we show that (M,g) is rotationally symmetric. In particular, our result…

We examine questions of geometric realizability for algebraic structures which arise naturally in affine and Riemannian geometry. Suppose given an algebraic curvature operator R at a point P of a manifold M and suppose given a real analytic…

A new method is presented for the determination of Ricci Collineations (RC) and Matter Collineations (MC) of a given spacetime, in the cases where the Ricci tensor and the energy momentum tensor are non-degenerate and have a similar form…

We study the prescribed Ricci curvature problem for homogeneous metrics. Given a (0,2)-tensor field $T$, this problem asks for solutions to the equation $\mathrm{Ric}(g)=cT$ for some constant $c$. Our approach is based on examining global…