Related papers: On Einstein-type manifold with cyclic parallel Ric…

In this short note, we prove that a Calabi extremal Kaehler-Ricci soliton on a compact toric Kaehler manifold is Einstein. This solves for the class of toric manifolds a general problem stated by the authors that they solved only under some…

A 3-dimensional Riemannian manifold equipped with a tensor structure of type $(1,1)$, whose third power is the identity, is considered. This structure and the metric have circulant matrices with respect to some basis, i.e., these structures…

The Eisenhart problem of finding parallel tensors is solved for the symmetric case in the regular $f$-Kenmotsu framework. On this way, the Olszack-Rosca example of Einstein manifolds provided by $f$-Kenmotsu manifolds via locally symmetric…

In this paper, we use the normalized Ricci-DeTurk flow to prove a stability result for strictly stable conformally compact Einstein manifolds. As an application, we show a local volume comparison of conformally compact manifolds with scalar…

On a smooth metric measure spacetime $(M,g,e^{-f} dvol_g)$, we define a weighted Einstein tensor. It is given in terms of the Bakry-\'Emery Ricci tensor as a tensor which is symmetric, divergence-free, concomitant of the metric and the…

We present several deformation and rigidity results within the classes of closed Riemannian manifolds which either are $2k$-Einstein (in the sense that their $2k$-Ricci tensor is constant) or have constant $2k$-Gauss-Bonnet curvature. The…

This paper is concerned with the study of generalized gradient Ricci-Yamabe solitons. We characterize the compact generalized gradient Ricci-Yamabe soliton and find certain conditions under which the scalar curvature becomes constant. The…

In this paper we obtain a simple upper bound for the infimum of the Ricci curvatures of a complete Riemannian manifold with nonzero injectivity radius i(M) depending only on of the i(M). In case of rigidity the Riemannian manifold must be…

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.

We show that any quantum irreducible flag manifold satisfies an analogue of the Einstein condition, expressing proportionality between the Ricci tensor and the metric, at least in a small open interval around the classical value of the…

It is observed that on a compact almost complex Calabi-Yau with torsion 6-manifold the Nijenhuis tensor is parallel with respect to the torsion connection. If the torsion is closed then the space is a compact generalized gradient Ricci…

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…

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 the paper, we study complete almost Ricci solitons using the concepts and methods of geometric dynamics and geometric analysis. In particular, we characterize Einstein manifolds in the class of complete almost Ricci solitons. Then, we…

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…

In this article, we investigate the geometry of compact quasi-Einstein manifolds with boundary. We show that a $3$-dimensional simply connected compact quasi-Einstein manifold with boundary and constant scalar curvature is isometric, up to…

We provide a classification of Einstein submanifolds in space forms with flat normal bundle and parallel mean curvature. This extends a previous result due to Dajczer and Tojeiro for isometric immersions of Riemannian manifolds with…

We prove that a compact stratied space satises the Riemannian curvature-dimension condition RCD(K, N) if and only if its Ricci tensor is bounded below by K $\in$ R on the regular set, the cone angle along the stratum of codimension two is…

Let g be a G-invariant Einstein metric on a compact homogeneous space M=G/K. We use a formula for the Lichnerowicz Laplacian of g at G-invariant TT-tensors to study the stability type of g as a critical point of the scalar curvature…

In this paper, we prove that a compact quasi-Einstein manifold $(M^n,\,g,\,u)$ of dimension $n\geq 4$ with boundary $\partial M,$ nonnegative sectional curvature and zero radial Weyl tensor is either isometric, up to scaling, to the…