Related papers: Half conformally flat generalized quasi-Einstein m…
We study the affine quasi-Einstein Equation for homogeneous surfaces. This gives rise through the modified Riemannian extension to new half conformally flat generalized quasi-Einstein neutral signature $(2,2)$ manifolds, to conformally…
We study the affine quasi-Einstein equation, a second order linear homogeneous equation, which is invariantly defined on any affine manifold. We prove that the space of solutions is finite-dimensional, and its dimension is a strongly…
In this short note we prove that any complete four dimensional anti-self-dual (or self-dual) quasi-Einstein manifolds is either Einstein or locally conformally flat. This generalizes a recent result of X. Chen and Y. Wang.
The holonomy algebras of Einstein not Ricci-flat pseudo-Riemannian manifolds of arbitrary signature are classified. As illustrating examples, the cases of Lorentzian manifolds, pseudo-Riemannian manifolds of signature $(2,n)$ and the…
Let $\mathcal{M}$ be a Type $\mathcal{A}$ affine surface. We show that $\mathcal{M}$ is linearly strongly projectively flat. We use the quasi-Einstein equation together with the condition that $\mathcal{M}$ is strongly projectively flat to…
A rigidity result for a class of compact generalized quasi-Einstein manifolds with constant scalar curvature is obtained. Moreover, under some geometric assumptions, the rigidity for the noncompact case is also proved. Considering non…
The aim of this paper is to present some structural equations for generalized quasi-Einstein metrics which was defined recently by Catino in [12]. In addition, supposing that the Riemannian manifold is Einstein we shall show that it is a…
An almost Einstein manifold satisfies equations which are a slight weakening of the Einstein equations; Einstein metrics, Poincare-Einstein metrics, and compactifications of certain Ricci-flat asymptotically locally Euclidean structures are…
We discuss smooth metric measure spaces admitting two weighted Einstein representatives of the same weighted conformal class. First, we describe the local geometries of such manifolds in terms of certain Einstein and quasi-Einstein warped…
We analyze the classic problem of existence of Einstein metrics in a given conformal structure for the class of conformal structures inducedf Nurowski's construction by (oriented) (2,3,5) distributions. We characterize in two ways such…
We examine the space of solutions to the affine quasi--Einstein equation in the context of homogeneous surfaces. As these spaces can be used to create gradient Yamabe solitions, conformally Einstein metrics, and warped product Einstein…
We classify weakly Einstein submanifolds in space forms that satisfy Chen's equality. We also give a classification of weakly Einstein hypersurfaces in space forms that satisfy the semisymmetric condition. In addition, we discuss some…
In this paper the geometry of normal metric contact pair manifolds is studied under the flatness of conformal, concircular and quasi-conformal curvature tensors. It is proved that a conformal flat normal metric contact pair manifold is an…
In this paper we construct a family of examples of self-dual Einstain metrics of neutral signature, which are not Ricci flat, nor locally homogenous. Curvature of these manifolds is studied in details. These are obtained by the…
We develop a geometric and explicit construction principle that generates classes of Poincare-Einstein manifolds, and more generally almost Einstein manifolds. Almost Einstein manifolds satisfy a generalisation of the Einstein condition;…
First, we show that a warped product of a line and a fiber manifold is weakly conformally flat and quasi Einstein if and only if the fiber is Einstein. Next, we characterize and classify contact (in particular, $K$-contact) Riemannian…
We review basic facts on the structure of nearly K\"ahler manifolds, focussing in particular on the six-dimensional case. A self-contained proof that nearly K\"ahler six-manifolds are Einstein is given by combining different known results.…
We obtain generalized Wintgen inequalities for submanifolds in conformally flat manifolds. We give some applications for submanifolds in a Riemannian manifold of quasi-constant curvature. Equality cases are also considered.
In this paper, we classify $n$-dimensional ($n\geq 5$) quasi-Einstein manifolds with harmonic Weyl curvature, thus extending the work of Shin \cite{Shin} in dimension four for quasi-Einstein manifolds and refining the work of…
We discuss Sasakian-Einstein geometry under a quasi-regularity assumption. It is shown that the space of all quasi-regular Sasakian-Einstein orbifolds has a natural multiplication on it. Furthermore, necessary and sufficient conditions are…