We give a smooth analogue of the classical Schläfli formula, relating the variation of the volume bounded by a hypersurface moving in a general Einstein manifold and the integral of the variation of the mean curvature. We extend it to variations of the metric in a Riemannian Einstein manifold with boundary, and apply it to Einstein cone-manifolds, to isometric deformations of Euclidean hypersur...

Abstract In this paper, we classify the compact locally homogeneous non-gradient m -quasi Einstein 3- manifolds. Along way, also prove that given a quotient of Lie group any dimension is Einstein, potential vector field X must be left invariant and Killing. We nontrivial metrics are product two metrics. show S 1 only manifold which admits metric nontrivially Einstein.

In this paper we describe the Einstein-Kähler metric for the Cartan-Hartogs domains which are the special case of the Hua domains. First of all, we reduce the Monge-Ampère equation for the metric to an ordinary differential equation in the auxiliary function X = X(z, w). This differential equation can be solved to give an implicit function in X. Secondly, for some cases, we obtained explicit fo...

In the present work, generalized recurrent and concircularly Weyl manifolds are examined. We define nearly quasi-Einstein we proved that if a or manifold admits special concircular vector field, then reduces to manifold. Also, some other results presented.

In this paper, we initiate the study of conformal $\eta$-Ricci soliton and almost within framework para-Sasakian manifold. We prove that if metric admits soliton, then manifold is $\eta$-Einstein either vector field $V$ Killing or it leaves $\phi$ invariant. Here, have shown characteristics scalar curvature when admitting pointwise collinear with characteristic $\xi$. Next, show a endowed an in...

Ricci solitons (RS) have an extensive background in modern physics and are extensively used cosmology general relativity. The focus of this work is to investigate almost (RAS) on Lorentzian manifolds with a special metric connection called semi-symmetric u-connection (SSM-connection). First, we show that any quasi-Einstein manifold having SSM-connection, whose RS, Einstein manifold. A similar c...

The aim of this paper is to study a contact Riemannian submersion ? : M B between almost metric manifolds such that its total space admits an ??Ricci soliton. Here, we obtain some necessary conditions for which any fiber and the manifold are soliton, Ricci generalized quasi-Einstein, ??Einstein or Einstein. Finally, equipped with torqued vector field give characterizations ?.

In this paper we consider pseudo projectively flat Riemannian manifold whose Ricci tensor S satisfies the condition S(X,Y ) = rT (X)T (Y ), where r is the scalar curvature, T is a non-zero 1-form defined by g(X, ξ) = T (X), ξ is a unit vector field and prove that the manifold is of pseudo quasi constant curvature, integral curves of the vector field ξ are geodesic and ξ is a proper concircular ...

We study canonical paracontact connection on a para-Sasakian manifold. We prove that a Ricci-flat para-Sasakian manifold with respect to canonical paracontact connection is an η-Einstein manifold.We also investigate some properties of curvature tensor, conformal curvature tensor,W2curvature tensor, concircular curvature tensor, projective curvature tensor, and pseudo-projective curvature tensor...

We determine a new class of paracontact paracomplex Riemannian manifolds derived from certain cone construction, called para-Sasaki-like manifolds, and give explicit examples. define hyperbolic extension paraholomorphic manifold, which is local product two spaces equal dimension, show that it manifold. If the original manifold complete Einstein space negative scalar curvature, then its curvatur...