The object of this paper is to define and study a new type of non-flat Riemannian manifold called nearly Einstein manifold. The notion of this nearly Einstein manifold has been established by an example and an existence theorem. Some geometric properties are obtained. AMS Mathematics Subject Classification (2010): 53C25.

In the present paper we have studied an N(k)-quasi Einstein manifold satisfying R(ξ, X).P̃ , where P̃ is the pseudo-projective curvature tensor. Among others, it is shown that if quasi-Einstein manifold with constant associated scalars is Ricci symmetric then the generator of the manifold is a Killing vector field. AMS Mathematics Subject Classification (2000): 53C25

The object of the present paper is to study some properties of a quasi Einstein manifold. A non-trivial concrete example of a quasi Einstein manifold is also given.

Suppose M a compact manifold which admits an Einstein metric g which is Kähler with respect to some complex structure J . Is every other Einstein metric h on M also Kähler-Einstein? If the complex dimension of (M,J) is ≥ 3, the answer is generally no; for example, CP3 admits both the FubiniStudy metric, which is Kähler-Einstein, and a non-Kähler Einstein metric [2] obtained by appropriately squ...

In this paper, we have proved that a projectively flat Sasakian manifold is an Einstein manifold. Also, if an Einstein-Sasakian manifold is projectively flat, then it is locally isometric with a unit sphere S(1). It has also been proved that if in an Einsten-Sasakian manifold the relation K(X, Y ).P = 0 holds, then it is locally isometric with a unit sphere S(1). AMS Mathematics Subject Classif...

We study the geometric properties of the base manifold for the unit tangent bundle satisfying the η-Einstein condition with the standard contact metric structure. One of the main theorems is that the unit tangent bundle of 4-dimensional Einstein manifold, equipped with the canonical contact metric structure, is ηEinstein manifold if and only if base manifold is the space of constant sectional c...

Among the eigenvalue problems of the Laplacian, the biharmonic operator eigenvalue problems are interesting projects because these problems root in physics and geometric analysis. The buckling problem is one of the most important problems in physics, and many studies have been done by the researchers about the solution and the estimate of its eigenvalue. In this paper, first, we obtain the evol...

It is shown that the first order (Palatini) variational principle for a generic nonlinear metric-affine Lagrangian depending on the (symmetrized) Ricci square invariant leads to an almost-product Einstein structure or to an almost-complex anti-Hermitian Einstein structure on a manifold. It is proved that a real anti-Hermitian metric on a complex manifold satisfies the Kähler condition on the sa...