It is proved that a compact Kähler manifold whose Ricci tensor has two distinct constant non-negative eigenvalues is locally the product of two Kähler-Einstein manifolds. A stronger result is established for the case of Kähler surfaces. Irreducible Kähler manifolds with two distinct constant eigenvalues of the Ricci tensor are shown to exist in various situations: there are homogeneous examples...

We consider a closed manifold M with a Riemannian metric gij(t) evolving by ∂t gij = −2Sij where Sij(t) is a symmetric two-tensor on (M, g(t)). We prove that if Sij satisfies the tensor inequality D(Sij , X) ≥ 0 for all vector fields X on M , where D(Sij , X) is defined in (1.6), then one can construct a forwards and a backwards reduced volume quantity, the former being non-increasing, the latt...

The aim of this paper is to determine left-invariant strictly almost Kähler structures on 4-dimensional Lie groups (g, J,Ω) such that the Ricci tensor is J-invariant.

We use the recent discovered x−ψ duality to parameterize spacetime in terms of wave functions and quantum prepotentials. The components of metric, connection, curvature, Ricci tensor and scalar curvature as well as Einstein equations in this parameterization are

We establish a link between a connection symmetry, called conformal collineation, and almost Ricci soliton (in particular Ricci soliton) in reducible Ricci symmetric semi-Riemannian manifolds. As a physical application, by investigating the kinematic and dynamic properties of almost Ricci soliton manifolds, we present a physical model of imperfect fluid spacetimes. This model gives a general re...

We construct a 2-parameter family FZ of Riemannian metrics on the twistor space Z of a positive quaternion Kähler manifold M satisfying the following properties : (1) the family FZ contains an Einstein metric gZ and its scalings, (2) the family FZ is closed under the operation of making the convex sums, (3) the Ricci map g 7→ Ric(g) defines a dynamical system on the family FZ, (4) the Ricci flo...

The object of the present paper is to characterize K-contact Einstein manifolds satisfying the curvature condition R · C = Q(S,C), where C is the conformal curvature tensor and R the Riemannian curvature tensor. Next we study K-contact Einstein manifolds satisfying the curvature conditions C ·S = 0 and S ·C = 0, where S is the Ricci tensor. Finally, we consider K-contact Einstein manifolds sati...