The objective of the present paper is to study real hyper surfaces of a complex projective space with generalized recurrent second fundamental tensor and it is shown that such type real hyper surface exist. Also, we study real hyper surfaces of a complex projective space with generalized recurrent Ricci tensor. It is proved that a real hyper surfaces of complex projective space is generalized R...

Given a convex body K ⊂ Rn with the barycenter at the origin we consider the corresponding Kähler-Einstein equation e = detDΦ. If K is a simplex, then the Ricci tensor of the Hessian metric DΦ is constant and equals n−1 4(n+1) . We conjecture that the Ricci tensor of D Φ for arbitrary K is uniformly bounded by n−1 4(n+1) and verify this conjecture in the two-dimensional case. The general case r...

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. Without the compactness assumption, irreducible Kähler manifolds with Ricci tensor having two distinct constant eigenvalues are shown to exist in various situat...

We characterize the Einstein metrics in such broad classes of as almost $$\eta $$ -Ricci solitons and on Kenmotsu manifolds, generalize some known results. First, we prove that a metric an soliton is if either it -Einstein or potential vector field V infinitesimal contact transformation collinear to Reeb field. Further, manifold admits gradient with leaving scalar curvature invariant, then mani...

The object of the present paper is to characterize class Kenmotsu manifolds which admits conformal $$\eta $$ -Ricci soliton. Here, we have investigated nature soliton within framework manifolds. It shown that an -Einstein manifold admitting Einstein one. Moving further, considered gradient on and established a relation between potential vector field Reeb field. Next, it proved under certain con...

In the context of paracontact geometry, η-Ricci solitons are considered on manifolds satisfying certain curvature conditions: R(ξ,X) · S = 0, S · R(ξ,X) = 0, W2(ξ,X) · S = 0 and S · W2(ξ,X) = 0. We prove that on a para-Kenmotsu manifold (M,φ, ξ, η, g), the existence of an η-Ricci soliton implies that (M, g) is quasi-Einstein and if the Ricci curvature satisfies R(ξ,X) · S = 0, then (M, g) is Ei...

Though the Weyl tensor is a linear combination of the curvature tensor, Ricci tensor and Ricci scalar, it does not have all and only the Lie symmetries of these tensors since it is possible, in principle, that “asymmetries cancel”. Here we investigate if, when and how the symmetries can be different. It is found that we can obtain a metric with a finite dimensional Lie algebra of Weyl symmetrie...

In this paper we introduce the concept of (ε)-almost paracontact manifolds, and in particular, of (ε)-para Sasakian manifolds. Several examples are presented. Some typical identities for curvature tensor and Ricci tensor of (ε)-para Sasakian manifolds are obtained. We prove that if a semi-Riemannian manifold is one of flat, proper recurrent or proper Ricci-recurrent, then it can not admit an (ε...