We prove that if a real hypersurface with constant mean curvature of a complex space form satisfying ▽S = 0 and Sξ = ξ for a smooth function , then the structure vector field ξ is principal, where S denotes the Ricci tensor of the hypersurface.

REAL HYPERSURFACES WITH ξ-PARALLEL RICCI TENSOR IN A COMPLEX SPACE FORM

Ricci-like solitons with arbitrary potential are introduced and studied on Sasaki-like almost contact B-metric manifolds. It is proved that the Ricci tensor of such a soliton is the vertical component of both B-metrics multiplied by a constant. It is established that gradient almost Ricci-like solitons have constant soliton coefficients. Explicit examples of Lie groups as manifolds of dimensions 3 and 5 equipped with the structures studied are provided.

Ricci-like solitons with arbitrary potential and gradient almost Ricci-like solitons on Sasaki-like almost contact B-metric manifolds

The object of the present research is to study the δ-Lorentzian  Trans Sasakian manifolds addmitting the η-Einstein Solitons and gradient Ein stein soliton. It is shown that a symmetric second order covariant tensor in  a δ-Lorentzian Trans Sasakian manifold is a constant multiple of metric ten sor. Also an example of η-Einstein soliton in 3-diemsional δ-Lorentzian Trans  Sasakian manifold is provided in the region where δ-Lorentzian Trans Sasakian  manifold expanding.

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η-Einstein Solitons in δ- Lorentzian Trans Sasakian Manifolds

The object of the present research is to study generalized η-Riccisoliton on trans-Sasakian manifolds. It shows that a trans-Sasakian manifoldwhich also satisfies the generalized gradient η-Ricci soliton equation satisfiessome conditions, is necessarily Einstein manifold.

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Generalized η-Ricci Solitons in Trans Sasakian Manifolds

In this paper, we consider an η-Ricci soliton on the (LCS)n-manifolds (M,φ, ξ, η, g) satisfying certain curvature conditions likes: R(ξ,X) · S = 0 and W2(ξ,X) · S = 0. We show that on the (LCS)n-manifolds (M,φ, ξ, η, g), the existence of η-Ricci soliton implies that (M, g) is a quasi-Einstein. Further, we discuss the existence of Ricci solitons with the potential vector field ξ. In the end, we construct the non-trivial examples of η-Ricci solitons on the (LCS)n-manifolds.

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Some results of η-Ricci solitons on (LCS) n -manifolds

The aim of this paper is to study the -Ricci solitons in 3-dimensionaltrans-Sasakian manifolds

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K Venu

Papers

η-Ricci solitons in trans-Sasakian manifolds