Real Hypersurfaces of Complex Space Forms in Terms of Ricci $*$-Tensor
01 Dec 2002-Tokyo Journal of Mathematics (Publication Committee for the Tokyo Journal of Mathematics)-Vol. 25, Iss: 2, pp 473-483
TL;DR: In this article, the authors classified the $*$-Einstein real hypersurfaces in complex space forms such that the structure vector is a principal curvature vector and the principal curvatures of the hypersurface can be computed with the K\"ahler metric.
Abstract: It is known that there are no Einstein real hypersurfaces in complex space forms equipped
with the K\"ahler metric. In the present paper we classified the $*$-Einstein real
hypersurfaces $M$ in complex space forms $M_{n}(c)$ and such that the structure vector is
a principal curvature vector.
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TL;DR: In this paper, the notion of *-Ricci soliton is introduced and real hypersurfaces in non-flat complex space forms admitting a *-ricci s soliton with potential vector field being the structure vector field.
53 citations
TL;DR: In this article, it was shown that if a complete Sasakian metric is an almost gradient ∗-Ricci soliton, then it is either positive or null-Sakian.
Abstract: We prove that if a Sasakian metric is a ∗-Ricci Soliton, then it is either positive Sasakian, or null-Sasakian. Next, we prove that if a complete Sasakian metric is an almost gradient ∗-Ricci Solit...
37 citations
TL;DR: In this paper, the authors considered the case of *-Ricci soliton in the framework of a Kenmotsu manifold and proved that soliton constant λ is zero.
Abstract: Abstract In this paper, we consider *-Ricci soliton in the frame-work of Kenmotsu manifolds. First, we prove that if (M, g) is a Kenmotsu manifold and g is a *-Ricci soliton, then soliton constant λ is zero. For 3-dimensional case, if M admits a *-Ricci soliton, then we show that M is of constant sectional curvature –1. Next, we show that if M admits a *-Ricci soliton whose potential vector field is collinear with the characteristic vector field ξ, then M is Einstein and soliton vector field is equal to ξ. Finally, we prove that if g is a gradient almost *-Ricci soliton, then either M is Einstein or the potential vector field is collinear with the characteristic vector field on an open set of M. We verify our result by constructing examples for both *-Ricci soliton and gradient almost *-Ricci soliton.
29 citations
TL;DR: In this paper, the authors studied ∗-η-Ricci soliton on Sasakian manifolds and obtained some significant curvature properties on the manifold admitting the soliton.
Abstract: In this paper we study ∗-η-Ricci soliton on Sasakian manifolds. Here, we have discussed some curvature properties on Sasakian manifold admitting ∗-η-Ricci soliton. We have obtained some significant...
25 citations
TL;DR: In this article, it was shown that if the metric g of M is a *-Ricci soliton, then either M is locally isometric to the product ℍn+1(−4)×ℝn or the potential vector field is strict infinitesimal contact transformation.
Abstract: Abstract Let (M, g) be a non-Kenmotsu (κ, μ)′-almost Kenmotsu manifold of dimension 2n + 1. In this paper, we prove that if the metric g of M is a *-Ricci soliton, then either M is locally isometric to the product ℍn+1(−4)×ℝn or the potential vector field is strict infinitesimal contact transformation. Moreover, two concrete examples of (κ, μ)′-almost Kenmotsu 3-manifolds admitting a Killing vector field and strict infinitesimal contact transformation are given.
24 citations
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84 citations
"Real Hypersurfaces of Complex Space..." refers background in this paper
...Proof. We know that we may write the shape operator A of a ruled real hypersurface M in Mn(c) [ 4 ]:...
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"Real Hypersurfaces of Complex Space..." refers background in this paper
...in complex hyperbolic space has constant principal curvatures fi and ‚. Such real hypersurfaces are completely classifled by J. Berndt [ 1 ]....
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