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.
Citations
More filters
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
References
More filters
01 Jan 2015
TL;DR: The study of real hypersurfaces in complex projective space CP n and complex hyperbolic space CH n began at approximately the same time as Munzner's work on isoparametric hypersurface in spheres as discussed by the authors.
Abstract: The study of real hypersurfaces in complex projective space CP n and complex hyperbolic space CH n began at approximately the same time as Munzner’s work on isoparametric hypersurfaces in spheres. A key early work was Takagi’s classification [669] in 1973 of homogeneous real hypersurfaces in CP n . These hypersurfaces necessarily have constant principal curvatures, and they serve as model spaces for many subsequent classification theorems. Later Montiel [501] provided a similar list of standard examples in complex hyperbolic space CH n . In this chapter, we describe these examples of Takagi and Montiel in detail, and later we prove many important classification results involving them. We also study Hopf hypersurfaces, focal sets, parallel hypersurfaces and tubes using both standard techniques of submanifold geometry and the method of Jacobi fields.
228 citations
225 citations
"Real Hypersurfaces of Complex Space..." refers result in this paper
...For complex hyperbolic space, the analogous result was proved by Montiel [ 8 ]....
[...]
206 citations
"Real Hypersurfaces of Complex Space..." refers background in this paper
...Lemma 2.1. ([ 7 ]) If » is a principal curvature vector, then the corresponding principal curvature fi is locally constant....
[...]
...Lemma 2.2. ([ 7 ]) Assume that » is a principal curvature vector and the corresponding principal curvature is fi. If AX = ‚X for X ? », then we have (2‚ i fi)A`X = (fi‚ + 2c)`X....
[...]
TL;DR: In this paper, a real hypersurface of a complex hyperbolic space #x2102 was constructed and a principal circle bundle over it was constructed, which is a Lorentzian hypergraph of the anti-De Sitter space H12n+1.
Abstract: Given a real hypersurface of a complex hyperbolic space #x2102;?Hn,we construct a principal circle bundle over it which is a Lorentzian hypersurface of the anti-De Sitter space H12n+1.Relations between the respective second fundamental forms are obtained permitting us to classify a remarkable family of real hypersurfaces of ℂHn.
184 citations
"Real Hypersurfaces of Complex Space..." refers background in this paper
...Proposition 2.3. ([6], [ 9 ]) Let M, where n ‚ 2, be a real hypersurface in Mn(c) of constant holomorphic sectional curvature 4c 6= 0. Then `A = A` if and only if M is an open subset of the following: (i) a geodesic hypersphere, (ii) a tube over totally geodesic complex space form Mk(c), where 0 < k • ni1....
[...]
Abstract: Ruled real hypersurfaces of complex space forms are investigated by using the fact that such hypersurfaces can be constructed by moving a 1-codimensional complex totally geodesic submanifold of the ambient space along a curve Among other results, a classification of minimal ruled real hypersurfaces and an example of a homogeneous ruled real hypersurface are given
86 citations
"Real Hypersurfaces of Complex Space..." refers methods in this paper
...For more details on ruled real hypersurfaces, we refer to [2], [ 6 ], [7]....
[...]