On real hypersurfaces of a complex space form
01 Jan 1995-Vol. 2, pp 1-14
About: The article was published on 1995-01-01 and is currently open access. It has received 77 citations till now. The article focuses on the topics: Complex space.
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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
TL;DR: In this article, it was shown that a real hypersurface in a non-flat complex space form does not admit a Ricci soliton whose potential vector field is the Reeb vector field.
Abstract: We prove that a real hypersurface in a non-flat complex space form does not admit a Ricci soliton whose potential vector field is the Reeb vector field. Moreover, we classify a real hypersurface admitting so-called “$\eta$-Ricci soliton” in a non-flat complex space form.
154 citations
TL;DR: In this article, the Ricci tensor of a real hypersurface M in complex two-plane Grassmannians G 2 (C m +2 ) was derived from the equation of Gauss.
Abstract: . In this paper, flrst we introduce the full expression of thecurvature tensor of a real hypersurface M in complex two-plane Grass-mannians G 2 (C m +2 ) from the equation of Gauss and derive a new formulafor the Ricci tensor of M in G 2 (C m +2 ). Next we prove that there do notexist any Hopf real hypersurfaces in complex two-plane Grassmannians G 2 (C m +2 ) with parallel and commuting Ricci tensor. Finally we showthat there do not exist any Einstein Hopf hypersurfaces in G 2 (C m +2 ). IntroductionIn the geometry of real hypersurfaces in complex space forms or in quater-nionic space forms it can be easily checked that there do not exist any realhypersurfaces with parallel shape operator A by virtue of the equation of Co-dazzi.But if we consider a real hypersurface with parallel Ricci tensor S in suchspace forms, the proof of its non-existence is not so easy. In the class of Hopfhypersurfaces Kimura [7] has asserted that there do not exist any real hyper-surfaces in a complex projective space C
98 citations
Journal Article•
TL;DR: An n-dimensional complex space form (M_n(c)) is a Kaehlerian manifold of constant holomorphic sectional curvature c as discussed by the authors, where c is the number of vertices in the manifold.
Abstract: An n-dimensional complex space form $M_n(c)$ is a Kaehlerian manifold of constant holomorphic sectional curvature c. As is well known, complete and simply connected complex space forms are a complex projective space $P_n C$, a complex Euclidean space $C_n$ or a complex hyperbolic space $H_n C$ according as c > 0, c = 0 or c
43 citations
TL;DR: In this article, the Ricci curvature tensor for the generalized Tanaka-Webster connection on the Levi subbundle is proportional to the Levi form, and a classification of pseudo-Einstein Hopf-hypersurfaces in a non-flat complex space form is given.
Abstract: We introduce the pseudo-Einstein structure on real hypersurfaces in a K\"ahlerian manifold, namely, the Ricci curvature tensor for the generalized Tanaka-Webster connection (restricted) on the Levi subbundle $D$ is proportional to the Levi form. In particular, we give a classification of pseudo-Einstein Hopf-hypersurfaces in a non-flat complex space form.
32 citations