Real hypersurfaces of quaternionic projective space satisfying ▽UiR = 0
TL;DR: In this paper, the authors classify real hypersurfaces of quaternionic projective space whose curvature tensor is parallel in the direction of a 3D distribution, and they show that there are real hypersurifaces with parallel curvature vectors in quaternion projective spaces.
Abstract: It is known that there do not exist real hypersurfaces with parallel curvature tensor in quaternionic projective spaces. In this paper we classify real hypersurfaces of quaternionic projective space whose curvature tensor is parallel in the direction of certain 3-dimensional distribution.
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01 Jan 2015TL;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.
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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
Cites background from "Real hypersurfaces of quaternionic ..."
...Then the formula concerned with the Ricci tensor mentioned above is not so simple if we consider a real hypersurface in complex two-plane Grassmannians G2(C) (See [3], [4], [10], [11], [12] and [13])....
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TL;DR: In this article, a characterization of real hypersurfaces of type B is given, that is, a tube over a totally real totally geodesic in complex two-plane Grassmannians with the shape operator A satisfying Af + fA = kf, k is non-zero constant.
Abstract: In this paper we give a characterization of real hypersurfaces of type B, that is, a tube over a totally real totally geodesic in complex two-plane Grassmannians with the shape operator A satisfying Af + fA = kf, k is non-zero constant, for the structure tensor f
81 citations
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TL;DR: It is shown that the real hypersurfaces with isometric Reeb flow in complex hyperbolic two-plane Grassmannians SU"2","m/S(U"[email protected]?U"m), m>=2.
70 citations
Cites background from "Real hypersurfaces of quaternionic ..."
...And some fundamental formulas related to the Codazzi and Gauss equations from the curvature tensor of complex hyperbolic two-plane Grassmannian SU2,m/S(U2·Um) will be recalled (see [6], [7], and [8])....
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TL;DR: In this article, the authors introduced the notion of parallel Ricci tensor for real hypersurfaces in the complex quadric Qm=SOm+2/SOmSO2.
64 citations
References
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TL;DR: Alekseevskii et al. as mentioned in this paper studied quaternion Kahlerian manifolds by using tensor calculus and obtained many interesting results. But they did not define a manifold as a Riemannian manifold which admits a bundle V of tensors of type (1, 1) having some properties.
Abstract: A quaternion Kahlerian manifold is defined as a Riemannian manifold whose holonomy group is a subgroup of Sp(m) Sp(l) . Recently, several authors (Alekseevskii [1], [2], Gray [3], Ishihara [4], Ishihara and Konishi [5], Krainse [6] and Wolf [10]) have studied quaternion Kahlerian manifolds and obtained many interesting results. In the present paper, we shall study those manifolds by using tensor calculus. To do so, it is rather convinient to define a quaternion Kahlerian manifold as a Riemannian manifold which admits a bundle V of tensors of type (1,1) having some properties. The bundle V is 3-dimensional as a vector bundle and admits an algebraic structure which is isomorphic to that of pure imaginary quaternions. In § 1, we define quaternion Kahlerian manifolds in our fashion and give some results proved in [6]. § 2 is devoted to the establishment of some formulas required in the following sections. In § 3, it is proved among some other theorems that any quaternion Kahlerian manifold is an Einstein space (Alekseevskii [1]). We prove in § 4 that a quaternion Kahlerian manifold, which is of constant curvature or conformally flat, is of zero curvature, if the manifold is of dimension > 8 . In §5, we define β-sectional curvatures and determine the form of the curvature tensor of a quaternion Kahlerian manifold when it has constant β-sectional curvature (See Alekseevskii [1]). Manifolds, mappings and geometric objects under discussion are assumed to be differentiable and of class C°°. The indices h, i, /, k, I, p, q, r, s, t, u, v run over the range {1, , n}, and the summation convention will be used with respect to this system of indices.
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TL;DR: In this paper, it was shown that in non-Euclidean spaces of constant holomorphic sectional curvature the curvature-adapted (real) hypersurfaces are exactly the Hopf hypersurface.
Abstract: Obviously, every totally umbilical hypersurface of a Riemannian manifold is curvature-adapted. In spaces of constant sectional curvature every hypersurface is curvature-adapted. But in other ambient spaces our definition is restrictive. For example, in non-Euclidean spaces of constant holomorphic sectional curvature the curvature-adapted (real) hypersurfaces are exactly the Hopf hypersurfaces (see [3] for the notion of Hopf hypersurfaces). In locally Symmetrie spaces it turns out that for the investigation of the geometry of curvature-adapted hypersurfaces Jacobi field theory may be very useful ( s can be seen in section 5).
113 citations
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TL;DR: In this article, a systematic study of real hypersurfaces of quaternionic projective space using focal set theory was made, and the Ricci tensor of such hypersurface was studied.
Abstract: This paper is devoted to make a systematic study of real hypersurfaces of quaternionic projective space using focal set theory. We obtain three types of such real hypersurfaces. Two of them are known. Third type is new and in its study the first example of proper quaternion CR-submanifold appears. We study real hypersurfaces with constant principal curvatures and classify such hypersurfaces with at most two distinct principal curvatures. Finally we study the Ricci tensor of a real hypersurface of quaternionic projective space and classify pseudo-Einstein, almost-Einstein and Einstein real hypersurfaces.
93 citations
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TL;DR: In this paper, real hypersurfaces of quaternionic projective space satisfying the following properties were classified: 1, 2, 3.1.2, 4.3.
Abstract: We classify real hypersurfaces of quaternionic projective space satisfying
$$
abla _{U_i } A = 0$$
, i=1,2,3.
25 citations