Real Hypersurfaces in Complex Two-Plane Grassmannians
TL;DR: In this paper, the complex two-plane Grassmannian with both a Kahler and a quaternionic Kahler structure was applied to the normal bundle of a real hypersurface M in G
Abstract: The complex two-plane Grassmannian G
2(C
m+2
in equipped with both a Kahler and a quaternionic Kahler structure. By applying these two structures to the normal bundle of a real hypersurface M in G
2(C
m+2
one gets a one- and a three-dimensional distribution on M. We classify all real hypersurfaces M in G
2
C
m+2
, m≥3, for which these two distributions are invariant under the shape operator of M.
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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 or methods from "Real Hypersurfaces in Complex Two-P..."
...Then by virtue of theorem due to Berndt and Suh [3], M is locally congruent to a real hypersurfaces of type (A), that is, M is a tube over a totally geodesic G2(Cm+1) in G2(Cm+2)....
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...Now let us introduce Proposition B due to Berndt and the second author [3] as follows: Proposition B....
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...Riemannian geometry of G2(C) In this section we summarize basic material about G2(C), for details we refer to [1], [2], [3] and [4]....
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...By using such kinds of geometric conditions Berndt and the second author [3] have proved the following: Theorem A....
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...Related to hypersurfaces of type (A) in Theorem A we introduce another Proposition due to Berndt and the second author [3] as follows: Proposition C....
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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
TL;DR: In this article, the authors give a characterization of real hyper-surface of type B, that is, a tube over a totally geodesic QP n in complex two-plane Grassmannians G2(C m+2 ), where m = 2n, with the Reeb vec- tor belonging to the distribution D, where D denotes a subdistribution in the tangent space such that TxM = D'D? for any point x 2 M and D? = Span{»1,»2,»3 }.
Abstract: In this paper we give a new characterization of real hyper- surfaces of type B, that is, a tube over a totally geodesicQP n in complex two-plane Grassmannians G2(C m+2 ), where m = 2n, with the Reeb vec- tor » belonging to the distribution D, where D denotes a subdistribution in the tangent space TxM such that TxM = D'D? for any point x 2 M and D? = Span{»1,»2,»3 }.
72 citations
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
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
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TL;DR: In this paper, the location of the focal points of a real submanifold is defined in terms of its second fundamental form, and the rank of a focal map onto a sheet of focal points corresponding to a principal curvature is computed using the Codazzi equation.
Abstract: Let M be a real submanifold of CPm, and let J denote the complex structure. We begin by finding a formula for the location of the focal points of M in terms of its second fundamental form. This takes a particularly tractable form when M is a complex submanifold or a real hypersurface on which Ji is a principal vector for each unit normal ( to M. The rank of the focal map onto a sheet of the focal set of M is also computed in terms of the second fundamental form. In the case of a real hypersurface on which JE is principal with corresponding principal curvature ,u, if the map onto a sheet of the focal set corresponding to ,u has constant rank, then that sheet is a complex submanifold over which M is a tube of constant radius (Theorem 1). The other sheets of the focal set of such a hypersurface are given a real manifold structure in Theorem 2. These results are then employed as major tools in obtaining two classifications of real hypersurfaces in cPm. First, there are no totally umbilic real hypersurfaces in cPm, but we show: THEOREM 3. Let M be a connected real hypersurface in CPm, m > 3, with at most two distinct principal curvatures at each point. Then M is an open subset of a geodesic hypersphere. Secondly, we show that there are no Einstein real hypersurfaces in cPm and characterize the geodesic hyperspheres and two other classes of hypersurfaces in terms of a slightly less stringent requirement on the Ricci tensor in Theorem 4. One of the first results in the geometry of submanifolds is that an umbilic hypersurface M in Euclidean space must be an open subset of a hyperplane or sphere. The proof goes as follows: assume that the shape operator is a scalar multiple of the identity, A = AX, and use the Codazzi equation to show that X is constant. Then either X = 0, in which case M lies on a hyperplane, or the focal points fx(x) = x + (l/X)t, ( the unit normal, all coincide, and M lies on the sphere of radius 1 /X centered at the unique focal point. This simple idea suggests a plan of attack for classifying hypersurfaces in terms of the nature of the principal curvatures. Under fairly general conditions, the set of focal points corresponding to a principal curvature X can be given a differentiable Received by the editors December 19, 1980. Presented to the Society at its annual meeting in San Francisco, January 10, 1981. 1980 Mathematics Subject Classificatiom Primary 53B25, 53C40.
296 citations
"Real Hypersurfaces in Complex Two-P..." refers background in this paper
...proved in [ 4 ] that these tubes are essentially characterized by this feature....
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Journal Article•
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
"Real Hypersurfaces in Complex Two-P..." refers background in this paper
...The first author proved in [ 2 ] that also every tube around a totally geodesic CP m in HP m satisfies AJO? MU JO? MU, and that there are no other ones....
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