Real hypersurfaces with constant principal curvatures in the complex hyperbolic plane

Berndt, Jürgen; Díaz-Ramos, José Carlos

doi:10.1090/s0002-9939-07-09012-0

Cited by 28 publications

(40 citation statements)

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“…Using Lemma 21.6 [14] we conclude that, under the assumptions of the Main Theorem, the shape operator with respect to a distinguished normal vector field has at most three distinct eigenvalues, which are constant. We mention here that Montiel [21] obtained the complete classification of all real hypersurfaces in complex hyperbolic spaces CH m , m ≥ 3 with two distinct constant principal curvatures, Berndt and Diaz-Ramos in [3] obtained this classification for three distinct constant principal curvatures, while in [4], these classifications have been obtained for the plane.…”

Section: Introductionmentioning

confidence: 94%

Certain condition on the second fundamental form of CR submanifolds of maximal CR dimension of complex hyperbolic space

Djorić

Okumura

2010

Ann Glob Anal Geom

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Section: Introductionmentioning

confidence: 94%

Certain condition on the second fundamental form of CR submanifolds of maximal CR dimension of complex hyperbolic space

Djorić

Okumura

2010

Ann Glob Anal Geom

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“…Proof (i) follows immediately from (6) and the fact that the only complex totally geodesic submanifolds of CH n are complex hyperbolic spaces.…”

Section: Propositionmentioning

confidence: 99%

“…Again, the second term can be calculated using (6). All in all this means (interchanging the roles of X and Y ) that…”

Section: If P ∈ M(r ) and Rh 3 Is The Totally Geodesic Real Hyperbolimentioning

confidence: 99%

“…Some time ago the first author demonstrated in [1] that for CH n the answer is yes within the class of Hopf hypersurfaces. For arbitrary real hypersurfaces, we obtained recently an affirmative answer in [6] in case of CH 2 , and for n ≥ 3 an affirmative answer was given in [13] for g ≤ 2 and in [5] for g = 3, where g is the number of distinct principal curvatures. It is a well-established fact that the geometry of hypersurfaces is intimately related to the geometry of their focal sets, (see e.g.…”

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Homogeneous hypersurfaces in complex hyperbolic spaces

Berndt

Díaz-Ramos

2008

Geom Dedicata

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“…Berndt and Ramos (2007) classified real hypersurfaces with constant principal curvatures in the complex hyperbolic plane.…”

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Non-existence of real hypersurfaces equipped with recurrent structure Jacobi operator in nonflat complex planes

Theofanidis

Xenos

2011

Beitr Algebra Geom

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The aim of the present paper is to prove the nonexistence of real hypersurfaces with D-recurrent structure Jacobi operator, in non-flat complex planes. Our results complement work of several other authors who worked in CP n and CH n for n ≥ 3.Keywords Real hypersurface · Structure Jacobi operator · Recurrent tensor field Mathematics Subject Classification (2010) 53C40 · 53D15 IntroductionAn n-dimensional Kaehlerian manifold of constant holomorphic sectional curvature c is called complex space form, which is denoted by M n (c). The complete and simply connected complex space form is complex analytically isometric to a projective space CP n if c > 0, a hyperbolic space CH n if c < 0, or a Euclidean space C n if c = 0. The induced almost contact metric structure of a real hypersurface M of M n (c) will be denoted by (φ, ξ, η, g). The vector field ξ is defined by ξ = −J N where J is the complex structure of M n (c) and N is a locally defined unit normal vector field.Real hypersurfaces in CP n which are homogeneous, were classified by Takagi (1973). Berndt (1989) classified real hypersurfaces with principal structure vector Th. Theofanidis · Ph. J. Xenos (

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Real hypersurfaces with constant principal curvatures in the complex hyperbolic plane

Abstract: Abstract. We classify real hypersurfaces with constant principal curvatures in the complex hyperbolic plane. It follows from this classification that all of them are open parts of homogeneous ones.

Cited by 28 publications

References 10 publications

Certain condition on the second fundamental form of CR submanifolds of maximal CR dimension of complex hyperbolic space

Certain condition on the second fundamental form of CR submanifolds of maximal CR dimension of complex hyperbolic space

Homogeneous hypersurfaces in complex hyperbolic spaces

Non-existence of real hypersurfaces equipped with recurrent structure Jacobi operator in nonflat complex planes

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