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Keywords = Levi-flat hypersurfaces

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13 pages, 286 KB  
Article
Nonexistence of Homogeneous Levi-Flat Hypersurfaces in CP2
by Abdel Rahman Al-Abdallah
Mathematics 2025, 13(17), 2742; https://doi.org/10.3390/math13172742 - 26 Aug 2025
Viewed by 623
Abstract
We investigate the longstanding question of whether compact Levi-flat hypersurfaces exist in the complex projective plane CP2. While the nonexistence of closed real-analytic Levi-flat hypersurfaces in CPn for n>2 is well known, the case n=2 remains [...] Read more.
We investigate the longstanding question of whether compact Levi-flat hypersurfaces exist in the complex projective plane CP2. While the nonexistence of closed real-analytic Levi-flat hypersurfaces in CPn for n>2 is well known, the case n=2 remains open. By combining techniques from the classification of homogeneous CR-manifolds with projective foliation geometry, we prove that no homogeneous Levi-flat hypersurfaces exist in CP2, thus partially resolving the problem under natural symmetry assumptions. Full article
(This article belongs to the Special Issue Advances in Differential Geometry and Its Applications, 2nd Edition)
12 pages, 285 KB  
Article
A Characterization of Ruled Real Hypersurfaces in Non-Flat Complex Space Forms
by George Kaimakamis, Konstantina Panagiotidou and Juan de Dios Pérez
Mathematics 2020, 8(4), 642; https://doi.org/10.3390/math8040642 - 21 Apr 2020
Cited by 1 | Viewed by 2463
Abstract
The Levi-Civita connection and the k-th generalized Tanaka-Webster connection are defined on a real hypersurface M in a non-flat complex space form. For any nonnull constant k and any vector field X tangent to M the k-th Cho operator [...] Read more.
The Levi-Civita connection and the k-th generalized Tanaka-Webster connection are defined on a real hypersurface M in a non-flat complex space form. For any nonnull constant k and any vector field X tangent to M the k-th Cho operator F X ( k ) is defined and is related to both connections. If X belongs to the maximal holomorphic distribution D on M, the corresponding operator does not depend on k and is denoted by F X and called Cho operator. In this paper, real hypersurfaces in non-flat space forms such that F X S = S F X , where S denotes the Ricci tensor of M and a further condition is satisfied, are classified. Full article
(This article belongs to the Special Issue Differential Geometry: Theory and Applications)
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