Some Problems on Ruled Hypersurfaces in Nonflat Complex Space Forms
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Some Problems on Ruled Hypersurfaces in Nonflat Complex Space Forms Olga P´erez-Barral Abstract. We study ruled real hypersurfaces whose shape operators have constant squared norm in nonflat complex space forms. In particular, we prove the nonexistence of such hypersurfaces in the projective case. We also show that biharmonic ruled real hypersurfaces in nonflat complex space forms are minimal, which provides their classification due to a known result of Lohnherr and Reckziegel. Mathematics Subject Classification. 53B25, 53C42, 53C55. Keywords. Complex projective space, complex hyperbolic space, ruled hypersurface, minimal hypersurface, strongly 2-Hopf hypersurface, biharmonic hypersurface.
1. Introduction A ruled real hypersurface in a nonflat complex space form, that is, in a complex projective or hyperbolic space, CP n or CH n , is a submanifold of real codimension one which is foliated by totally geodesic complex hypersurfaces of CP n or CH n . Ruled hypersurfaces in nonflat complex space forms constitute a very large class of real hypersurfaces. It becomes then an interesting problem to classify these objects under some additional geometric properties. For example, Lohnherr and Reckziegel classified ruled minimal hypersurfaces in nonflat complex space forms into three classes [8]: Kimura type hypersurfaces in CP n or CH n , bisectors in CH n and Lohnherr hypersurfaces in CH n . Moreover, they proved that Lohnherr hypersurfaces in CH n are the only complete ruled hypersurfaces with constant principal curvatures in nonflat complex space forms [8]. The author acknowledges support by projects MTM2016-75897-P (AEI/FEDER, Spain), PID2019105138GB-C21 (AEI/FEDER, Spain), ED431F 2020/04 (Xunta de Galicia, Spain), and ED431C 2019/10 (Xunta de Galicia, Spain), and by a research grant under the Ram´ on y Cajal project RYC-2017-22490 (AEI/FSE, Spain). 0123456789().: V,-vol
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Results Math
Another important notion in the context of real hypersurfaces is that of Hopf hypersurface, which is defined as a real hypersurface whose Reeb vector field is an eigenvector of the shape operator at every point. Ruled hypersurfaces in nonflat complex space forms are never Hopf; indeed, the smallest tangent distribution invariant under the shape operator and containing the Reeb vector field has rank two. In particular, the minimal ruled hypersurfaces mentioned above have an additional property, which has been introduced in [4]: they are strongly 2-Hopf, that is, the smallest distribution invariant under the shape operator and containing the Reeb vector field is integrable and has rank two, and the principal curvatures associated with the principal directions defining such distribution are constant along its integral submanifolds. This concept is also important since it characterizes, at least in the complex projective and hyperbolic planes CP 2 and CH 2 , the real hypersurfaces of cohomogeneity one that can be constructed as union of principal orbits of a polar action of cohomogeneity two
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