Commuting Conditions of the k-th Cho operator with the structure Jacobi operator of real hypersurfaces in complex space forms

Konstantina Panagiotidou; Juan de Dios Pérez

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Displaying similar documents to “Commuting Conditions of the k-th Cho operator with the structure Jacobi operator of real hypersurfaces in complex space forms”

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The aim of the present paper is to classify real hypersurfaces with pseudo-𝔻-parallel structure Jacobi operator, in non-flat complex space forms.

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We prove the non-existence of real hypersurfaces in complex two-plane Grassmannians whose normal Jacobi operator is of Codazzi type.

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We introduce the new notion of pseudo- $𝔻$ -parallel real hypersurfaces in a complex projective space as real hypersurfaces satisfying a condition about the covariant derivative of the structure Jacobi operator in any direction of the maximal holomorphic distribution. This condition generalizes parallelness of the structure Jacobi operator. We classify this type of real hypersurfaces.

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Considering the notion of Jacobi type vector fields for a real hypersurface in a complex two-plane Grassmannian, we prove that if a structure vector field is of Jacobi type it is Killing. As a consequence we classify real hypersurfaces whose structure vector field is of Jacobi type.

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Regarding the generalized Tanaka-Webster connection, we considered a new notion of $𝔇^{⊥}$ -parallel structure Jacobi operator for a real hypersurface in a complex two-plane Grassmannian G 2(ℂm+2) and proved that a real hypersurface in G 2(ℂm+2) with generalized Tanaka-Webster $𝔇^{⊥}$ -parallel structure Jacobi operator is locally congruent to an open part of a tube around a totally geodesic quaternionic projective space ℍP n in G 2(ℂm+2), where m = 2n.

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Bulletin of the Polish Academy of Sciences. Mathematics

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We characterize totally η-umbilic real hypersurfaces in a nonflat complex space form M̃ₙ(c) (= ℂPⁿ(c) or ℂHⁿ(c)) and a real hypersurface of type (A₂) of radius π/(2√c) in ℂPⁿ(c) by observing the shape of some geodesics on those real hypersurfaces as curves in the ambient manifolds (Theorems 1 and 2).