Abstract
We prove the non-existence of Hopf real hypersurfaces in complex two-plane Grassmannians whose Jacobi operators corresponding to the directions in the distribution D are of Codazzi type if they satisfy a further condition. We obtain that that they must be either of type (A) or of type (B) (see [1]), but no one of these satisfies our condition. As a consequence, we obtain the non-existence of Hopf real hypersurfaces in such ambient spaces whose Jacobi operators corresponding to D -directions are parallel with the same further condition.
Highlights
The geometry of real hypersurfaces in complex space forms or in quaternionic space forms is one of interesting parts in the field of differential geometry
We prove the non-existence of Hopf real hypersurfaces in complex two-plane Grassmannians whose Jacobi operators corresponding to the directions in the distribution D are of Codazzi type if they satisfy a further condition
= 1, 2,3, are -component parallel if the disof the Reeb vector field is invariant by the shape operator
Summary
The geometry of real hypersurfaces in complex space forms or in quaternionic space forms is one of interesting parts in the field of differential geometry. We prove the non-existence of Hopf real hypersurfaces in complex two-plane Grassmannians whose Jacobi operators corresponding to the directions in the distribution D are of Codazzi type if they satisfy a further condition. We obtain the non-existence of Hopf real hypersurfaces in such ambient spaces whose Jacobi operators corresponding to D -directions are parallel with the same further condition.
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