Hypersurfaces of a sphere with parallel shape operator

Part of the Developments in Mathematics book series (DEVM, volume 19)


In [53] P. J. Ryan considered hypersurfaces of real space forms and specifically, he gave a complete classification of hypersurfaces in the sphere which satisfy a certain condition. The condition that the shape operator is parallel is its special case. In this section we give the proof of this classification (in the specific case \(\nabla_XA=0\)) and furthermore, we show that the algebraic condition (13.5) on the shape operator implies that it is parallel.


Riemannian Manifold Orthonormal Basis Constant Curvature Shape Operator Complex Projective Space 


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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.University of BelgradeBelgradeSerbia
  2. 2.Saitama UniversitySaitamaJapan

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