Pseudo-parallel submanifolds of a space form (Q2769464)

Denoting the Riemann curvature tensor by \(R,\) a manifold is locally symmetric if \( \nabla R =0 \) and semi-symmetric if \( R\cdot R =0 .\) Various authors have now turned their attention to a further generalization, that of pseudo-symmetric manifolds. The defining condition for pseudo-symmetry is that for all tangent vectors \(X\) and \(Y\) one has \( R(X \wedge Y) = \phi (X\wedge Y) \cdot R \) for some smooth function \(\phi .\) In particular, one wants to understand submanifolds of space forms which possess these properties. NEWLINENEWLINENEWLINEA pseudo-parallel immersion \( M \hookrightarrow \overline{M} \) is one for which \( \overline{R} (X\wedge Y) \cdot \alpha = \phi (X\wedge Y) \cdot \alpha ,\) where \(\alpha \) is the second fundamental form. They study pseudo-parallel hypersurfaces of space forms, pseudo-parallel surfaces inside space forms, and other special cases. As an example of their geometrically appealing classification results, they show that a pseudo-parallel hypersurface of a space form is either quasi-umbilic or a cyclid of Dupin. They also give examples that illustrate the various objects defined and that demonstrate the relationships among them.