Invariant submanifolds of real hypersurfaces of complex space forms

Abstract

In Remark 15.1 we recalled that real hypersurfaces of a complex manifold admit a naturally induced almost contact structure F ′ from the almost complex structure of the ambient manifold. In Theorem 21.3 we proved that if M is a complete n-dimensional CR submanifold of maximal CR dimension of a complex projective space $${\bf P}^{\frac{n+p}{2}}({\bf C})$$ satisfying the condition (21.1) then M is congruent to a geodesic hypersphere $$M_{0,k}^C$$ for $$k=\frac{n-1}{2}$$ , or to $$M(n,\theta)$$ , or there exists a geodesic hypersphere s of $${\bf P}^{\frac{n+p}{2}}({\bf C})$$ M such that is an invariant submanifold by the almost contact structure F of S. It is easy to check that for the geodesic hypersphere $$M_{0,k}^C$$ for $$k=\frac{n-1}{2}$$ , the following relation is satisfied