Invariant Submanifolds of Real Hypersurfaces of Complex Manifolds

Abstract

Real hypersurfaces of a complex manifold admit a naturally induced almost contact structure F′ from the almost complex structure of the ambient manifold. We prove that for any F′-invariant submanifold M of a geodesic hypersphere in a non-flat complex space form and of a horosphere in a complex hyperbolic space, its second fundamental form h satisfies the condition h(FX,Y)-h(X, FY)=g(FX,Y)η,X,Y ∈ T{box drawings light up and horizontal}(M), 0 ≠ ∈ T(M), which has been considered in [2] and [3]. © 2010 Springer Basel AG.