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 \({h(FX,Y ) - h(X, FY) = g(FX, Y )\eta, X,Y \in T(M), 0 \ne \eta \in {T^\perp}(M)}\), which has been considered in [2] and [3].