Abstract.We characterize real hypersurfaces of type (A) and ruled real hypersurfaces in a complex projective space in terms of two φ-invariances of their shape operators, and give geometric meanings of these real hypersurfaces by observing their some geodesics.2010 Mathematics Subject Classification. Primary 53B25, Secondary 53C40, 53C22.
Introduction.The theory of Riemannian submanifolds in a Euclidean sphere is one of the most interesting objects in differential geometry. It is known that an isometric immersion f of a Kähler manifold M with Kähler structure J into a sphere has parallel second fundamental form σ if and only if σ is J-invariant, that is σ (JX, JY ) = σ (X, Y ) holds for each vector X, Y on M (Proposition 3).In this context, we consider a real hypersurface M 2n−1 in an n-dimensional complex projective space ރP n (c) of constant holomorphic sectional curvature c(> 0), furnished with the almost contact metric structure (φ, ξ, η, , ) on M induced from the Kähler structure J of the ambient space ރP n (c). In this case the structure tensor φ behaves on M similarly to a Kähler structure on a Kähler manifold, and on the other hand there exists no real hypersurface with parallel second fundamental form in ރP n (c). So, we introduce the following conditions concerning φ-invariances of the shape operator A of M.The shape operator A of M is called strongly φ-invariant if A satisfies