Publ. Math. Debrecen 62/1-2 (2003), 213–225 Certain application of an integral formula to CR submanifold of complex projective space By MIRJANA DJORI ´ C (Belgrade) and MASAFUMI OKUMURA (Urawa) Abstract. Let M be an n-dimensional CR submanifold of CR dimension n-1 2 of complex projective space. In this case M is necessarily odd-dimensional and there exists a unit vector field ξ 1 normal to M such that JT (M ) ⊂ T (M ) ⊕ξ 1 . Under the assumption that ξ 1 is parallel with respect to the normal connection, we bring into use an integral formula which leads to an inequality between the Ricci tensor, the scalar curvature and the mean curvature of M . Using this inequality, we provide a sufficient condition for the submanifold M to be a tube over a totally geodesic complex subspace of P n+k 2 (C). 0. Introduction The study of hypersurfaces of complex projective space has been a fertile field for differential geometricians for many years now. Much of this work has involved finding sufficient conditions for a hypersurface to be one of the “standard examples”. However, contrary to the case of hypersurfaces of a Euclidean space where totally geodesic hypersurfaces and totally umbilical hypersurfaces characterize hyperplanes and hyperspheres, respectively, and to the case of a sphere as an ambient space where they characterize great and small spheres, respectively, in complex projective space there exist neither to- tally geodesic real hypersurfaces nor totally umbilical real hypersurfaces. Mathematics Subject Classification: 53C15, 53C25, 53C40. Key words and phrases: CR submanifold, complex projective space, Ricci tensor, scalar curvature, mean curvature.