CR-HYPERSURFACES OF COMPLEX PROJECTIVE SPACE

We consider compact n-dimensional minimal foliate CR-real submanifolds of a complex projective space. We show that these submanifolds are great circles on a 2-dimensional sphere provided that the square of the length of the second fundamental form is less than or equal to n-1.

Now let / be the complex projective space, which is a Kaehler manifold with constant holomorphic sectional curvature 4. Let g be the Hermitian metric tensor field of/.Suppose that M is an n-dimensional CR-hypersurface of .We denote by the same g the Riemannian metric tensor field induced on M from that of/.Let V, , X7 be the Riemannian connections on M, / and the normal bundle respectively.Then we have Gauss formula and Weingarten formula; where h(X,Y) and ANX are the second fundamental forms which are related by (2.1) where X and Y are vector fields on M.
Let H 1 (trace h) be the mean curvature vector.Then M is said to be minimal if H 0.
A CR-submanifold is said to be mixed foliate if (a) the holomorphic distribution D is integrable.
(b) h(X,O 0 for X E D and . For mixed foliate submanifolds of a complex spa:e form .(c) (i.e., a Kaehler manifold of constant holomorphic sectional curvature c), the following result is well known THEOREM 2.[3] If M is a mixed foliate proper CR-submanifold of a complex space form .(c), then we have c < 0.

C/-HYPERSURFACES OF A COMPLEX PROJECTIVE SPACE.
We consider an n-dimensional proper CR-hypersurface M of a complex projective space .
NOTE: It has been pointed out to us that the result in this theorem might be in conflict with Proposition 2.3 of Maeda, Y., "On real hypersurfaces of a complex projective space," J. Math.Soc.Japan, Vol. 28, No. 3.3 (1976), 529-540.We could not detect any mistakes in our proof, but we shall investigate this point later.