We consider a complex projective space with the almost complex structure J. Let M be a real submanifold of the space. If at each point x of M the tangent space T(M) satisfies JT(M)~ T(M), M is called an invariant submanifold under J. It is well known that an invariant submanifold is a complex submanifold. We consider the more general case that at each point x of M, T(M) and the normal space Nx(M ) satisfies condition that dim T(M)m JNx(M ) is independent of x. Such real submanifolds involve complex submanifolds as a special case.
The main purpose of the paper is to study relations between the dimension of T(M)c~JN(M) and the codimension of the real submanifold and to prove that, under some additional conditions, if dim T(M)~ JNx(M ) is less than that codimension there exists such a totally geodesic, complex submanifold M' that M ~ M'.
The method used here is the standard submersion method which is established by H. B. Lawson Jr [3]. Hence the first three sections of the paper are devoted to investigation of a certain class of submanifolds of an odddimensional sphere and then we push the results obtained on the sphere down to the complex projective space.
In ยง1, we establish some formulas which follow from the Sasakian structure of the sphere. In ยง2 we define a function f on M and show under certain condition f is constant and so its Laplacian is zero. An explicit formula for this Laplacian, furnished in ยง2, is used in ยง3 to establish codimension reducing results for submanifolds of spheres. We apply in ยง4, the theorem obtained in ยง3 to submanifolds of complex projective space.
l. SUBMANIFOLDS OF AN ODD-DIMENSIONAL SPHERE Let S "~-p be an odd-dimensional sphere of radius 1 and (q~, 4, q,g') the natural Sasakian structure on S n+p, where ~p, 4, ~7 denote an endomorphism of the tangent bundle T(S"+P), a vector field, a 1-form on S n+p, and g' the associated Riemannian metric to (~, ~., ~). Then they satisfy (p2X' = -X' + r/(X')~, q)~ = O, r/((pX') = O, r/(~_) = 1,
(1.1) '" x' "x' gt~ o ,(PY')=gt ,Y')-rI(X')rI(Y'), tI(X')=g(~,X'), for any vector fields X',Y'6T(S"+P). Since the structure (q~,~.,r/,g') is