Let M be a submanifold of a Riemannian manifold ]fir and P an endomorphism of the tangent bundle T M of M. If F T x M 3_T ,M for each point x¢ M, we say that M is an anti-invariant submanifold of M under F. If P is an almost complex structure, then such M are usually called totally real submanifolds and have been studied by various authors [1, 3, 5-7]. The purpose of the present paper is to study the case, in which P is the endomorphism q) of an almost contact metric structure (~p, ~, r/, 9) on AT1, in particular, that of a Sasakian structure (see § 2). The authors will study the cases in which the structure vector field ~ is either tangent to M (see § 4) or normal to M (see § 5). In both cases the computation of the Laplacian of the square of the norm of the second fundamental form of M in AS1 plays an important role. The Theorems 8, 15, 16 present typical examples of our main results, which say, roughly speaking, that compact, minimal, anti-invariant submanifolds M of a Sasakian space form ~/, which are of not too large relative curvature (measured in terms of the norm I[cq[ of the second fundamental form ~ of M in .g/) have to be already totally geodesic in hT/.