SUBMANIFOLDS IN A RIEMANNIAN PRODUCT MANIFOLD

This paper studies invariant submanifolds and CR submanifolds in a Riem-annian product manifold of two Sasakian manifold, and obtains the two main results as follows:I) Let N2m+1 and W2n+1 be Sasakian manifolds and M be a complete submanifolds in a Riemannian product manifold N=N2m+1×N2n+1. If M is totally umbilical,a) M is j-invariant; b) M is Φ-invariant.II) Let M, (i=1,2) be a contact CR submanifold of Sasakian manifold Ni with respect to the distribution Di. If , then M=M1×M2 is a CR submanifold of Hermitian manifold N=N1×N2 with respect to distribution D=D1×D2. Conversely, if M=M1×M2 is a CR submanifold of N=N1×N2 with respect to distribution D=D1×D2 and , then Mi(i= 1,2) is a contact CR submanifold of Ni with respect to distribution Di.