WARPED PRODUCT SEMI-INVARIANT SUBMANIFOLDS IN LOCALLY DECOMPOSABLE RIEMANNIAN MANIFOLDS

Warped product semi-invariant submanifolds were defined and studied in locally Riemannian product manifolds in (K. Matsumoto, On sub- manifolds of locally product Riemannian manifolds, TRU Math. 18 (2), 145–157, 1982). In this paper, we go on to study warped product semi-invariant submanifolds in locally decomposable Riemannian man- ifolds. We prove several fundamental properties of warped product semi-invariant submanifolds, and establish a general inequality for a warped product semi-invariant submanifold in a locally decomposable Riemannian manifold. After that, we investigate warped product semi- invariant submanifolds in a locally decomposable Riemannian manifold which satisfy the equality case of the inequality and obtain some new results.