Abstract: The present paper deals with the compact submanifolds with paralled me an curvature in locally symmetric δ-Pinching Riemannian manifold, on the basis of which some theorems about the square of modulus-length, the pinch of section curvature and the reduction of codimension are obtained. The similar problems in the space of constant curvature are popularized in the space of local symmetry.

Key words: locally symmetric, parallel mean curvature, totally geodesic