The paper discusses the relations between parallel vector fields and Jacobi fields in a complete Riemannian manifold, and proves that there exists infinite closed geodesics in a local symmetric space which is orientable, compact, nonsimply connected and with nonnegative sectional curvature.

The properties of both parallelogram identity vector and isometric reflection vector in a real Banach space were studied.

In the first part, we consider the compact totally real pseudoumbilical submanifolds Mn that have nonzero parallel normalized mean curvature vector in complex space forms M(n+p)(c).

Triangulation mesh generation of scatter points based on normal field;