Involution Tensor And Product Manifold

A linear transformation a on a vector space is called a involution if it satisfies condi-tion σ2=I, where I is identity mapping. Let M be a smooth manifold and F is a (1,1)-type smooth tensor field on M. F is called involution tensor if F is involution on tangent space TpM for each point p∈M.We not only investigate local structure of (M,F), but also give a sufficient and necessary condition for a Riemannian manifold can be decomposed into two Riemannian manifolds.As a generalization, we also present a condition of which a Riemannian manifold can be decomposed into a Warped product.