Submanifolds with Parallel Mean Curvature Vector Field in Product Spaces

Abstract

By Moving Frame Method, we firstly derive some Simons type equations for an n-dimensional submanifold M n with parallel mean curvature vector field μ in \(M^{m}(c)\times \mathbb {R}\), where M m (c) is an m-dimensional space form of constant sectional curvature c and obtain a lower bound of the squared norm of the covariant differential of the second fundamental form h of M n . Then, we use these results to prove some gap theorems on |h|2 and |ϕ|2=|h|2−n|μ|2.