Vietnam Journal of Mathematics

, Volume 43, Issue 4, pp 705–723

# Submanifolds with Parallel Mean Curvature Vector Field in Product Spaces

• Zhong Hua Hou
• Wang-Hua Qiu
Article

## 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|2n|μ|2.

## Keywords

Simons type equation Parallel mean curvature Product spaces Gap theorems

## Mathematics Subject Classification (2010)

Primary 53C20 Secondary 52C42 53A10

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