Quasi-conformally flat submanifolds with parallel mean curvature vector

This paper studied the conditions such that an n-dimensional complete space-like submanifold M with a parallel mean curvature vector field in an indefinite space form S
n+p
(c)(p ≥ 2, n≥ 3) is a totally umbilical submanifold. By means of Laplacian estimation and the choice of a diagonal frame field, the following theorem is proved: if Mis quasi-conformally flat and H
²
≤ c, Ric
$(M) \leq \delta_{M} < [(n-1)-(1)/(t)](c-H^{2})$
, then M is a totally umbilical submanifold.