共形空间${mathbb Q}^n_s$中的正则Blaschke拟全脐子流形

[Nie C X,Wu C X,Regular submanifolds in the conformal space Q_p~n,ChinAnn Math,2012,33B(5):695-714]中研究了共形空间Q_s~n中的正则子流形,并引入了共形空间Q_s~n中的子流形理论.本文作者将分类共形空间Q_s~n中的Blaschke拟全脐子流形,证明伪Riemann空间形式中具有常数量曲率和平行的平均曲率向量场的正则子流形是共形空间中的Blaschke拟全脐子流形;反之,共形空间中的Blaschke拟全脐子流形共形等价于伪Riemann空间形式中具有常数量曲率和平行的平均曲率向量场的正则子流形.这一结论可看作是共形空间Q_s~n中共形迷向子流形分类定理的推广.

Regular Blaschke Quasi-umbilical Submanifolds in the Conformal Space ${mathbb Q}^n_s$

In [Nie C X, Wu C X, Regular submanifolds in the conformal space ${mathbb Q}^n_p$, {it Chin Ann Math}, 2012, 33B(5):695--714],the authors studied the regular submanifolds in the conformal space ${mathbb Q}^n_s$ and introduced the submanifold theory in the conformal space ${mathbb Q}^n_s$. This paper classifies the Blaschke quasi-umbilical submanifolds in the conformal space ${mathbb Q}^n_s$. It is proved that regular submanifolds in pseudo-Riemann space forms with constant scalar curvature and parallel mean curvature vector field are Blaschke quasi-umbilical submanifolds in the conformal space, and that any Blaschke quasi-umbilical submanifold in the conformal space is conformal equivalent to a regular submanifold withconstant scalarcurvature and parallel mean curvature vector field in pseudo-Riemann space forms.These results may be regarded as an extension of the classification of the conformal isotropic submanifolds in the conformal space ${mathbb Q}^n_s$.