  We consider a complete, totally umbilical hypersurface $M$ of Riemannian
space $(\hat{R}^n, \hat{g})$ induced by a Minkowski space $(R^n, F)$. Under
certain conditions we prove that $M$ is isometric to a "round" hypersphere of
the $(n + 1)-$dimensional Euclidean space. We also prove that the Minkowski
norm $F$ must be arised from an inner product if there exist a non-zero vector
field, which is parallel according to Levi-Civita connection of the metric
tensor $\hat{g}$.
