Some remarks on a minkowski space \((r^n, f)\)

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 \(\hat{g}\).