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

TL;DR

This paper investigates the geometric properties of hypersurfaces in Minkowski spaces, showing conditions under which they are spherical and characterizing Minkowski norms derived from inner products based on parallel vector fields.

Contribution

It establishes that certain totally umbilical hypersurfaces are spherical and characterizes Minkowski norms arising from inner products via parallel vector fields.

Findings

Hypersurfaces are isometric to Euclidean spheres under specific conditions.

Minkowski norm F originates from an inner product if a parallel vector field exists.

Conditions for hypersurfaces to be totally umbilical in Minkowski spaces.

Abstract

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 $\hat{g}$.