A reflection on Tingley's problem and some applications

TL;DR

This paper investigates Tingley's problem by exploring metric properties of normed space spheres, proving conditions under which isometries are linear, and extending curvature concepts to normed spaces.

Contribution

It establishes that isometries between spheres of strictly convex spaces are linear if they agree on a relative open set, and extends the Mazur-Ulam Property to certain two-dimensional spaces.

Findings

Isometries are linear if they agree on a relative open subset.

Two-dimensional non strictly convex spaces have the Mazur-Ulam Property.

Generalization of curvature to normed spaces.

Abstract

In this paper we show how some metric properties of the unit sphere of a normed space can help to approach a solution to Tingley's problem. In our main result we show that if an onto isometry between the spheres of strictly convex spaces is the identity when restricted to some relative open subset, then it is the identity. This implies that an onto isometry between the unit spheres of strictly convex finite dimensional spaces is linear if and only it is linear on a relative open set. We prove the same for arbitrary two-dimensional spaces and obtain that every two-dimensional, non strictly convex, normed space has the Mazur-Ulam Property. We also include some other less general, yet interesting, results, along with a generalisation of curvature to normed spaces.