We prove that if $M$ is a $\text{JBW}^{\ast }$-triple and not a Cartan factor of rank two, then $M$ satisfies the Mazur–Ulam property, that is, every surjective isometry from the unit sphere of $M$ onto the unit sphere of another real Banach space $Y$ extends to a surjective real linear isometry from $M$ onto $Y$.

In this paper, it is proved that every isometry between the unit spheres of two real Banach spaces preserves the frames of the unit balls. As a consequence, if $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}X$ and $Y$ are $n$-dimensional Banach spaces and $T_0$ is an isometry from the unit sphere of $X$ onto that of $Y$ then it maps the set of all $(n-1)$-extreme points of the unit ball of $X$ onto that of $Y$.

This paper gives a characterization of nonexpansive mappings from the unit sphere of ℓβ (Γ) onto the unit sphere of ℓβ (Δ) where 0<β≤1. By this result, we prove that such mappings are in fact isometries and give an affirmative answer to Tingley’s problem in ℓβ (Γ) spaces. We also show that the same result holds for expansive mappings between unit spheres of ℓβ (Γ) spaces without the surjectivity assumption.