Resumen
Consider the following question:
Let $\Delta: \mathcal{S}_X \rightarrow \mathcal{S}_Y$ be a surjective isometry between the unit spheres of two normed spaces $X$ and $Y$. Does there exists a surjective real linear isometry $T : X \rightarrow Y$ such that $T|\mathcal{S}_X= \Delta$?
This question is known as Tingley's problem, and it remains unsolved since 1987, even for the case of two 2-dimensional normed spaces.
The aim of this work is to completely determine the form of a surjective isometry defined on the the unit sphere of n \times n complex matrices, $\mathcal{S}_{M_n(\mathbb{C})}$, with respect to the operator norm. As an immediate consequence, this result gives a positive answer to Tingley's problem for the case of a surjective isometry between the unit spheres of two finite dimensional normed spaces of complex matrices.