The origin of Tingley's problem is the celebrated Mazur-Ulam theorem that states every surjective isometry between two normed spaces is automatically affine. In 1972, Mankiewicz provided a localization of this result by showing that every surjective isometry between open and connected subsets of normed spaces extends uniquely to an affine isometry between the whole spaces; and it guarantees, in particular, that each surjective isometry between the unit balls of two normed spaces is the restriction of an isometric isomorphism between those spaces. This observation motivated Tingley [3] to propose the following problem: does a surjective isometry between the unit spheres of two normed spaces extend to an isometric isomorphism? This problem is also known as the isometric extension problem; and it has been extensively studied especially since 2000. However, Tingley's problem is still open even in the two-dimensional case, because of the lack of interior points and appropriate algebraic structure.

The purpose of this talk is to present the solution to Tingley's problem on matrix algebras given in [2]. In this case, we can use some operator algebraic techniques. The key ingredient is the result of Hatori and Molnár [1] that determines the forms of (surjective) isometries of the unitary group of the algebra of all operators acting on a Hilbert space. To activate this result, we prepare some tools from Banach space geometry.