On the Isometric Extension Problem: A Survey

[[abstract]]This is a short survey on the extension of isometric mappings be- tween the unit spheres of Banach spaces. Some important results in the development of this problem are outlined in section 1. In the sec- tion 2, we raise the approximate isometric extension problem and non- surjective isometric extension problem for the future research. In the last part, we rst discuss the metric properties of isometries between the unit spheres of general Banach spaces, which is a new idea to work on this topic. And the classical Mazur-Ulam theorem has been indepen- dently generalized in several directions by Th.M. Rassias, D. Tingley, A. Vogt in a number of papers