Properties Preserved By An Isometry Map Between Unit Spheres

The isometry is an important field in space theory and operator theory.Linear is also important for a mapping.Mazur-Ulam theorem has showed the relationship between linearity and isometry.Since it have been published,there are mathematician that have generalized this theorem in different aspects.In 1987,Tingley posed the isometric extension problem on the unit sphere as follow:if there is an surjective isometry between two unit spheres,and then can it be extended to an affine mapping on the whole space? Considering the properties holding by an isometry map itself,there is an method indeed to approach the Tingley’s problem,which is focusing on those geometric characteristics and metric properties of unit sphere in domain that can be preserving in range space.At the beginning,we give some basic geometric properties of the unit sphere,for further,the properties preservation under an isometric mapping of subsets which belong to an unit sphere have been studied.Specifically,we define the concept of facet of unit sphere in the case of dimension is finite,and prove that some properties of facet which are preserved by an isometry mapping.Moreover,we proved that surjective operates between spheres can hold arc length equality,which means surjective operates preserve arc length orthogonal.Meanwhile,we porved that a point in domain is an extreme point if and only if the image of this point is also an extreme point in range.After that,we obtain some basic properties of two-dimensional strictly convex normed space.Through the concept and properties of generalized orthogonality,we than proved that the isometric mapping between spheres,which is required to keep the isosceles orthogonality unchanged,can be extended on the whole space under certain conditions.