ON B 3 1 A family of pseudo metrics on B 3 and its application

There is no one to one 1-Lipschitz function from S2 to the interval [0,1] because of 1.2. Is there finite to one 1-Lipschitz function from S2 to the interval [0,1]? Is there countable to one 1-Lipschitz fuction from S2 to the interval [0,1]? The answers are no for both questions becase of 5.3. In this paper, I define a family of new metrics on the Euclidean 2-sphere of radius r in which the distances between antipodal points are closer than in the standard Euclidean metric. Note that the distances between two points which are near antipodal should be changed, too to satisfy triangle inequalities of new metrics. Consider all possible isometric embeddings of any family of Euclidean 2-spheres with not necessarily different radii into B3 = {(x, y, z) ∈ R | x2+y2+z2 ≤ 1} up to rotations. In this paper I restrict my interest on the isometric embeddings of Euclidean 2-spheres which cover B3 and in which any two point in B3 can be connected with finite number of embedded 2-spheres. Therefore I get a family of path connected metrics on B3 in cannonical way. I change metrics of embedded 2-sphere from the standard Euclidean metric to the new metrics which I already defined above in arbitrary way. Therefore I get a family of pseudo metrics on B3 in cannonical way. Although I study on these pseudo metrics on B3 in this paper, they can be cannonically generalized to a bigger family of pseudo metrics on B3. I define the bigger family of pseudo metrics and the quantum structure in the last section for futher study.