Locally Euclidean Metrics on $S^2$ in Which Some Open Balls Are Not Connected
Young Deuk Kim · Rocky Mountain Journal of Mathematics · 2006
Let S 2 r ⊂ R 3 be the 2-sphere with center O and radius r.For all 0 0. We write d E to denote the Euclidean metric on S 2 r .A metric d on the set S 2 r is called locally Euclidean if, for all P ∈ S 2 r , there exists t > 0 such that