ar X iv : m at h / 07 02 45 3 v 2 [ m at h . D G ] 1 6 Fe b 20 07 Locally Euclidean metrics on R 3

For all 0 < t ≤ 1, we define a locally Euclidean metric ρt on R . These metrics are invariant under Euclidean isometries and, if t increases to 1, converge to the Euclidean metric dE . This research is motivated by expanding universe. key words. locally Euclidean metric PACS number(s). 98.80.Jk Mathematics Subject Classifications (2000). 85A40, 57M50 1 The metric ρt Let dE denote the Euclidean metric on R . As in [5], a nonnegative function d : R ×R → R is called a locally Euclidean metric if (i) d(P,Q) = 0 if and only if P = Q (ii) d(P,Q) = d(Q,P ) for all P,Q ∈ R (iii) d(P,Q) + d(Q,R) ≥ d(P,R) for all P,Q,R ∈ R (iv) For all P ∈ R, there exists r > 0 such that d(Q,R) = dE(Q,R) for all Q,R ∈ Br(P ) = {S ∈ R 3 | d(P, S) < r}. Let S ⊂ R be the unit 2-sphere with center O = (0, 0, 0) and radius 1. A locally Euclidean metric on S is defined in the same way as on R. In the following theorem of the author, −P is the antipodal point of P ∈ S and 0 ≤ α < π/4.