Let T1 and T2 be two flat tori
(i.e., provided with a complete Riemannian metric of vanishing curvature). Since
they are locally Euclidean each pair of points P1,P2,Pi∈ Ti, has isometric
neighborhoods. In general it is not possible, however, to join these separate isometries
of neighborhoods to produce a single isometry T1→ T2 or T2→ T1; indeed there
may not even exist a locally isometric map (of the whole surfaces). Necessary and
sufficient conditions for the existence of such maps are deduced, making use of a
recent conformal classification of maps between tori. As expected “ample” and
nonample tori behave differently, and the determination of all local isometries leads
to number-theoretic problems. Finally, for two given tori, the local isometries are
compared with respect to homotopy by analyzing their effect on the fundamental
groups.
