Vol. 32, No. 1, 1970

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Local isometries of flat tori

Heinz Helfenstein

Vol. 32 (1970), No. 1, 113–117

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.

Mathematical Subject Classification
Primary: 30.45
Secondary: 53.00
Received: 9 July 1969
Published: 1 January 1970
Heinz Helfenstein