What is the isometry group of a torus given a flat metric? I know $ O(1) \times O(1) $ should be a subgroup of it. Is there any other possible isometries? What if the metric is not flat?
2026-03-26 21:25:22.1774560322
Isometry of Torus
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Consider $$ (\theta, \phi) \mapsto (\pm\phi, \pm\theta), $$
In short: consider all affine maps on the plane that map the integer lattice (including the lattice "edges") in the plane to itself in a 1-1 way, and project them to the torus, and that gets you some more isometries.
(I'm assuming your flat torus is the quotient of $\mathbb R^2$ by the integer grid $\mathbb Z^2$, with the metric inherited from the quotient map.)
I have this feeling that rotations of the plane should work in some form as well, but haven't worked it out (and after screwing up once today, I'm gun-shy.)