Isometry of Torus 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?  
 A: I'm going to consider the flat torus $\mathbb T = \mathbb R^2 / \Gamma$ for some plane lattice $$\Gamma_{e_1, e_2} = \{ z_1 e_1 + z_2 e_2 : z_1, z_2 \in \mathbb Z \}.$$ 
(Here $e_1, e_2$ is any basis of $\mathbb R^2$.) Since any isometry of the torus lifts to an isometry of $\mathbb R^2,$ we just need to determine which isometries of the plane descend to diffeomorphisms of $\mathbb T.$ 
A map $\phi: \mathbb R^2 \to \mathbb R^2$ descends iff it satisfies $$\phi(x+z) - \phi(x) \in \mathbb \Gamma \textrm{ for all } x \in \mathbb R^2, z \in \mathbb \Gamma.\tag 1 $$ Since the isometries of the plane can be written as $\phi(x) = Ax + c$ for $A \in O(2), c \in \mathbb R^2,$ this condition reduces to the fact that $Az = z$ for all $z,$ i.e. $A(\Gamma) \subset \Gamma.$ Finally, to have an inverse, $\phi^{-1}$ must also descend, and thus the condition is $A(\Gamma) = \Gamma;$ so $\phi(x) = Ax + c$ is an isometry of $\mathbb T$ iff the linear isometry $A$ is also an automorphism of the lattice $\Gamma.$
Note that two isometries induce identical maps on $\mathbb T$ if their constant terms differ by an element of $\Gamma,$ and composition of translations corresponds to addition of constant terms; so we can conclude that $$\mathrm{Isom}(\mathbb T) = (O(2)\cap \operatorname{Aut}(\Gamma)) \ltimes (\mathbb T, +). $$ (Compare to $\operatorname{Isom}(\mathbb R^2) =O(2) \ltimes (\mathbb R^2, +).$)
The nature of the normal subgroup $G = O(2)\cap \operatorname{Aut}(\Gamma)$ depends on the isometry class of the lattice (or equivalently on the shape of the fundamental domain):


*

*For a square, $G$ is the isometry group of the square, $D_4.$

*For a rhombus with angle $\pi/3$ (which fits on a hexagonal grid), $G$ is the isometry group of the hexagon, $D_6.$

*For any other rectangle, rhombus, or parallelogram with $\pi/3$ angle, $G$ is the product $C_2 \times C_2$ of cyclic groups (acting as orthogonal reflections).

*For a generic parallelogram, $G = C_2$ (acting as the antipodal map).

A: 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.)
