I read that the possible conformal structures on the 2-torus are characterized by two real numbers (or one complex number, the "modulus"): a representative metric for a particular conformal structure is that induced on a parallelogram immersed in the Euclidean plane, with the opposite sides identified. One of the two real numbers of the modulus then is related to the ratio between the two sides of the parallelogram, and the other is related to one of the internal angles of the parallelogram. Then, if I understood well, the space of all the values of these two numbers, modulo the discrete "large" diffeomorphisms of the torus, is called Teichmüller space.
My question regards the analogue of this construction for the 3-torus, in particular I'd like to understand if there's an analogue of the internal angle mentioned above. Now, in 3D, the space of conformal structures on the torus is infinite-dimensional: in 2D it's finite-dimensional because all surfaces are conformally flat, so one it's left with a discrete number of parameters to characterize the possible conformal geometries, depending on the global topology that is assumed. In 3D not all metrics are conformally flat, so I have an infinite-dimensional space of conformal metrics. To consider an analog of Teichmüller space for 3D tori, I need to assume something more: I will consider only homogeneous metrics. The space of homogeneous metrics on a given 3D topology is indeed finite-dimensional. These have already been classified by Bianchi: for the 3-torus, the homogeneous metrics are those whose structure Lie algebra is Abelian. The most generic homogeneous metric takes the form:
$ds^2 = L_1^2 \, dx^2 + L_2^2 \, dy^2 + L_3^2 \, dz^2 \,, $
where $L_i$ are three real (assume them positive) parameters, and $x,y,z$ are periodic coordinates of period 1. The space of homogeneous conformal metrics is then two-dimensional, and it is parametrized, for example, by the twi ratios $L_1/L_2$ and $L_2/L_3$. Now, the metric above is the one induced by the ambient Euclidean space on a rectangular parallelepiped (or box) of sides $L_i$, whose opposite faces have been identified.
So, what happens if, instead of a rectangular parallelepiped, I consider a non-rectangular one, with some of the internal angles of its faces different from zero? Isn't the metric induced on it (and therefore on the 3-torus) still homogeneous? Aren't two such metrics non-isometric to each other? Then what's the dimension of the space of homogeneous metrics on the 3-torus? Is it larger than three?