What will be the sensible extension of the 2D Ising to some curved surface - for instance, for a sphere or even something non-orientable?
For the flat space energy is given by well-known expression: $$ E = \sum_{i, j} J s_i s_j + \sum_ih_i s_i $$ What would make sense for the curved surface, should I imagine the spin as an arrow pointing along the $z$-axis, or the vector normal to the given surface. Also in the term, describing the interaction between the nearest neighbors, the spins $s_i $ now belong to different vector spaces, so it seems that for this expression to make sense, the neighbor should be parallely transported to the $i_{th}$ site. Or this construction doesn't make sense, and one has to work with the full Heisenberg model?