If your model for the torus is the unit square with edges identified together, then the metric is the usual Euclidean metric, dx² + dy². (Alternatively, the me

If your model for the torus is the unit square with edges identified together, then the metric is the usual Euclidean metric, dx² + dy². (Alternatively, the metric induced from the embedding into Euclidean 4-space, with a parameterization like φ=atan2(y,x), θ=atan2(z,w), should give you the same answer give or take some factors of 2π.) (Also, Mathoverflow relevant to the double torus question: http://mathoverflow.net/questions/194451/canonical-immersion-of-the-double-torus. http://mathoverflow.net/questions/194451/canonical-immersion...)