Consider the double-torus, i.e., the orientable genus-$2$ surface. By embedding it into $\mathbb{R}^3$, we can get an induced metric from the usual Euclidean metric.
What's the isometry group of this double-torus? How does one think of it?
2026-03-30 09:55:05.1774864505
isometry group of double torus
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Generically, the isometry group will be trivial.
In any case, the isometry group will be finite (unlike the genus zero or one case where a surface of revolution is possible). See, for example, the book Finite groups of mapping classes of surfaces by Zieschang.
The fact that your surface is embedded in three-space should give a further non-trivial restriction on the allowable isometry groups. Points realising the diameter will be permuted by any isometry. This should prevent the surface from turning "inside out".