Let M and N be two Riemannian manifolds. A diffeomorphism g : M → N is an isometry if (v1, v2)p = (dgp(v1), dgp(v2))g(p) (1.65) for every p ∈ M and v1, v2 ∈ TpM. If N = M, g is called an isometry of M.
Let M = R be endowed with the Euclidean metric. The isometry group of Rcontains translations, rotations and reflections.
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