3
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About Palais' remark that an isometry of Riemannian manifolds does not induce an isometry of the Hilbert manifolds of curves
Turning the comments by Dick Palais and me into an answer:
The problem with the argument in the question lies in defining $g_\sigma(\lambda,\mu)$ as metric on $H_1([0,1];V)$. It is the metric for $H_0$...
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