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let $(M,g)$ be a smooth compact Riemannian manifold and $$\phi:M\rightarrow M$$ an isometry. Let $u\in C^{\infty}(M)$, are there general conditions on $u$ and $\phi$ such that the following relation holds? $$\Delta_{g}\left( u\circ \phi \right)=(\Delta_{g}u)\circ\phi$$

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1 Answer 1

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Always: the definition of $\Delta_g$ is in terms of $g$, so any diffeomorphism preserving $g$ preserves $\Delta_g$, and this is what ``preserves'' means.

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