If for Riemannian charts $(P, g)$, $(Q, h)$ and $(R, i)$ I have two Riemannian isometries (differentiable bijection with differentiable inverse), one $\phi: (P, g) \to (Q,h)$ such that $g = \phi^{*}h$ and one $\rho: (Q,h) \to (R, i)$ such that $h = \rho^{*}i$, how do I show that the composition of these two isometries is again an isometry?
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3$\phi^\rho^=(\rho\circ\phi)^*$ – user10354138 Feb 22 '21 at 22:44
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Prove it for metric spaces. – Deane Feb 23 '21 at 01:43