Differential properties of manifolds with an intrinsic metric (Q1363912)

The author claims that for a manifold \(M^n\) with intrinsic metric \(\rho\) the existence of an isometry, \(\alpha>0\), at the point \(A\), i.e., of a map \(f:M^n\to \mathbb{R}^n\) such that \[ \rho_{\mathbb{R}^n}(f(X), f(Y))=\rho_{M^n}(X,Y)(1 + \varepsilon_f(X,Y)), \] where \(\varepsilon_f(X,Y)=o(\max\{\rho^\alpha_{M^n}(A,X),\rho^\alpha_{M^n}(A,Y)\})\), is equivalent to the existence of a metric tensor \(G=g_{ij}\) in some coordinate chart \(\phi : M^n\to \mathbb{R}^n\) (which is an open neighborhood of \(A\)) with the similar property \[ \rho_{M^n}(X, Y)=\sqrt{\langle G(f(X)-f(Y)),(f(X)-f(Y))\rangle}(1+ \varepsilon_\phi(X,Y)), \] where \(\varepsilon_\phi(X,Y)=o(\max \{\rho^\alpha_{M^n}(A,X),\rho^\alpha_{M^n}(A,Y)\})\).