Let $U$ be a normal neighbourhood of a point $p$ in a Riemannian manifold $M$. Can we say that the exponential map is an isometric map from an open subset of the tangent space $T_pM$ to the manifold $M$? Isometric in the sense that each point in the tangent space $T_pM$ can also be considered to have a tangent space at that point, and the exponential map preserves the inner product.
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1Hint: Not every Riemannian manifold is locally isometric to a Euclidean space. Think about curvature. – Moishe Kohan Mar 28 '19 at 21:21
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If you put on $T_pM$ the pull back metric $\exp_p^*g$, then locally yes. – Yuval Mar 30 '19 at 05:55