Is the exponential map an isometric map from an open subset of the tangent space to the Riemannian manifold?

Question

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.