Let \((M,g)\) be a Riemannian manifold and \(K\subset M\) a compact subset. Then there is a constant \(r_{K}>0\) such that for all \(p\in K\), we have \(B(0_{p},r_{K})\subset\mathcal{D}_{p}\) and
\begin{equation} \exp_{p}\bigg{|}_{B(0_{p},r_{K})}:B(0_{p},r_{K})\to\exp_{p}(B(0_{p},r_{K}))\nonumber \\ \end{equation}is a diffeomorphism.
Inverse (or Implicit) Funciton Theorem. ∎