In Chap. 12 we define the Exponential map (and restricted exponential map) from points in the fibre \(T_{\boldsymbol {p}}\mathcal {M}\) of the tangent bundle \(T\mathcal {M}\) of a manifold \(\mathcal {M}\) to a neighbourhood of \(\boldsymbol {p}\in \mathcal {M}\) . The properties of this map are developed and used to establish the Gauss Lemma for n-dimensional pseudo-Riemannian manifolds using properties of Levi-Civita geodesic curves starting at \(\boldsymbol {p}\) . These properties are then employed to define a particular local chart that has been found to be of both practical and mathematical value in a number of situations in differential geometry.