exponential map

On a differentiable manifold, the exponential map is the map from the tangent space to the manifold.

For each vector \(v\) in the tangent space \(T_x M\), there is a unique geodesic \(\gamma\) with initial derivative \(v\) and \(\gamma(0) = x\). The exponential is the unique solution you get from solving this differential equation. That is, you start at \(x\) and follow the curve that satisfies \(\gamma'(0) = v\). (Question for myself: how is the derivative defined after \(t=0\)?).