Definition
Let M be a differentiable manifold and p a point of M. An affine connection on M allows one to define the notion of a geodesic through the point p.
Let v ∈ TpM be a tangent vector to the manifold at p. Then there is a unique geodesic γv satisfying γv(0) = p with initial tangent vector γ′v(0) = v. The corresponding exponential map is defined by expp(v) = γv(1). In general, the exponential map is only locally defined, that is, it only takes a small neighborhood of the origin at TpM, to a neighborhood of p in the manifold. This is because it relies on the theorem on existence and uniqueness for ordinary differential equations which is local in nature. An affine connection is called complete if the exponential map is well-defined at every point of the tangent bundle.
Read more about this topic: Exponential Map
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