The exponential map exp_p : TpM → M sends a tangent vector v to the point reached by following the geodesic from p with initial velocity v for unit time. It is a local diffeomorphism near the origin of TpM, providing "normal coordinates" centered at p in which geodesics through p are straight lines and the metric is Euclidean to first order. The exponential map connects the linear algebra of the tangent space to the nonlinear geometry of the manifold and is the fundamental tool for local geometric analysis.
The tangent space TpM is a vector space — linear, flat, and easy to work with. The manifold M is curved and nonlinear. The exponential map is the bridge between them: it takes a tangent vector v ∈ TpM and maps it to the point in M you reach by "walking along the geodesic" in the direction v for unit time. For small v, this is a diffeomorphism, and the inverse map provides normal coordinates — a coordinate system centered at p where geodesics through p are straight lines.
Precisely, for v ∈ TpM with |v| small, let γ_v(t) be the unique geodesic with γ_v(0) = p and γ_v'(0) = v. Then exp_p(v) = γ_v(1). By rescaling: exp_p(tv) = γ_v(t) — the exponential map sends rays through the origin in TpM to geodesic rays from p in M. The map d(exp_p)_0 : T_0(TpM) → TpM is the identity, so by the inverse function theorem, exp_p is a diffeomorphism from a neighborhood of 0 in TpM to a neighborhood of p in M. This neighborhood, described in the linear coordinates of TpM, gives normal coordinates.
In normal coordinates, the metric is optimally simple. At the center point p: gij(0) = δij (the metric looks Euclidean) and Γᵏij(0) = 0 (the Christoffel symbols vanish). The Taylor expansion gij(x) = δij - ⅓Rikjl(p) xᵏxˡ + O(|x|³) shows that the curvature tensor is the leading correction to flatness. This is why curvature controls local geometry: within normal coordinates, the manifold looks like Euclidean space up to first order, with curvature appearing at second order.
The injectivity radius inj(p) is the largest r such that exp_p is injective on the ball {v ∈ TpM : |v| < r}. Beyond this radius, geodesics from p may cross each other or form conjugate points. On the sphere Sⁿ of radius 1, the injectivity radius is π — geodesics from any point refocus at the antipodal point. On hyperbolic space, the injectivity radius is infinite. The exponential map also connects to Lie theory: on a Lie group G with Lie algebra 𝔤 = TeG, the Riemannian exponential map (for a bi-invariant metric) coincides with the Lie group exponential map, which is why both share the name. This is the historical origin of the term "exponential map" in Riemannian geometry.