the Riemannian exponential map
Stand at a point p of a curved space, pick a direction and a speed (a tangent vector v), and fire a geodesic. After unit time you land somewhere. The Riemannian exponential map is the bookkeeping of exactly this: it takes a tangent vector at p and returns the point you reach by following the geodesic in that direction for the corresponding distance. It is the canonical way to chart a neighborhood of p using the flat tangent space as a template.
Formally, for p in M and v in T_p M, let gamma_v be the unique geodesic with gamma_v(0) = p and gamma_v'(0) = v. Then exp_p(v) := gamma_v(1), defined for v small enough that the geodesic exists up to time 1. By the homogeneity of geodesics, gamma_v(t) = gamma_(tv)(1), so following v for time t is the same as following tv for unit time; thus exp_p sends the ray t -> tv to the geodesic through p in direction v. Its differential at the origin is the identity, so by the inverse function theorem exp_p is a diffeomorphism from a star-shaped neighborhood of 0 in T_p M onto a neighborhood of p — a geodesic ball.
This is the gateway to normal coordinates and the Gauss lemma, and it is how one transfers linear-algebra intuition from T_p M onto the manifold. The single most important caution: the Riemannian exp_p (built from geodesics of the Levi-Civita connection) is NOT the same as the Lie-group exponential exp: g -> G (built from one-parameter subgroups). They share a name and a tangent-to-manifold flavor but are different constructions, agreeing only for bi-invariant metrics on a Lie group. Also exp_p is generally only a LOCAL diffeomorphism: globally it can fail to be injective (antipodes on a sphere) or onto (incomplete manifolds).
On the unit sphere, exp_p(v) for v in T_p M of length |v| is the point reached by going distance |v| along the great circle in direction v. Every geodesic from the north pole reconverges at the south pole, so exp_p collapses an entire circle of tangent vectors of length pi to one point — a vivid example of exp_p failing to be globally injective.
exp_p is a local chart near p but globally can fold many tangent vectors onto one point (here the south pole).
Do not conflate the Riemannian exp_p with the Lie-group exp: g -> G; they coincide only for bi-invariant metrics. And exp_p is generally only a local diffeomorphism — global injectivity fails wherever geodesics refocus (conjugate points).