Exponential map

The exponential map sends a tangent vector at a point on a manifold to the endpoint reached by following the geodesic determined by that vector for unit time, when that geodesic exists.

Geodesic: A curve that locally extremizes length and generalizes a straight line to curved spaces. The map’s output is the point reached along a geodesic launched with the input vector.
Smooth manifold: A space locally modeled on Euclidean space with smoothly compatible coordinate charts. The exponential map carries tangent vectors into points of this underlying space.
Normal coordinates: Local coordinates on a Riemannian manifold obtained by applying an exponential map to a tangent-space basis. They use the map to make geodesics through the coordinate origin appear as radial lines.
Lie group exponential map: A map from a Lie algebra to its Lie group that sends an infinitesimal generator to the corresponding one-parameter subgroup element. It also turns infinitesimal data into finite motion, but uses group multiplication rather than geodesic travel.
Tangent space: A vector space that represents the possible directions of motion at a point on a smooth manifold. Each exponential map begins with a vector in the tangent space at its base point.
Affine connection: A rule for differentiating vector fields along directions on a manifold. It specifies parallel transport and determines which curves count as geodesics.
Hopf–Rinow theorem: A theorem relating metric completeness, geodesic completeness, and compactness of closed bounded sets in connected Riemannian manifolds. It connects global completeness with geodesics reaching points and the exponential map’s domain.
Matrix exponential: The matrix function defined by the power series exponential, converting a square matrix into an invertible matrix. It is a related algebraic construction and often realizes the Lie group exponential for matrix groups.
Geodesic equation: A differential equation whose solutions describe geodesics on a manifold with a connection. Solving this equation determines the curve whose endpoint the map returns.
Riemannian metric: A smoothly varying inner product on each tangent space of a manifold. A metric induces the Levi-Civita connection used to define Riemannian geodesics.

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