Linked from
The 12 pages that link to Exponential map, each with the reason it gives.
GeodesicBroader topic: It turns initial directions and speeds into geodesic endpoints.
Lie groupRelated: It connects algebraic directions to continuous motions in the group.
Lie algebraRelated: It converts infinitesimal generators into finite transformations, locally.
Cartan–Hadamard theoremRelated: The theorem's central map is the exponential map at each base point.
Hopf–Rinow theoremRelated: Geodesic completeness means this map is defined on every tangent vector.