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  "body": "\n## Summary\n\nLet $p$ be a point of a [[Riemannian manifold|7CBT4SM83Y]].\nThe exponential map at $p$ sends $v \\in T_pM$ to the point reached at time $1$ by the [[geodesic|NSYMQMHPDN]] from $p$ with velocity $v$.\nIt maps a small ball around $0$ in the tangent space diffeomorphically onto a neighbourhood of $p$.\nSo it gives coordinates adapted to the geometry, the [[normal coordinates|RRHKMGNKG9]].\nIt is central to the study of completeness, conjugate points and the cut locus.\nFor a bi-invariant metric on a Lie group, it agrees at the identity with the exponential map of the Lie group.\n\n## Explanations\n\n### L1 human\n\nStand at a point $p$, choose a direction and a distance, and walk straight ahead, along a geodesic, for that distance.\nThe point where you stop is the exponential map of the vector with that direction and length.\nOn the unit sphere, starting at the north pole, the vectors of length $\\pi/2$ go to the equator and all vectors of length $\\pi$ go to the south pole.\nThis is like wrapping the flat tangent plane around the sphere so that straight lines through the origin become great circles through $p$.\n",
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