The Exponential Map

Differential Geometry · Axiom Academy

Understanding how tangent vectors generate geodesics and map the tangent plane to the manifold 1. Definition of the Exponential Map For a Riemannian manifold M and a point p ∈ M, the exponential map at p is defined as: where γ is the unique geodesic satisfying the initial conditions γ(0) = p and γ'(0) = v. In other words, we follow the geodesic with initial velocity v for unit time. 2. Mapping the Tangent Plane to the Surface The exponential map exp p : T p M → M takes vectors in the tangent space (a flat, linear space) and maps them to points on the curved manifold. This mapping is smooth and respects the geometry: straight lines through the origin in T p M (of the form tv for fixed v) map to geodesics through p on the manifold. 3. Local Diffeomorphism Property Near the origin of T p M, the exponential map is a local diffeomorphism. This means: The differential of exp p at the origin (0 ∈ T p M) is the identity map. By the inverse function theorem, exp p is a diffeomorphism on some neighborhood of 0. Using the exponential map, we can define geodesic polar coordinates (r, θ) around p. Choose an orthonormal basis e 1 , ..., e n for T p M, then: Here r = ||v|| is the radial coordinate (distance from origin in T p M) and θ represents angular coordinates. Points at constant r map to equidistant curves from p along geodesics. 5. Relation to Distance from p

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