Geodesics on Riemannian Manifolds

Differential Geometry · Axiom Academy

LESSON Geodesics in Riemannian Geometry Understanding curves of extremal length and parallel transport on Riemannian manifolds 1. Geodesics as Curves with Parallel Tangent A curve γ(t) on a Riemannian manifold (M,g) is a geodesic if its tangent vector γ'(t) is parallel-transported along the curve. This means the covariant derivative of the tangent vector along itself vanishes. Geometrically, this condition means the tangent vector maintains constant direction (in the intrinsic geometry of the manifold) as we move along the curve. There is no "turning" or acceleration perpendicular to the curve. 2. The Geodesic Equation in Local Coordinates When we express the geodesic condition in local coordinates (x 1 , ..., x n ), we obtain a system of second-order ordinary differential equations involving the Christoffel symbols Γ k ij . This is a system of n coupled second-order ODEs. The Christoffel symbols encode how coordinates "curve" in the manifold and are given by: 3. Geodesics as Length-Minimizing Curves Geodesics are critical points of the energy functional and are locally length-minimizing curves. For sufficiently close points, a geodesic provides the shortest path. While geodesics minimize length locally , globally they may not be the shortest path. For example, on a sphere, two antipodal points are connected by infinitely many geodesics (great circles), and "long" portions of these great circles are not minimizing. 4. The Exponential Map: exp p : T p M → M

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