Geodesic Completeness

Differential Geometry · Axiom Academy

LESSON Completeness and the Hopf-Rinow Theorem Understanding geodesic and metric completeness on Riemannian manifolds and their profound equivalence Intuitively, geodesic completeness means that if you "walk straight" along any geodesic, you can continue walking forever without "falling off" the manifold. Every geodesic extends indefinitely in both directions. Formally, for every point p ∈ M and every tangent vector v ∈ T_p M, the geodesic γ(t) with γ(0) = p and γ'(0) = v must be defined for all t ∈ ℝ. The exponential map exp_p: T_p M → M is defined on the entire tangent space. where γ ranges over all piecewise smooth curves from p to q, and the length L(γ) is: Metric completeness is a purely topological/analytic condition: it says that the manifold has "no missing points" with respect to its distance function. A sequence (p_n) is Cauchy if d(p_m, p_n) → 0 as m, n → ∞. (M, g) is geodesically complete There exists p ∈ M such that exp_p is defined on all of T_p M Closed and bounded subsets of M are compact Moreover, if any (hence all) of these conditions hold, then for every pair of points p, q ∈ M, there exists a minimizing geodesic γ connecting them, i.e., a geodesic with L(γ) = d(p, q). One of the most important consequences of the Hopf-Rinow theorem is that complete manifolds are geodesically connected : any two points can be joined by a length-minimizing geodesic.

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