Geodesic manifold

Geodesic manifold

In mathematics, a geodesic manifold (or geodesically complete manifold) is a "surface" on which any two points can be joined by a shortest path, called a geodesic.

Definition

Let (M, g) be a (connected) (pseudo-) Riemannian manifold, and let gamma : [a, b] o M be some differentiable path. Recall that the length of the curve is defined by

:ell (gamma) := int_{a}^{b} sqrt{pm g (dot{gamma} (t), dot{gamma} (t))} , mathrm{d} t.

Given two points x, y in M, a path gamma_{0} : [a, b] o M is called a geodesic (in M) if its length attains the infimum over all differentiable paths gamma : [a, b] o M such that gamma (a) = x and gamma (b) = y.

The manifold (M, g) is called geodesic (or geodesically complete) if any two (distinct) points of the manifold can be joined by a geodesic path (in M).

Examples

Euclidean space mathbb{R}^{n}, the spheres mathbb{S}^{n} and the tori mathbb{T}^{n} (with their usual Riemannian metrics) are all geodesic manifolds. Geodesics in Euclidean space are unique; for certain pairs of points on spheres or tori, the choice of geodesic is not unique (e.g. antipodal points on a sphere).

A simple example of a non-geodesic manifold is given by the punctured plane M := mathbb{R}^{2} setminus { 0 } (with its usual metric). If x eq 0 is any point of M and y := - x is its antipodal point, then although the distance from x to y is 2 | x |, and this is the infimum over the lengths of all possible paths gamma from x to y, there is no path gamma_{0} that attains this infimum: any minimizing path would have to pass through the origin, and geodesic paths are explicitly required to take values only within the space M.

Path-connectedness, completeness and geodesic completeness

It can be shown that a path-connected, complete Riemannian manifold is necessarily geodesically complete. The example of a non-geodesic manifold (the punctured plane) given above fails to be geodesically complete because, although it is path-connected, it is not a complete space: the origin lies in the closure of the space, but not in the space itself.


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