Geodesic manifold

From Wikipedia, the free encyclopedia

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.

[edit] Definition

Let (M,g) be a (connected) (pseudo-) Riemannian manifold, and let \gamma : [a, b] \to 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] \to M is called a geodesic (in M) if its length attains the infimum over all differentiable paths \gamma : [a, b] \to M such that γ(a) = x and γ(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).

[edit] 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 \neq 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 γ from x to y, there is no path γ0 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.

[edit] 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.