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.