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 $γ 0$ 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 $M$ .

[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.

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Category: Riemannian geometry

Geodesic manifold

From Wikipedia, the free encyclopedia

[edit] Definition

[edit] Examples

[edit] Path-connectedness, completeness and geodesic completeness

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