nLab geodesic

Idea

On a Riemannian manifold $(X,g)$, a geodesic (or geodesic line, geodesic path) is a path $x : I \to X$, for some (possibly infinite) interval $I$, which locally minimizes distance.

Definition

One way to define a geodesic is as a critical point of the functional of length

$$(x : I \to M) \mapsto \int_I \sqrt{g(\stackrel{\cdot}{x}(t),\stackrel{\cdot}{x}(t))} dt$$

on the appropriate space of curves $I \to X$, where $g$ is the metric tensor. This implies that it satisfies the corresponding Euler-Lagrange equations, which in this case means that the covariant derivative (for the Levi-Civita connection)

$$\nabla_{\stackrel{\cdot}{x}} \stackrel{\cdot}{x} = 0$$

vanishes.

In local coordinates, with Christoffel symbols $\Gamma^i_{jk}$ the Euler-Lagrange equations for geodesics form a system

$$\frac{d^2 x^i}{dt^2} + \sum_{jk} \Gamma^i_{jk}(x) \frac{d x^j}{dt} \frac{d x^k}{dt} = 0.$$

So this means that a curve is a geodesic if at every point its tangent vector is the parallel transport of the tangent vector at the start point along the curve.

Minimizing geodesics

A geodesic may not globally minimize the distance between its end points. For instance, on a 2-dimensional sphere, geodesics are arcs of great circle?s. Any two distinct non-antipodal points are connected by exactly two such geodesics, one shorter than the other (you can go from Los Angeles to Boston directly across North America, or the long way around the world).

A geodesic which does globally minimize distance between its end points is called a minimizing geodesic. The length of a minimizing geodesic between two points defines a distance function for any Riemannian manifold which makes it into a metric space.

Related concepts

• prime geodesic

• geodesic completeness

• totally geodesic submanifold

• variational calculus

References

• Wikipedia: Geodesic

• Springer eom: Yu. A. Volkov, Geodesic line; Yu. A. Volkov, Geodesic coordinates; Geodesic distance; V.A. Zalgaller, Geodesic geometry; D.V. Anosov, Geodesic flow

• Sh. Kobayashi, K. Nomidzu, Foundations of differential geometry, vol 1, 1963, vol 2, 1969, Wiley Interscience; reedition 1996 in series Wiley Classics; Russian ed.: Nauka, Moscow 1981.

• Arthur L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 10, Springer-Verlag 1987, xii + 510 pp. (for a review see Bull. AMS and MR88f:53087); reprinted 2008, Springer Classics in Math.

On Maslov indices and stability of geodesics:

• Paolo Piccione, Alessandro Portaluri, Daniel V. Tausk: Spectral Flow, Maslov Index and Bifurcation of Semi-Riemannian Geodesics, Annals of Global Analysis and Geometry 25 (2004) 121–149 [doi:10.1023/B:AGAG.0000018558.65790.db]

• Alessandro Portaluri, Li Wu, Ran Yang: Linear instability for periodic orbits of non-autonomous Lagrangian systems, Nonlinearity 34 1 (2021) 237 [arXiv:1907.05864]