nLab
geodesic

Context

Riemannian geometry

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Idea

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

Definition

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

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

on the appropriate space of curves IXI \to X, where gg 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)

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

vanishes.

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

d 2x idt 2+ jkΓ jk i(x)dx jdtdx kdt=0. \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.

Revised on December 10, 2014 16:39:22 by Urs Schreiber (163.1.81.28)