Editing Geodesic (section)

==Affine geodesics==
{{See also|Geodesics in general relativity}}
A '''geodesic''' on a [[Differentiable manifold|smooth manifold]] <math>M</math> with an [[affine connection]] <math>\nabla</math> is defined as a [[curve]] <math>\gamma(t)</math> such that [[parallel transport]] along the curve preserves the tangent vector to the curve, so
{{NumBlk|:|<math> \nabla_{\dot\gamma} \dot\gamma= 0</math>|{{EquationRef|1}}}}
at each point along the curve,  where <math>\dot\gamma</math> is the derivative with respect to <math>t</math>.  More precisely, in order to define the covariant derivative of <math>\dot\gamma</math> it is necessary first to extend <math>\dot\gamma</math> to a continuously differentiable [[vector field]] in an [[open set]].  However, the resulting value of ({{EquationNote|1}}) is independent of the choice of extension.

Using [[local coordinates]] on <math>M</math>, we can write the '''geodesic equation''' (using the [[summation convention]]) as
:<math>\frac{d^2\gamma^\lambda }{dt^2} + \Gamma^{\lambda}_{\mu \nu }\frac{d\gamma^\mu }{dt}\frac{d\gamma^\nu }{dt} = 0\ ,</math>
where <math>\gamma^\mu = x^\mu \circ \gamma (t)</math> are the coordinates of the curve <math>\gamma(t)</math> and <math>\Gamma^{\lambda }_{\mu \nu }</math> are the [[Christoffel symbol]]s of the connection <math>\nabla</math>.  This is an [[ordinary differential equation]] for the coordinates.  It has a unique solution, given an initial position and an initial velocity.  Therefore, from the point of view of [[classical mechanics]], geodesics can be thought of as trajectories of [[free particle]]s in a manifold. Indeed, the equation <math> \nabla_{\dot\gamma} \dot\gamma= 0</math> means that the [[Acceleration (differential geometry)|acceleration vector]] of the curve has no components in the direction of the surface (and therefore it is perpendicular to the tangent plane of the surface at each point of the curve). So, the motion is completely determined by the bending of the surface. This is also the idea of general relativity where particles move on geodesics and the bending is caused by gravity.

===Existence and uniqueness===
The ''local existence and uniqueness theorem'' for geodesics states that geodesics on a smooth manifold with an [[affine connection]] exist, and are unique.  More precisely:

:For any point ''p'' in ''M'' and for any vector ''V'' in ''T<sub>p</sub>M'' (the [[tangent space]] to ''M'' at ''p'') there exists a unique geodesic <math>\gamma \,</math> : ''I'' &rarr; ''M'' such that
::<math>\gamma(0) = p \,</math> and
::<math>\dot\gamma(0) = V,</math>
:where ''I'' is a maximal [[open interval]] in '''R''' containing 0.

The proof of this theorem follows from the theory of  [[ordinary differential equation]]s, by noticing that the geodesic equation is a second-order ODE. Existence and uniqueness then follow from the [[Picard&ndash;Lindelöf theorem]] for the solutions of ODEs with prescribed initial conditions. γ depends [[smooth function|smoothly]] on both ''p'' and&nbsp;''V''.

In general, ''I'' may not be all of '''R''' as for example for an open disc in '''R'''<sup>2</sup>. Any {{mvar|γ}} extends to all of {{mvar|ℝ}} if and only if {{mvar|M}} is [[geodesic manifold|geodesically complete]].

===Geodesic flow{{anchor|Flow}}===
'''Geodesic [[Flow (mathematics)|flow]]''' is a local '''R'''-[[Group action (mathematics)|action]] on the [[tangent bundle]] ''TM'' of a manifold ''M'' defined in the following way

:<math>G^t(V)=\dot\gamma_V(t)</math>

where ''t''&nbsp;∈&nbsp;'''R''', ''V''&nbsp;∈&nbsp;''TM'' and <math>\gamma_V</math> denotes the geodesic with initial data <math>\dot\gamma_V(0)=V</math>. Thus,  ''<math>G^t(V)=\exp(tV)</math> is the [[exponential map (Riemannian geometry)|exponential map]] of the vector ''tV''.  A closed orbit of the geodesic flow corresponds to a [[closed geodesic]] on&nbsp;''M''.

On a (pseudo-)Riemannian manifold, the geodesic flow is identified with a [[Hamiltonian flow]] on the cotangent bundle.  The [[Hamiltonian mechanics|Hamiltonian]] is then given by the inverse of the (pseudo-)Riemannian metric, evaluated against the [[canonical one-form]]. In particular the flow preserves the (pseudo-)Riemannian metric <math>g</math>, i.e.

: <math>g(G^t(V),G^t(V))=g(V,V). \, </math>

In particular, when ''V'' is a unit vector, <math>\gamma_V</math> remains unit speed throughout, so the geodesic flow is tangent to the [[unit tangent bundle]].  [[Liouville's theorem (Hamiltonian)|Liouville's theorem]] implies invariance of a kinematic measure on the unit tangent bundle.

===Geodesic spray===
{{further|Spray (mathematics)#Geodesic}}
The geodesic flow defines a family of curves in the [[tangent bundle]]. The derivatives of these curves define a [[vector field]]  on the [[total space]] of the tangent bundle, known as the '''geodesic [[spray (mathematics)|spray]]'''.

More precisely, an affine connection gives rise to a splitting of the [[double tangent bundle]] TT''M'' into [[horizontal bundle|horizontal]] and [[vertical bundle]]s:
:<math>TTM = H\oplus V.</math>
The geodesic spray is the unique horizontal vector field ''W'' satisfying
:<math>\pi_* W_v = v\,</math>
at each point ''v''&nbsp;∈&nbsp;T''M''; here {{pi}}<sub>∗</sub>&nbsp;:&nbsp;TT''M''&nbsp;→&nbsp;T''M'' denotes the [[pushforward (differential)]] along the projection {{pi}}&nbsp;:&nbsp;T''M''&nbsp;→&nbsp;''M'' associated to the tangent bundle.

More generally, the same construction allows one to construct a vector field for any [[Ehresmann connection]] on the tangent bundle.  For the resulting vector field to be a spray (on the deleted tangent bundle T''M''&nbsp;\&nbsp;{0}) it is enough that the connection be equivariant under positive rescalings: it need not be linear.  That is, (cf. [[Ehresmann connection#Vector bundles and covariant derivatives]]) it is enough that the horizontal distribution satisfy
:<math>H_{\lambda X} = d(S_\lambda)_X H_X\,</math>
for every ''X''&nbsp;∈&nbsp;T''M''&nbsp;\&nbsp;{0} and λ&nbsp;>&nbsp;0.  Here ''d''(''S''<sub>λ</sub>) is the [[pushforward (differential)|pushforward]] along the scalar homothety <math>S_\lambda: X\mapsto \lambda X.</math>  A particular case of a non-linear connection arising in this manner is that associated to a [[Finsler manifold]].

===Affine and projective geodesics===
Equation ({{EquationNote|1}}) is invariant under affine reparameterizations; that is, parameterizations of the form
:<math>t\mapsto at+b</math>
where ''a'' and ''b'' are constant real numbers. Thus apart from specifying a certain class of embedded curves, the geodesic equation also determines a preferred class of parameterizations on each of the curves. Accordingly, solutions of ({{EquationNote|1}}) are called geodesics with '''affine parameter'''.

An affine connection is ''determined by'' its family of affinely parameterized geodesics, up to [[torsion tensor|torsion]] {{harv|Spivak|1999|loc=Chapter 6, Addendum I}}.  The torsion itself does not, in fact, affect the family of geodesics, since the geodesic equation depends only on the symmetric part of the connection.  More precisely, if <math>\nabla, \bar{\nabla}</math> are two connections such that the difference tensor
:<math>D(X,Y) = \nabla_XY-\bar{\nabla}_XY</math>
is [[skew-symmetric matrix|skew-symmetric]], then <math>\nabla</math> and <math>\bar{\nabla}</math> have the same geodesics, with the same affine parameterizations. Furthermore, there is a unique connection having the same geodesics as  <math>\nabla</math>, but with vanishing torsion.

Geodesics without a particular parameterization are described by a [[projective connection]].