Editing Geodesic (section)

===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.