Geodesic flow

A flow whose phase space is the manifold of vectors tangent to a Riemannian (more generally, a Finsler) manifold (the so-called configuration manifold of the flow), while the motion is defined as follows. Let be a vector tangent to at a point and let its length be . Let a geodesic line on be drawn through in the direction and let be the point on the distance of which from along is (where that direction on is considered to be positive which is identical with the direction of the vector at ). One then has . In case , one has . It turns out that , and for this reason the vectors of unit length form a submanifold in that is invariant with respect to . A geodesic flow is often understood to mean the restriction of the flow to . In local coordinates a geodesic flow is described by a system of ordinary second-order differential equations, which, in the Riemannian case, have the form

where is the -th coordinate of the point and the are the Christoffel symbols (cf. Christoffel symbol) of the second kind. A geodesic flow preserves the natural symplectic structure on , while its restriction to preserves the corresponding contact structure. Geodesic flows obviously play an important role in geometry (see also Variational calculus in the large). If, in addition, a certain change of time is made, then it is possible to reduce the description of the motion of a mechanical system, in accordance with the Maupertuis principle, to a geodesic flow.

Comments

For the application to mechanical systems, see, for example, Section 45D and Appendices 1J and 4F in [a2]. The geodesic flows on (compact) manifolds of negative curvature have interesting dynamical properties (cf. Hyperbolic set; -system). See [a1]. For applications of geodesic flows in differential geometry, see [a3], Chapt. 3.

References

[a1]	D.V. Anosov, "Geodesic flows on compact Riemannian manifolds of negative curvature" Proc. Steklov Inst. Math. , 90 (1969) Trudy Mat. Inst. Steklov. , 90 (1969)
[a2]	V.I. Arnol'd, "Mathematical methods of classical mechanics" , Springer (1978) (Translated from Russian)
[a3]	W. Klingenberg, "Riemannian geometry" , Springer (1982) (Translated from German)