Closed geodesic

In differential geometry and dynamical systems, a closed geodesic on a Riemannian manifold is a geodesic that returns to its starting point with the same tangent direction. It may be formalized as the projection of a closed orbit of the geodesic flow on the tangent space of the manifold.

Contents

• 
  1 Definition
  


• 
  2 Examples
  


• 
  3 See also
  


• 
  4 References
  

Definition

In a Riemannian manifold (M,g), a closed geodesic is a curve γ:R→M{displaystyle gamma :mathbb {R} rightarrow M} that is a geodesic for the metric g and is periodic.

Closed geodesics can be characterized by means of a variational principle. Denoting by ΛM{displaystyle Lambda M} the space of smooth 1-periodic curves on M, closed geodesics of period 1 are precisely the critical points of the energy function E:ΛM→R{displaystyle E:Lambda Mrightarrow mathbb {R} }, defined by

E(γ)=∫01gγ(t)(γ˙(t),γ˙(t))dt.{displaystyle E(gamma )=int _{0}^{1}g_{gamma (t)}({dot {gamma }}(t),{dot {gamma }}(t)),mathrm {d} t.}

If γ{displaystyle gamma } is a closed geodesic of period p, the reparametrized curve t↦γ(pt){displaystyle tmapsto gamma (pt)} is a closed geodesic of period 1, and therefore it is a critical point of E. If γ{displaystyle gamma } is a critical point of E, so are the reparametrized curves γm{displaystyle gamma ^{m}}, for each m∈N{displaystyle min mathbb {N} }, defined by γm(t):=γ(mt){displaystyle gamma ^{m}(t):=gamma (mt)}. Thus every closed geodesic on M gives rise to an infinite sequence of critical points of the energy E.

Examples

On the unit sphere Sn⊂Rn+1{displaystyle S^{n}subset mathbb {R} ^{n+1}} with the standard round Riemannian metric, every great circle is an example of a closed geodesic. Thus, on the sphere, all geodesics are closed. On a smooth surface topologically equivalent to the sphere, this may not be true, but there are always at least three simple closed geodesics; this is the theorem of the three geodesics.^[1] Manifolds all of whose geodesics are closed have been thoroughly investigated in the mathematical literature. On a compact hyperbolic surface, whose fundamental group has no torsion, closed geodesics are in one-to-one correspondence with non-trivial conjugacy classes of elements in the Fuchsian group of the surface.

See also

• Curve-shortening flow


• Selberg trace formula


• Selberg zeta function


• Zoll surface

References

1. 
   ^ Grayson, Matthew A. (1989), "Shortening embedded curves" (PDF), Annals of Mathematics, Second Series, 129 (1): 71–111, doi:10.2307/1971486, MR 0979601.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}.
   

• 
  Besse, A.: "Manifolds all of whose geodesics are closed", Ergebisse Grenzgeb. Math., no. 93, Springer, Berlin, 1978.


• 
  Klingenberg, W.: "Lectures on closed geodesics", Grundlehren der Mathematischen Wissenschaften, Vol. 230. Springer-Verlag, Berlin-New York, 1978. x+227 pp.
  ISBN 3-540-08393-6
  

This page is only for reference, If you need detailed information, please check here