differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...

and

dynamical systems In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a ...

, a closed geodesic on a

Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ...

is a

geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connecti ...

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 In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...

on the

tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...

of the manifold.

Definition

In a

(''M'',''g''), a closed geodesic is a curve

\gamma:\mathbb R\rightarrow M

that is a

for the metric ''g'' and is periodic. Closed geodesics can be characterized by means of a variational principle. Denoting by

\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:\Lambda M\rightarrow\mathbb R

, defined by :

E(\gamma)=\int_0^1 g_(\dot\gamma(t),\dot\gamma(t))\,\mathrmt.

\gamma

is a closed geodesic of period ''p'', the reparametrized curve

t\mapsto\gamma(pt)

is a closed geodesic of period 1, and therefore it is a critical point of ''E''. If

\gamma

is a critical point of ''E'', so are the reparametrized curves

\gamma^m

, for each

m\in\mathbb N

, defined by

\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

S^n\subset\mathbb R^

with the standard round Riemannian metric, every

great circle In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geome ...

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. 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.

References

{{Reflist * Besse, A.: "Manifolds all of whose geodesics are closed", ''Ergebisse Grenzgeb. Math.'', no. 93, Springer, Berlin, 1978. Differential geometry Dynamical systems Geodesic (mathematics)

Definition

Examples

See also

References