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 multili ...

the theorem of the three geodesics, also known as Lyusternik–Schnirelmann theorem, states that every

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

with the topology of a

sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

has at least three simple

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 flo ...

s (i.e. three embedded geodesic circles). The result can also be extended to quasigeodesics on a convex polyhedron, and to closed geodesics of reversible Finsler 2-spheres. The theorem is sharp: although every Riemannian 2-sphere contains infinitely many distinct closed geodesics, only three of them are guaranteed to have no self-intersections. For example, by a result of

Morse Morse may refer to: People * Morse (surname) * Morse Goodman (1917-1993), Anglican Bishop of Calgary, Canada * Morse Robb (1902–1992), Canadian inventor and entrepreneur Geography Antarctica * Cape Morse, Wilkes Land * Mount Morse, Churchi ...

if the lengths of three principal axes of an ellipsoid are distinct, but sufficiently close to each other, then the ellipsoid has only three simple closed geodesics.

History and proof

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 connection. ...

, on a Riemannian surface, is a curve that is locally straight at each of its points. For instance, on the

Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...

the geodesics are

lines Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Arts ...

, and on the surface of a sphere the geodesics are

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 geomet ...

s. The shortest path in the surface between two points is always a geodesic, but other geodesics may exist as well. A geodesic is said to be a

if it returns to its starting point and starting direction; in doing so it may cross itself multiple times. The theorem of the three geodesics says that for surfaces

homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...

to the sphere, there exist at least three non-self-crossing closed geodesics. There may be more than three, for instance, the sphere itself has infinitely many. This result stems from the mathematics of ocean navigation, where the surface of the earth can be modeled accurately by an

ellipsoid An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface; that is, a surface that may be defined as the ...

, and from the study of the

geodesics on an ellipsoid The study of geodesics on an ellipsoid arose in connection with geodesy specifically with the solution of triangulation networks. The figure of the Earth is well approximated by an ''oblate ellipsoid'', a slightly flattened sphere. A ''geodes ...

, the shortest paths for ships to travel. In particular, a nearly-spherical triaxial ellipsoid has only three simple closed geodesics, its equators.. In 1905,

Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...

conjectured that every smooth surface topologically equivalent to a sphere likewise contains at least three simple closed geodesics, and in 1929

Lazar Lyusternik Lazar Aronovich Lyusternik (also Lusternik, Lusternick, Ljusternik; ; 31 December 1899, in Zduńska Wola, Congress Poland, Russian Empire – 23 July 1981, in Moscow, Soviet Union) was a Soviet mathematician. He is famous for his work in topol ...

and Lev Schnirelmann published a proof of the conjecture; while the general topological argument of the proof was correct, it employed a deformation result that was later found to be flawed. Several authors proposed unsatisfactory solutions of the gap. A universally accepted solution was provided in the 1980s by Grayson, by means of the

curve shortening flow In mathematics, the curve-shortening flow is a process that modifies a smooth curve in the Euclidean plane by moving its points perpendicularly to the curve at a speed proportional to the curvature. The curve-shortening flow is an example of a ge ...

Generalizations

A strengthened version of the theorem states that, on any Riemannian surface that is topologically a sphere, there necessarily exist three simple closed geodesics whose length is at most proportional to the diameter of the surface. The number of closed geodesics of length at most ''L'' on a smooth topological sphere grows in proportion to ''L''/log ''L'', but not all such geodesics can be guaranteed to be simple. On compact

hyperbolic Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry. The following phenomena are described as ''hyperbolic'' because they ...

Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...

s, there are infinitely many simple closed geodesics, but only finitely many with a given length bound. They are encoded analytically by the

Selberg zeta function The Selberg zeta-function was introduced by . It is analogous to the famous Riemann zeta function : \zeta(s) = \prod_ \frac where \mathbb is the set of prime numbers. The Selberg zeta-function uses the lengths of simple closed geodesics inste ...

. The growth rate of the number of simple closed geodesics, as a function of their length, was investigated by

Maryam Mirzakhani Maryam Mirzakhani ( fa, مریم میرزاخانی, ; 12 May 1977 – 14 July 2017) was an Iranian mathematician and a professor of mathematics at Stanford University. Her research topics included Teichmüller theory, hyperbolic geometry, ...

. The existence of three simple closed geodesics also holds for any reversible

Finsler metric In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold where a (possibly asymmetric) Minkowski functional is provided on each tangent space , that enables one to define the length of any smooth curve ...

on the 2-sphere.

Non-smooth metrics

It is also possible to define geodesics on some surfaces that are not smooth everywhere, such as

convex polyhedra A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...

. The surface of a convex polyhedron has a metric that is locally Euclidean except at the vertices of the polyhedron, and a curve that avoids the vertices is a geodesic if it follows straight line segments within each face of the polyhedron and stays straight across each polyhedron edge that it crosses. Although some polyhedra have simple closed geodesics (for instance, the

regular tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...

and

disphenoid In geometry, a disphenoid () is a tetrahedron whose four faces are congruent acute-angled triangles. It can also be described as a tetrahedron in which every two edges that are opposite each other have equal lengths. Other names for the same sh ...

s have infinitely many closed geodesics, all simple). others do not. In particular, a simple closed geodesic of a convex polyhedron would necessarily bisect the total

angular defect In geometry, the (angular) defect (or deficit or deficiency) means the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the Euclidean plane would. The opposite notion is the excess. Classically the defe ...

of the vertices, and

almost all In mathematics, the term "almost all" means "all but a negligible amount". More precisely, if X is a set, "almost all elements of X" means "all elements of X but those in a negligible subset of X". The meaning of "negligible" depends on the math ...

polyhedra do not have such bisectors. Nevertheless, the theorem of the three geodesics can be extended to convex polyhedra by considering quasigeodesics, curves that are geodesic except at the vertices of the polyhedra and that have angles less than on both sides at each vertex they cross. A version of the theorem of the three geodesics for convex polyhedra states that all polyhedra have at least three simple closed quasigeodesics; this can be proved by approximating the polyhedron by a smooth surface and applying the theorem of the three geodesics to this surface. It is an open problem whether any of these quasigeodesics can be constructed in

polynomial time In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by ...

References

{{reflist, 30em Theorems in differential geometry Geodesic (mathematics)