Geodesic

This article is about geodesics in general. For geodesics in general relativity, seeGeodesic (general relativity).For the study of Earth's shape, seeGeodesy.For the application on Earth, seeEarth geodesic.For other uses, seeGeodesic (disambiguation).

Klein quarticwith 28 geodesics(marked by 7 colors and 4 patterns)

Ingeometry,ageodesic(/ˌdʒiː.əˈdɛsɪk,-oʊ-,-ˈdiːsɪk,-zɪk/)is acurverepresenting in some sense the shortestpath (arc) between two points in asurface,or more generally in aRiemannian manifold.The term also has meaning in anydifferentiable manifoldwith aconnection.It is a generalization of the notion of a "straight line".

The noungeodesicand the adjectivegeodeticcome fromgeodesy,the science of measuring the size and shape ofEarth,though many of the underlying principles can be applied to anyellipsoidalgeometry. In the original sense, a geodesic was the shortest route between two points on the Earth'ssurface.For aspherical Earth,it is asegmentof agreat circle(see alsogreat-circle distance). The term has since been generalized to more abstract mathematical spaces; for example, ingraph theory,one might consider ageodesicbetween twovertices/nodes of agraph.

In aRiemannian manifoldor submanifold, geodesics are characterised by the property of having vanishinggeodesic curvature.More generally, in the presence of anaffine connection,a geodesic is defined to be a curve whosetangent vectorsremain parallel if they aretransportedalong it. Applying this to theLevi-Civita connectionof aRiemannian metricrecovers the previous notion.

Geodesics are of particular importance ingeneral relativity.Timelikegeodesics in general relativitydescribe the motion offree fallingtest particles.

Introduction

A locally shortest path between two given points in a curved space, assumedto be aRiemannian manifold,can be defined by using theequationfor thelengthof acurve(a functionffrom anopen intervalofRto the space), and then minimizing this length between the points using thecalculus of variations.This has some minor technical problems because there is an infinite-dimensional space of different ways to parameterize the shortest path. It is simpler to restrict the set of curves to those that are parameterized "with constant speed" 1, meaning that the distance fromf(s) tof(t) along the curve equals |s−t|. Equivalently, a different quantity may be used, termed the energy of the curve; minimizing the energy leads to the same equations for a geodesic (here "constant velocity" is a consequence of minimization).Intuitively, one can understand this second formulation by noting that anelastic bandstretched between two points will contract its width, and in so doing will minimize its energy. The resulting shape of the band is a geodesic.

It is possible that several different curves between two points minimize the distance, as is the case for two diametrically opposite points on a sphere. In such a case, any of these curves is a geodesic.

A contiguous segment of a geodesic is again a geodesic.

In general, geodesics are not the same as "shortest curves" between two points, though the two concepts are closely related. The difference is that geodesics are onlylocallythe shortest distance between points, and are parameterized with "constant speed". Going the "long way round" on agreat circlebetween two points on a sphere is a geodesic but not the shortest path between the points. The map$t\to t^{2}$from the unit interval on the real number line to itself gives the shortest path between 0 and 1, but is not a geodesic because the velocity of the corresponding motion of a point is not constant.

Geodesics are commonly seen in the study ofRiemannian geometryand more generallymetric geometry.Ingeneral relativity,geodesics inspacetimedescribe the motion ofpoint particlesunder the influence of gravity alone. In particular, the path taken by a falling rock, an orbitingsatellite,or the shape of aplanetary orbitare all geodesicsin curved spacetime. More generally, the topic ofsub-Riemannian geometrydeals with the paths that objects may take when they are not free, and their movement is constrained in various ways.

This article presents the mathematical formalism involved in defining, finding, and proving the existence of geodesics, in the case ofRiemannian manifolds.The articleLevi-Civita connectiondiscusses the more general case of apseudo-Riemannian manifoldandgeodesic (general relativity)discusses the special case of general relativity in greater detail.

Examples

Ageodesic on a triaxial ellipsoid.

If an insect is placed on a surface and continually walks "forward", by definition it will trace out a geodesic.

The most familiar examples are the straight lines inEuclidean geometry.On asphere,the images of geodesics are thegreat circles.The shortest path from pointAto pointBon a sphere is given by the shorterarcof the great circle passing throughAandB.IfAandBareantipodal points,then there areinfinitely manyshortest paths between them.Geodesics on an ellipsoidbehave in a more complicated way than on a sphere; in particular, they are not closed in general (see figure).

Triangles

See also:Gauss–Bonnet theorem § For triangles,andToponogov's theorem

A geodesic triangle on the sphere.

Ageodesic triangleis formed by the geodesics joining each pair out of three points on a given surface. On the sphere, the geodesics aregreat circlearcs, forming aspherical triangle.

Geodesic triangles in spaces of positive (top), negative (middle) and zero (bottom) curvature.

Metric geometry

Inmetric geometry,a geodesic is a curve which is everywherelocallyadistanceminimizer. More precisely, acurveγ:I→Mfrom an intervalIof the reals to themetric spaceMis ageodesicif there is aconstantv≥ 0such that for anyt∈Ithere is a neighborhoodJoftinIsuch that for anyt₁, t₂∈Jwe have

$d(\gamma (t_{1}),\gamma (t_{2}))=v\left|t_{1}-t_{2}\right|.$

This generalizes the notion of geodesic for Riemannian manifolds. However, in metric geometry the geodesic considered is often equipped withnatural parameterization,i.e. in the above identityv= 1 and

$d(\gamma (t_{1}),\gamma (t_{2}))=\left|t_{1}-t_{2}\right|.$

If the last equality is satisfied for allt₁,t₂∈I,the geodesic is called aminimizing geodesicorshortest path.

In general, a metric space may have no geodesics, except constant curves. At the other extreme, any two points in alength metric spaceare joined by a minimizing sequence ofrectifiable paths,although this minimizing sequence need not converge to a geodesic.

Riemannian geometry

In aRiemannian manifoldMwithmetric tensorg,the lengthLof a continuously differentiable curve γ: [a,b] →Mis defined by

$L(\gamma )=\int _{a}^{b}{\sqrt {g_{\gamma (t)}({\dot {\gamma }}(t),{\dot {\gamma }}(t))}}\,dt.$

The distanced(p, q) between two pointspandqofMis defined as theinfimumof the length taken over all continuous, piecewise continuously differentiable curves γ: [a,b] →Msuch that γ(a) =pand γ(b) =q.In Riemannian geometry, all geodesics are locally distance-minimizing paths, but the converse is not true. In fact, only paths that are both locally distance minimizing and parameterized proportionately to arc-length are geodesics. Another equivalent way of defining geodesics on a Riemannian manifold, is to define them as the minima of the followingactionorenergy functional

$E(\gamma )={\frac {1}{2}}\int _{a}^{b}g_{\gamma (t)}({\dot {\gamma }}(t),{\dot {\gamma }}(t))\,dt.$

All minima ofEare also minima ofL,butLis a bigger set since paths that are minima ofLcan be arbitrarily re-parameterized (without changing their length), while minima ofEcannot. For a piecewise$C^{1}$curve (more generally, a$W^{1,2}$curve), theCauchy–Schwarz inequalitygives

$L(\gamma )^{2}\leq 2(b-a)E(\gamma )$

with equality if and only if$g(\gamma ',\gamma ')$is equal to a constant a.e.; the path should be travelled at constant speed. It happens that minimizers of$E(\gamma )$also minimize$L(\gamma )$,because they turn out to be affinely parameterized, and the inequality is an equality. The usefulness of this approach is that the problem of seeking minimizers ofEis a more robust variational problem. Indeed,Eis a "convex function" of$\gamma$,so that within each isotopy class of "reasonable functions", one ought to expect existence, uniqueness, and regularity of minimizers. In contrast, "minimizers" of the functional$L(\gamma )$are generally not very regular, because arbitrary reparameterizations are allowed.

TheEuler–Lagrange equationsof motion for the functionalEare then given in local coordinates by

${\frac {d^{2}x^{\lambda }}{dt^{2}}}+\Gamma _{\mu \nu }^{\lambda }{\frac {dx^{\mu }}{dt}}{\frac {dx^{\nu }}{dt}}=0,$

where$\Gamma _{\mu \nu }^{\lambda }$are theChristoffel symbolsof the metric. This is thegeodesic equation,discussedbelow.

Calculus of variations

Techniques of the classicalcalculus of variationscan be applied to examine the energy functionalE.Thefirst variationof energy is defined in local coordinates by

$\delta E(\gamma )(\varphi )=\left.{\frac {\partial }{\partial t}}\right|_{t=0}E(\gamma +t\varphi ).$

Thecritical pointsof the first variation are precisely the geodesics. Thesecond variationis defined by

$\delta ^{2}E(\gamma )(\varphi,\psi )=\left.{\frac {\partial ^{2}}{\partial s\,\partial t}}\right|_{s=t=0}E(\gamma +t\varphi +s\psi ).$

In an appropriate sense, zeros of the second variation along a geodesic γ arise alongJacobi fields.Jacobi fields are thus regarded as variations through geodesics.

By applying variational techniques fromclassical mechanics,one can also regardgeodesics as Hamiltonian flows.They are solutions of the associatedHamilton equations,with(pseudo-)Riemannian metrictaken asHamiltonian.

Affine geodesics

See also:Geodesics in general relativity

Ageodesicon asmooth manifoldMwith anaffine connection∇ is defined as acurveγ(t) such thatparallel transportalong the curve preserves the tangent vector to the curve, so

$\nabla _{\dot {\gamma }}{\dot {\gamma }}=0$(1)

at each point along the curve, where${\dot {\gamma }}$is the derivative with respect to$t$.More precisely, in order to define the covariant derivative of${\dot {\gamma }}$it is necessary first to extend${\dot {\gamma }}$to a continuously differentiablevector fieldin anopen set.However, the resulting value of (1) is independent of the choice of extension.

Usinglocal coordinatesonM,we can write thegeodesic equation(using thesummation convention) as

${\frac {d^{2}\gamma ^{\lambda }}{dt^{2}}}+\Gamma _{\mu \nu }^{\lambda }{\frac {d\gamma ^{\mu }}{dt}}{\frac {d\gamma ^{\nu }}{dt}}=0\,$

where$\gamma ^{\mu }=x^{\mu }\circ \gamma (t)$are the coordinates of the curve γ(t) and$\Gamma _{\mu \nu }^{\lambda }$are theChristoffel symbolsof the connection ∇. This is anordinary differential equationfor the coordinates. It has a unique solution, given an initial position and an initial velocity. Therefore, from the point of view ofclassical mechanics,geodesics can be thought of as trajectories offree particlesin a manifold. Indeed, the equation$\nabla _{\dot {\gamma }}{\dot {\gamma }}=0$means that theacceleration vectorof the curve has no components in the direction of the surface (and therefore it is perpendicular to the tangent plane of the surface at each point of the curve). So, the motion is completely determined by the bending of the surface. This is also the idea of general relativity where particles move on geodesics and the bending is caused by gravity.

Existence and uniqueness

Thelocal existence and uniqueness theoremfor geodesics states that geodesics on a smooth manifold with anaffine connectionexist, and are unique. More precisely:

For any pointpinMand for any vectorVinT_pM(thetangent spacetoMatp) there exists a unique geodesic$\gamma \,$:I→Msuch that

$\gamma (0)=p\,$and

${\dot {\gamma }}(0)=V,$

whereIis a maximalopen intervalinRcontaining 0.

The proof of this theorem follows from the theory ofordinary differential equations,by noticing that the geodesic equation is a second-order ODE. Existence and uniqueness then follow from thePicard–Lindelöf theoremfor the solutions of ODEs with prescribed initial conditions. γ dependssmoothlyon bothpandV.

In general,Imay not be all ofRas for example for an open disc inR².Anyγextends to all ofℝif and only ifMisgeodesically complete.

Geodesic flow

Geodesicflowis a localR-actionon thetangent bundleTMof a manifoldMdefined in the following way

$G^{t}(V)={\dot {\gamma }}_{V}(t)$

wheret∈R,V∈TMand$\gamma _{V}$denotes the geodesic with initial data${\dot {\gamma }}_{V}(0)=V$.Thus,$G^{t}(V)=\exp(tV)$is theexponential mapof the vectortV.A closed orbit of the geodesic flow corresponds to aclosed geodesiconM.

On a (pseudo-)Riemannian manifold, the geodesic flow is identified with aHamiltonian flowon the cotangent bundle. TheHamiltonianis then given by the inverse of the (pseudo-)Riemannian metric, evaluated against thecanonical one-form.In particular the flow preserves the (pseudo-)Riemannian metric$g$,i.e.

$g(G^{t}(V),G^{t}(V))=g(V,V).\,$

In particular, whenVis a unit vector,$\gamma _{V}$remains unit speed throughout, so the geodesic flow is tangent to theunit tangent bundle.Liouville's theoremimplies invariance of a kinematic measure on the unit tangent bundle.

Geodesic spray

Further information:Spray (mathematics) § Geodesic

The geodesic flow defines a family of curves in thetangent bundle.The derivatives of these curves define avector fieldon thetotal spaceof the tangent bundle, known as thegeodesicspray.

More precisely, an affine connection gives rise to a splitting of thedouble tangent bundleTTMintohorizontalandvertical bundles:

$TTM=H\oplus V.$

The geodesic spray is the unique horizontal vector fieldWsatisfying

$\pi _{*}W_{v}=v\,$

at each pointv∈ TM;hereπ_∗:TTM→ TMdenotes thepushforward (differential)along the projectionπ:TM→Massociated to the tangent bundle.

More generally, the same construction allows one to construct a vector field for anyEhresmann connectionon the tangent bundle. For the resulting vector field to be a spray (on the deleted tangent bundle TM\ {0}) it is enough that the connection be equivariant under positive rescalings: it need not be linear. That is, (cf.Ehresmann connection#Vector bundles and covariant derivatives) it is enough that the horizontal distribution satisfy

$H_{\lambda X}=d(S_{\lambda })_{X}H_{X}\,$

for everyX∈ TM\ {0} and λ > 0. Hered(S_λ) is thepushforwardalong the scalar homothety$S_{\lambda }:X\mapsto \lambda X.$A particular case of a non-linear connection arising in this manner is that associated to aFinsler manifold.

Affine and projective geodesics

Equation (1) is invariant under affine reparameterizations; that is, parameterizations of the form

$t\mapsto at+b$

whereaandbare constant real numbers. Thus apart from specifying a certain class of embedded curves, the geodesic equation also determines a preferred class of parameterizations on each of the curves. Accordingly, solutions of (1) are called geodesics withaffine parameter.

An affine connection isdetermined byits family of affinely parameterized geodesics, up totorsion(Spivak 1999,Chapter 6, Addendum I). The torsion itself does not, in fact, affect the family of geodesics, since the geodesic equation depends only on the symmetric part of the connection. More precisely, if$\nabla,{\bar {\nabla }}$are two connections such that the difference tensor

$D(X,Y)=\nabla _{X}Y-{\bar {\nabla }}_{X}Y$

isskew-symmetric,then$\nabla$and${\bar {\nabla }}$have the same geodesics, with the same affine parameterizations. Furthermore, there is a unique connection having the same geodesics as$\nabla$,but with vanishing torsion.

Geodesics without a particular parameterization are described by aprojective connection.

Computational methods

Efficient solvers for the minimal geodesic problem on surfaces have been proposed by Mitchell,Kimmel,Crane,and others.

Ribbon test

A ribbon "test" is a way of finding a geodesic on a physical surface.The idea is to fit a bit of paper around a straight line (a ribbon) onto a curved surface as closely as possible without stretching or squishing the ribbon (without changing its internal geometry).

For example, when a ribbon is wound as a ring around a cone, the ribbon would not lie on the cone's surface but stick out, so that circle is not a geodesic on the cone. If the ribbon is adjusted so that all its parts touch the cone's surface, it would give an approximation to a geodesic.

Mathematically the ribbon test can be formulated as finding a mapping$f:N(\ell )\to S$of aneighborhood$N$of a line$\ell$in a plane into a surface$S$so that the mapping$f$"doesn't change the distances around$\ell$by much "; that is, at the distance$\varepsilon$from$l$we have$g_{N}-f^{*}(g_{S})=O(\varepsilon ^{2})$where$g_{N}$and$g_{S}$aremetricson$N$and$S$.

Applications

Geodesics serve as the basis to calculate:

• geodesic airframes; seegeodesic airframeorgeodetic airframe
• geodesic structures – for examplegeodesic domes
• horizontal distances on or near Earth; seeEarth geodesics
• mapping images on surfaces, for rendering; seeUV mapping
• robotmotion planning(e.g., when painting car parts); seeShortest path problem
• geodesic shortest path (GSP) correction overPoisson surface reconstruction(e.g. indigital dentistry); without GSP reconstruction often results in self-intersections within the surface

See also

• Introduction to the mathematics of general relativity– non-technical introduction to the mathematics of general relativityPages displaying wikidata descriptions as a fallback
• Clairaut's relation– Formula in classical differential geometryPages displaying short descriptions of redirect targets
• Differentiable curve– Study of curves from a differential point of view
• Differential geometry of surfaces
• Geodesic circle
• Hopf–Rinow theorem– Gives equivalent statements about the geodesic completeness of Riemannian manifolds
• Intrinsic metric– Concept in geometry/topology
• Isotropic line
• Jacobi field
• Morse theory– Analyzes the topology of a manifold by studying differentiable functions on that manifold
• Zoll surface– Surface homeomorphic to a sphere
• The spider and the fly problem– Recreational geodesics problem

References

• Spivak, Michael(1999),A Comprehensive introduction to differential geometry (Volume 2),Houston, TX: Publish or Perish,ISBN978-0-914098-71-3

Further reading

• Adler, Ronald; Bazin, Maurice; Schiffer, Menahem (1975),Introduction to General Relativity(2nd ed.), New York:McGraw-Hill,ISBN978-0-07-000423-8.See chapter 2.
• Abraham, Ralph H.;Marsden, Jerrold E.(1978),Foundations of mechanics,London: Benjamin-Cummings,ISBN978-0-8053-0102-1.See section 2.7.
• Jost, Jürgen (2002),Riemannian Geometry and Geometric Analysis,Berlin, New York:Springer-Verlag,ISBN978-3-540-42627-1.See section 1.4.
• Kobayashi, Shoshichi; Nomizu, Katsumi (1996),Foundations of Differential Geometry,vol. 1 (New ed.), Wiley-Interscience,ISBN0-471-15733-3.
• Landau, L. D.;Lifshitz, E. M.(1975),Classical Theory of Fields,Oxford: Pergamon,ISBN978-0-08-018176-9.See section 87.
• Misner, Charles W.;Thorne, Kip;Wheeler, John Archibald(1973),Gravitation,W. H. Freeman,ISBN978-0-7167-0344-0
• Ortín, Tomás (2004),Gravity and strings,Cambridge University Press,ISBN978-0-521-82475-0.Note especially pages 7 and 10.
• Volkov, Yu.A. (2001) [1994],"Geodesic line",Encyclopedia of Mathematics,EMS Press.
• Weinberg, Steven(1972),Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity,New York:John Wiley & Sons,ISBN978-0-471-92567-5.See chapter 3.

External links

• Geodesics Revisited— Introduction to geodesics including two ways of derivation of the equation of geodesic with applications in geometry (geodesic on a sphere and on atorus), mechanics (brachistochrone) and optics (light beam in inhomogeneous medium).
• Totally geodesic submanifoldat the Manifold Atlas