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50-979: The latest version is 3.0d3. Marc Culler , Nathan Dunfield and collaborators have extended the SnapPea kernel and written Python extension modules which allow the kernel to be used in a Python program or in the interpreter. They also provide a graphical user interface written in Python which runs under most operating systems (see external links below). The following people are credited in SnapPea 2.5.3's list of acknowledgments: Colin Adams , Bill Arveson , Pat Callahan , Joe Christy , Dave Gabai , Charlie Gunn , Martin Hildebrand , Craig Hodgson , Diane Hoffoss , A. C. Manoharan , Al Marden , Dick McGehee , Rob Meyerhoff , Lee Mosher , Walter Neumann , Carlo Petronio , Mark Phillips , Alan Reid , and Makoto Sakuma . The C source code

100-682: A W 1 , 2 {\displaystyle W^{1,2}} curve), the Cauchy–Schwarz inequality gives with equality if and only if g ( γ ′ , γ ′ ) {\displaystyle g(\gamma ',\gamma ')} is equal to a constant a.e.; the path should be travelled at constant speed. It happens that minimizers of E ( γ ) {\displaystyle E(\gamma )} also minimize L ( γ ) {\displaystyle L(\gamma )} , because they turn out to be affinely parameterized, and

150-451: A connection . It is a generalization of the notion of a " straight line ". The noun geodesic and the adjective geodetic come from geodesy , the science of measuring the size and shape of Earth , though many of the underlying principles can be applied to any ellipsoidal geometry. In the original sense, a geodesic was the shortest route between two points on the Earth's surface . For

200-400: A minimizing geodesic or shortest path . In general, a metric space may have no geodesics, except constant curves. At the other extreme, any two points in a length metric space are joined by a minimizing sequence of rectifiable paths , although this minimizing sequence need not converge to a geodesic. The metric Hopf-Rinow theorem provides situations where a length space is automatically

250-426: A spherical Earth , it is a segment of a great circle (see also great-circle distance ). The term has since been generalized to more abstract mathematical spaces; for example, in graph theory , one might consider a geodesic between two vertices /nodes of a graph . In a Riemannian manifold or submanifold, geodesics are characterised by the property of having vanishing geodesic curvature . More generally, in

300-474: A Fellow of the American Mathematical Society . Geodesic In geometry , a geodesic ( / ˌ dʒ iː . ə ˈ d ɛ s ɪ k , - oʊ -, - ˈ d iː s ɪ k , - z ɪ k / ) 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

350-405: A Riemannian manifold, is to define them as the minima of the following action or energy functional All minima of E are also minima of L , but L is a bigger set since paths that are minima of L can be arbitrarily re-parameterized (without changing their length), while minima of E cannot. For a piecewise C 1 {\displaystyle C^{1}} curve (more generally,

400-417: A curved space, assumed to be a Riemannian manifold , can be defined by using the equation for the length of a curve (a function f from an open interval of R to the space), and then minimizing this length between the points using the calculus of variations . This has some minor technical problems because there is an infinite-dimensional space of different ways to parameterize the shortest path. It

450-414: A falling rock, an orbiting satellite , or the shape of a planetary orbit are all geodesics in curved spacetime. More generally, the topic of sub-Riemannian geometry deals 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

500-490: A family of curves in the tangent bundle . The derivatives of these curves define a vector field on the total space of the tangent bundle, known as the geodesic spray . More precisely, an affine connection gives rise to a splitting of the double tangent bundle TT M into horizontal and vertical bundles : The geodesic spray is the unique horizontal vector field W satisfying at each point v  ∈ T M ; here π ∗  : TT M  → T M denotes

550-458: A geodesic space. Common examples of geodesic metric spaces that are often not manifolds include metric graphs , (locally compact) metric polyhedral complexes , infinite-dimensional pre-Hilbert spaces , and real trees . In a Riemannian manifold M with metric tensor g , the length L of a continuously differentiable curve γ : [ a , b ] →  M is defined by The distance d ( p ,  q ) between two points p and q of M

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600-401: A given surface. On the sphere, the geodesics are great circle arcs, forming a spherical triangle . In metric geometry , a geodesic is a curve which is everywhere locally a distance minimizer. More precisely, a curve γ  : I → M from an interval I of the reals to the metric space M is a geodesic if there is a constant v ≥ 0 such that for any t ∈ I there is

650-502: A hyperbolic structure on the Dehn-filled manifold. Its volume is the sum of the volumes of the adjusted tetrahedra. SnapPea is usually able to compute the canonical decomposition of a cusped hyperbolic 3-manifold from a given ideal triangulation. If not, then it randomly retriangulates and tries again. This has never been known to fail. The canonical decomposition allows SnapPea to tell two cusped hyperbolic 3-manifolds apart by turning

700-405: A manifold. Indeed, the equation ∇ γ ˙ γ ˙ = 0 {\displaystyle \nabla _{\dot {\gamma }}{\dot {\gamma }}=0} means that the acceleration vector of 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

750-422: A method for describing the geometric shape of each hyperbolic tetrahedron by a complex number and a set of nonlinear equations of complex variables whose solution would give a complete hyperbolic metric on the 3-manifold. These equations consist of edge equations and cusp (completeness) equations . SnapPea uses an iterative method utilizing Newton's method to search for solutions. If no solution exists, then this

800-400: A neighborhood J of t in I such that for any t 1 ,  t 2 ∈ J we have This generalizes the notion of geodesic for Riemannian manifolds. However, in metric geometry the geodesic considered is often equipped with natural parameterization , i.e. in the above identity v  = 1 and If the last equality is satisfied for all t 1 , t 2 ∈ I , the geodesic is called

850-420: A sphere is given by the shorter arc of the great circle passing through A and B . If A and B are antipodal points , then there are infinitely many shortest paths between them. Geodesics on an ellipsoid behave in a more complicated way than on a sphere; in particular, they are not closed in general (see figure). A geodesic triangle is formed by the geodesics joining each pair out of three points on

900-445: Is determined by its family of affinely parameterized geodesics, up to torsion ( 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 ∇ , ∇ ¯ {\displaystyle \nabla ,{\bar {\nabla }}} are two connections such that

950-594: Is geodesically complete . Geodesic flow is a local R - action on the tangent bundle TM of a manifold M defined in the following way where t  ∈  R , V  ∈  TM and γ V {\displaystyle \gamma _{V}} denotes the geodesic with initial data γ ˙ V ( 0 ) = V {\displaystyle {\dot {\gamma }}_{V}(0)=V} . Thus, G t ( V ) = exp ⁡ ( t V ) {\displaystyle G^{t}(V)=\exp(tV)}

1000-499: Is defined as a curve γ( t ) such that parallel transport along the curve preserves the tangent vector to the curve, so at each point along the curve, where γ ˙ {\displaystyle {\dot {\gamma }}} is the derivative with respect to t {\displaystyle t} . More precisely, in order to define the covariant derivative of γ ˙ {\displaystyle {\dot {\gamma }}} it

1050-494: Is defined as the infimum of the length taken over all continuous, piecewise continuously differentiable curves γ : [ a , b ] →  M such that γ( a ) =  p and 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

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1100-404: Is enough that the horizontal distribution satisfy for every X  ∈ T M  \ {0} and λ > 0. Here d ( S λ ) is the pushforward along the scalar homothety S λ : X ↦ λ X . {\displaystyle S_{\lambda }:X\mapsto \lambda X.} A particular case of a non-linear connection arising in this manner

1150-424: Is extensively commented by Jeffrey Weeks and contains useful descriptions of the mathematics involved with references. The SnapPeaKernel is released under GNU GPL 2+ as is SnapPy. At the core of SnapPea are two main algorithms. The first attempts to find a minimal ideal triangulation of a given link complement . The second computes the canonical decomposition of a cusped hyperbolic 3-manifold . Almost all

1200-580: Is necessary first to extend γ ˙ {\displaystyle {\dot {\gamma }}} to a continuously differentiable vector field in an open set . However, the resulting value of ( 1 ) is independent of the choice of extension. Using local coordinates on M , we can write the geodesic equation (using the summation convention ) as where γ μ = x μ ∘ γ ( t ) {\displaystyle \gamma ^{\mu }=x^{\mu }\circ \gamma (t)} are

1250-462: Is one of the authors of a 1994 paper called "Plane curves associated to character varieties of 3-manifolds" which introduced the A-polynomial of a knot or, more generally, of a 3-manifold with one torus boundary component. Culler is an editor of The New York Journal of Mathematics . He was a Sloan Foundation Research Fellow (1986–1988) and a UIC University Scholar (2008). In 2014, he became

1300-400: Is reported to the user. The local minimality of the triangulation is meant to increase the likelihood that such a solution exists, since heuristically one might expect such a triangulation to be "straightened" without causing degenerations or overlapping of tetrahedra. From this description of the hyperbolic structure on a link complement, SnapPea can then perform hyperbolic Dehn filling on

1350-606: Is simpler to restrict the set of curves to those that are parameterized "with constant speed" 1, meaning that the distance from f ( s ) to f ( 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 an elastic band stretched between two points will contract its width, and in so doing will minimize its energy. The resulting shape of

1400-461: Is that associated to a Finsler manifold . Equation ( 1 ) is invariant under affine reparameterizations; that is, parameterizations of the form where a and b are 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 with affine parameter . An affine connection

1450-414: Is that geodesics are only locally the shortest distance between points, and are parameterized with "constant speed". Going the "long way round" on a great circle between two points on a sphere is a geodesic but not the shortest path between the points. The map t → t 2 {\displaystyle t\to t^{2}} from the unit interval on the real number line to itself gives

1500-410: Is the exponential map of the vector tV . A closed orbit of the geodesic flow corresponds to a closed geodesic on  M . On a (pseudo-)Riemannian manifold, the geodesic flow is identified with a Hamiltonian flow on the cotangent bundle. The Hamiltonian is then given by the inverse of the (pseudo-)Riemannian metric, evaluated against the canonical one-form . In particular the flow preserves

1550-636: The Bass–Serre theory , applied to the function field of the SL(2,C)- Character variety of a 3-manifold, to obtain information about incompressible surfaces in the manifold. Based on this work, Shalen, Cameron Gordon , John Luecke , and Culler proved the cyclic surgery theorem . Another important contribution by Culler came in a 1986 paper with Karen Vogtmann called "Moduli of graphs and automorphisms of free groups". This paper introduced an object that came to be known as Culler–Vogtmann Outer space . Culler

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1600-472: The Christoffel symbols of the metric. This is the geodesic equation , discussed below . Techniques of the classical calculus of variations can be applied to examine the energy functional E . The first variation of energy is defined in local coordinates by The critical points of the first variation are precisely the geodesics. The second variation is defined by In an appropriate sense, zeros of

1650-777: The University of California at Santa Barbara and his graduate work at Berkeley where he graduated in 1978. He is now at the University of Illinois at Chicago . Culler is the son of Glen Jacob Culler who was an important early innovator in the development of the Internet. Culler specializes in group theory , low dimensional topology, 3-manifolds , and hyperbolic geometry . Culler frequently collaborates with Peter Shalen and they have co-authored many papers. Culler and Shalen did joint work that related properties of representation varieties of hyperbolic 3-manifold groups to decompositions of 3-manifolds. In particular, Culler and Shalen used

1700-537: The pushforward (differential) along the projection π  : T M  →  M associated to the tangent bundle. More generally, the same construction allows one to construct a vector field for any Ehresmann connection on the tangent bundle. For the resulting vector field to be a spray (on the deleted tangent bundle T M  \ {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

1750-417: The (pseudo-)Riemannian metric g {\displaystyle g} , i.e. In particular, when V is a unit vector, γ V {\displaystyle \gamma _{V}} remains unit speed throughout, so the geodesic flow is tangent to the unit tangent bundle . Liouville's theorem implies invariance of a kinematic measure on the unit tangent bundle. The geodesic flow defines

1800-435: The algorithm to cull lists of duplicates. Additionally, from the canonical decomposition, SnapPea is able to: SnapPea has several databases of hyperbolic 3-manifolds available for systematic study. Marc Culler Marc Edward Culler (born November 22, 1953) is an American mathematician who works in geometric group theory and low-dimensional topology . A native Californian, Culler did his undergraduate work at

1850-429: 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

1900-485: The case of Riemannian manifolds . The article Levi-Civita connection discusses the more general case of a pseudo-Riemannian manifold and geodesic (general relativity) discusses the special case of general relativity in greater detail. The most familiar examples are the straight lines in Euclidean geometry . On a sphere , the images of geodesics are the great circles . The shortest path from point A to point B on

1950-495: The coordinates of the curve γ( t ) and Γ μ ν λ {\displaystyle \Gamma _{\mu \nu }^{\lambda }} are the Christoffel symbols of the connection ∇. This is an ordinary differential equation for the coordinates. It has a unique solution, given an initial position and an initial velocity. Therefore, from the point of view of classical mechanics , geodesics can be thought of as trajectories of free particles in

2000-411: 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. The local existence and uniqueness theorem for geodesics states that geodesics on a smooth manifold with an affine connection exist, and are unique. More precisely: The proof of this theorem follows from

2050-509: The cusps to obtain more hyperbolic 3-manifolds. SnapPea does this by taking any given slopes which determine certain Dehn filling equations (also explained in Thurston's notes), and then adjusting the shapes of the ideal tetrahedra to give solutions to these equations and the edge equations. For almost all slopes, this gives an incomplete hyperbolic structure on the link complement, whose completion gives

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2100-453: The difference tensor is skew-symmetric , then ∇ {\displaystyle \nabla } and ∇ ¯ {\displaystyle {\bar {\nabla }}} have the same geodesics, with the same affine parameterizations. Furthermore, there is a unique connection having the same geodesics as ∇ {\displaystyle \nabla } , but with vanishing torsion. Geodesics without

2150-425: The functional L ( γ ) {\displaystyle L(\gamma )} are generally not very regular, because arbitrary reparameterizations are allowed. The Euler–Lagrange equations of motion for the functional E are then given in local coordinates by where Γ μ ν λ {\displaystyle \Gamma _{\mu \nu }^{\lambda }} are

2200-414: The inequality is an equality. The usefulness of this approach is that the problem of seeking minimizers of E is a more robust variational problem. Indeed, E is a "convex function" of γ {\displaystyle \gamma } , so that within each isotopy class of "reasonable functions", one ought to expect existence, uniqueness, and regularity of minimizers. In contrast, "minimizers" of

2250-446: The other functions of SnapPea rely in some way on one of these decompositions. SnapPea inputs data in a variety of formats. Given a link diagram , SnapPea can ideally triangulate the link complement . It then performs a sequence of simplifications to find a locally minimal ideal triangulation. Once a suitable ideal triangulation is found, SnapPea can try to find a hyperbolic structure. In his Princeton lecture notes, Thurston noted

2300-525: The presence of an affine connection , a geodesic is defined to be a curve whose tangent vectors remain parallel if they are transported along it. Applying this to the Levi-Civita connection of a Riemannian metric recovers the previous notion. Geodesics are of particular importance in general relativity . Timelike geodesics in general relativity describe the motion of free falling test particles . A locally shortest path between two given points in

2350-545: The problem of recognition into a combinatorial question, i.e. checking if the two manifolds have combinatorially equivalent canonical decompositions. SnapPea is also able to check if two closed hyperbolic 3-manifolds are isometric by drilling out short geodesics to create cusped hyperbolic 3-manifolds and then using the canonical decomposition as before. The recognition algorithm allow SnapPea to tell two hyperbolic knots or links apart. Weeks, et al., were also able to compile different censuses of hyperbolic 3-manifolds by using

2400-445: The second variation along a geodesic γ arise along Jacobi fields . Jacobi fields are thus regarded as variations through geodesics. By applying variational techniques from classical mechanics , one can also regard geodesics as Hamiltonian flows . They are solutions of the associated Hamilton equations , with (pseudo-)Riemannian metric taken as Hamiltonian . A geodesic on a smooth manifold M with an affine connection ∇

2450-401: 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 of Riemannian geometry and more generally metric geometry . In general relativity , geodesics in spacetime describe the motion of point particles under the influence of gravity alone. In particular, the path taken by

2500-481: The theory of ordinary differential equations , by noticing that the geodesic equation is a second-order ODE. Existence and uniqueness then follow from the Picard–;Lindelöf theorem for the solutions of ODEs with prescribed initial conditions. γ depends smoothly on both p and  V . In general, I may not be all of R as for example for an open disc in R . Any γ extends to all of ℝ if and only if M

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