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1190:. 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 following
2751:
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 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
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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.
362:|. 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
338:. 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 from
1813:
4095:
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).
1989:
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2008:
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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
369:
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.
2745:
2276:
3994:
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3882:, 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
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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.
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1602:, 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 of
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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 only
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5247:— Introduction to geodesics including two ways of derivation of the equation of geodesic with applications in geometry (geodesic on a sphere and on a
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5449:
5299:
5697:
2575:{\displaystyle {\frac {d^{2}\gamma ^{\lambda }}{dt^{2}}}+\Gamma _{\mu \nu }^{\lambda }{\frac {d\gamma ^{\mu }}{dt}}{\frac {d\gamma ^{\nu }}{dt}}=0\ ,}
2143:{\displaystyle \delta ^{2}E(\gamma )(\varphi ,\psi )=\left.{\frac {\partial ^{2}}{\partial s\,\partial t}}\right|_{s=t=0}E(\gamma +t\varphi +s\psi ).}
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stretched between two points will contract its width, and in so doing will minimize its energy. The resulting shape of the band is a geodesic.
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This generalizes the notion of geodesic for
Riemannian manifolds. However, in metric geometry the geodesic considered is often equipped with
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have the same geodesics, with the same affine parameterizations. Furthermore, there is a unique connection having the same geodesics as
2687:
for the coordinates. It has a unique solution, given an initial position and an initial velocity. Therefore, from the point of view of
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This article presents the mathematical formalism involved in defining, finding, and proving the existence of geodesics, in the case of
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6666:
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3641: \ {0}) it is enough that the connection be equivariant under positive rescalings: it need not be linear. That is, (cf.
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1808:{\displaystyle {\frac {d^{2}x^{\lambda }}{dt^{2}}}+\Gamma _{\mu \nu }^{\lambda }{\frac {dx^{\mu }}{dt}}{\frac {dx^{\nu }}{dt}}=0,}
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Efficient solvers for the minimal geodesic problem on surfaces have been proposed by
Mitchell, Kimmel, Crane, and others.
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is formed by the geodesics joining each pair out of three points on a given surface. On the sphere, the geodesics are
452:
deals with the paths that objects may take when they are not free, and their movement is constrained in various ways.
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967:
In general, a metric space may have no geodesics, except constant curves. At the other extreme, any two points in a
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1984:{\displaystyle \delta E(\gamma )(\varphi )=\left.{\frac {\partial }{\partial t}}\right|_{t=0}E(\gamma +t\varphi ).}
1821:
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the shortest distance between points, and are parameterized with "constant speed". Going the "long way round" on a
1330:{\displaystyle E(\gamma )={\frac {1}{2}}\int _{a}^{b}g_{\gamma (t)}({\dot {\gamma }}(t),{\dot {\gamma }}(t))\,dt.}
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of the length taken over all continuous, piecewise continuously differentiable curves γ : →
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If an insect is placed on a surface and continually walks "forward", by definition it will trace out a geodesic.
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behave in a more complicated way than on a sphere; in particular, they are not closed in general (see figure).
175:
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2944:, by noticing that the geodesic equation is a second-order ODE. Existence and uniqueness then follow from the
1132:{\displaystyle L(\gamma )=\int _{a}^{b}{\sqrt {g_{\gamma (t)}({\dot {\gamma }}(t),{\dot {\gamma }}(t))}}\,dt.}
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is equal to a constant a.e.; the path should be travelled at constant speed. It happens that minimizers of
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on the tangent bundle. For the resulting vector field to be a spray (on the deleted tangent bundle T
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geometry. In the original sense, a geodesic was the shortest route between two points on the Earth's
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under the influence of gravity alone. In particular, the path taken by a falling rock, an orbiting
421:
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between two points on a sphere is a geodesic but not the shortest path between the points. The map
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4623: – Analyzes the topology of a manifold by studying differentiable functions on that manifold
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Geodesic triangles in spaces of positive (top), negative (middle) and zero (bottom) curvature.
275:
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231:). The term has since been generalized to more abstract mathematical spaces; for example, in
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A particular case of a non-linear connection arising in this manner is that associated to a
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972:
330:
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4600: – Gives equivalent statements about the geodesic completeness of Riemannian manifolds
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4170:
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5211:
Gravitation and
Cosmology: Principles and Applications of the General Theory of Relativity
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216:
179:
4612: – Line along which a quadratic form applied to any two points' displacement is zero
4802:
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3807:) is invariant under affine reparameterizations; that is, parameterizations of the form
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is then given by the inverse of the (pseudo-)Riemannian metric, evaluated against the
2153:
In an appropriate sense, zeros of the second variation along a geodesic γ arise along
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More generally, the same construction allows one to construct a vector field for any
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1356:
can be arbitrarily re-parameterized (without changing their length), while minima of
991:
325:
263:
53:
30:
This article is about geodesics in general. For geodesics in general relativity, see
4556:); without GSP reconstruction often results in self-intersections within the surface
304:
A locally shortest path between two given points in a curved space, assumed to be a
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are generally not very regular, because arbitrary reparameterizations are allowed.
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or submanifold, geodesics are characterised by the property of having vanishing
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to the space), and then minimizing this length between the points using the
290:
193:
187:
806:{\displaystyle d(\gamma (t_{1}),\gamma (t_{2}))=v\left|t_{1}-t_{2}\right|.}
4830:
3269:
On a (pseudo-)Riemannian manifold, the geodesic flow is identified with a
926:{\displaystyle d(\gamma (t_{1}),\gamma (t_{2}))=\left|t_{1}-t_{2}\right|.}
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309:
64:
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for the solutions of ODEs with prescribed initial conditions. γ depends
6537:
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610:
199:
35:
4103:
Mathematically the ribbon test can be formulated as finding a mapping
3474:
More precisely, an affine connection gives rise to a splitting of the
3438:
implies invariance of a kinematic measure on the unit tangent bundle.
3430:
remains unit speed throughout, so the geodesic flow is tangent to the
6863:
6455:
5973:
4703:
975:, although this minimizing sequence need not converge to a geodesic.
504:
4870:
4766:
2157:. Jacobi fields are thus regarded as variations through geodesics.
471:
discusses the special case of general relativity in greater detail.
448:
are all geodesics in curved spacetime. More generally, the topic of
4075:
Geodesics without a particular parameterization are described by a
5248:
3281:. In particular the flow preserves the (pseudo-)Riemannian metric
2343:. More precisely, in order to define the covariant derivative of
2210:
1001:
of a continuously differentiable curve γ : →
817:
618:
591:
568:
490:
478:
317:
204:
155:
48:
2764:
for geodesics states that geodesics on a smooth manifold with an
207:, though many of the underlying principles can be applied to any
4913:
A Comprehensive introduction to differential geometry (Volume 2)
6428:
5551:
5268:
4933:
4470:
2221:
along the curve preserves the tangent vector to the curve, so
125:
83:
3643:
Ehresmann connection#Vector bundles and covariant derivatives
2054:
1924:
113:
98:
89:
6424:
2740:{\displaystyle \nabla _{\dot {\gamma }}{\dot {\gamma }}=0}
2271:{\displaystyle \nabla _{\dot {\gamma }}{\dot {\gamma }}=0}
80:
27:
Straight path on a curved surface or a
Riemannian manifold
4998:
Adler, Ronald; Bazin, Maurice; Schiffer, Menahem (1975),
4584: – Study of curves from a differential point of view
3989:{\displaystyle D(X,Y)=\nabla _{X}Y-{\bar {\nabla }}_{X}Y}
3537:
The geodesic spray is the unique horizontal vector field
1998:
of the first variation are precisely the geodesics. The
143:
131:
104:
3711:{\displaystyle H_{\lambda X}=d(S_{\lambda })_{X}H_{X}\,}
3645:) it is enough that the horizontal distribution satisfy
3257:. A closed orbit of the geodesic flow corresponds to a
373:
A contiguous segment of a geodesic is again a geodesic.
4577:
Pages displaying short descriptions of redirect targets
4487:
2633:{\displaystyle \gamma ^{\mu }=x^{\mu }\circ \gamma (t)}
4358:{\displaystyle g_{N}-f^{*}(g_{S})=O(\varepsilon ^{2})}
3874:
its family of affinely parameterized geodesics, up to
2940:
The proof of this theorem follows from the theory of
4568:
Introduction to the mathematics of general relativity
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The most familiar examples are the straight lines in
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101:
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are two connections such that the difference tensor
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The geodesic flow defines a family of curves in the
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4747:Mitchell, J.; Mount, D.; Papadimitriou, C. (1987).
1489:{\displaystyle L(\gamma )^{2}\leq 2(b-a)E(\gamma )}
119:
92:
5255:) and optics (light beam in inhomogeneous medium).
5208:
5086:, vol. 1 (New ed.), Wiley-Interscience,
4575: – Formula in classical differential geometry
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4851:Crane, K.; Weischedel, C.; Wardetzky, M. (2017).
2691:, geodesics can be thought of as trajectories of
203:, the science of measuring the size and shape of
3782:{\displaystyle S_{\lambda }:X\mapsto \lambda X.}
3087:{\displaystyle G^{t}(V)={\dot {\gamma }}_{V}(t)}
2752:geodesics and the bending is caused by gravity.
1877:can be applied to examine the energy functional
4790:Proceedings of the National Academy of Sciences
4532:mapping images on surfaces, for rendering; see
3729: \ {0} and λ > 0. Here
1606:is a more robust variational problem. Indeed,
1352:is a bigger set since paths that are minima of
5082:Kobayashi, Shoshichi; Nomizu, Katsumi (1996),
3389:{\displaystyle g(G^{t}(V),G^{t}(V))=g(V,V).\,}
158:representing in some sense the shortest path (
6440:
5563:
5280:
4548:geodesic shortest path (GSP) correction over
2672:{\displaystyle \Gamma _{\mu \nu }^{\lambda }}
2415:) is independent of the choice of extension.
1885:of energy is defined in local coordinates by
1846:{\displaystyle \Gamma _{\mu \nu }^{\lambda }}
178:. It is a generalization of the notion of a "
8:
4673:The path is a local maximum of the interval
609:, a geodesic is a curve which is everywhere
420:Geodesics are commonly seen in the study of
262:, a geodesic is defined to be a curve whose
4526:horizontal distances on or near Earth; see
4506:Geodesics serve as the basis to calculate:
3456:. The derivatives of these curves define a
86:
6886:
6477:
6447:
6433:
6425:
5978:
5570:
5556:
5548:
5450:Fundamental theorem of Riemannian geometry
5287:
5273:
5265:
5057:Riemannian Geometry and Geometric Analysis
4853:"The Heat Method for Distance Computation"
936:If the last equality is satisfied for all
281:Geodesics are of particular importance in
4984:Learn how and when to remove this message
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3184:{\displaystyle {\dot {\gamma }}_{V}(0)=V}
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2301:
2299:
2251:
2250:
2238:
2237:
2231:
2089:
2075:
2062:
2056:
2016:
2010:
1945:
1926:
1893:
1837:
1829:
1823:
1779:
1769:
1752:
1742:
1736:
1728:
1712:
1697:
1687:
1680:
1678:
1635:
1615:
1578:
1549:
1504:
1447:
1432:
1398:
1392:
1371:
1365:
1317:
1294:
1293:
1270:
1269:
1251:
1241:
1236:
1222:
1205:
1119:
1094:
1093:
1070:
1069:
1051:
1045:
1039:
1034:
1013:
909:
896:
872:
850:
832:
789:
776:
749:
727:
709:
561:Gauss–Bonnet theorem § For triangles
401:
389:
6804:Covariance and contravariance of vectors
4947:This article includes a list of general
4652:
4650:
4629: – Surface homeomorphic to a sphere
3910:{\displaystyle \nabla ,{\bar {\nabla }}}
2768:exist, and are unique. More precisely:
2168:. They are solutions of the associated
2160:By applying variational techniques from
258:. More generally, in the presence of an
4782:"Computing Geodesic Paths on Manifolds"
4690:
4646:
3140:denotes the geodesic with initial data
1670:are then given in local coordinates by
971:are joined by a minimizing sequence of
4635: – Recreational geodesics problem
3879:
2913:{\displaystyle {\dot {\gamma }}(0)=V,}
2762:local existence and uniqueness theorem
2294:at each point along the curve, where
34:. For the study of Earth's shape, see
4664:, the definition is more complicated.
4542:(e.g., when painting car parts); see
4227:"doesn't change the distances around
1862:
463:discusses the more general case of a
7:
5084:Foundations of Differential Geometry
4606: – Concept in geometry/topology
3464:of the tangent bundle, known as the
2695:in a manifold. Indeed, the equation
2409:. However, the resulting value of (
2223:
1537:{\displaystyle g(\gamma ',\gamma ')}
523:of the great circle passing through
519:on a sphere is given by the shorter
170:. The term also has meaning in any
38:. For the application on Earth, see
4780:Kimmel, R.; Sethian, J. A. (1998).
4247:by much"; that is, at the distance
3448:Spray (mathematics) § Geodesic
2971:as for example for an open disc in
2640:are the coordinates of the curve γ(
507:, the images of geodesics are the
58:(marked by 7 colors and 4 patterns)
6667:Tensors in curvilinear coordinates
5181:. Note especially pages 7 and 10.
5000:Introduction to General Relativity
4953:it lacks sufficient corresponding
4915:, Houston, TX: Publish or Perish,
4520:geodesic structures – for example
4059:
4033:
4010:
3968:
3949:
3898:
3889:
3630:associated to the tangent bundle.
2703:
2652:
2489:
2323:is the derivative with respect to
2234:
2076:
2069:
2059:
1932:
1928:
1826:
1725:
573:A geodesic triangle on the sphere.
25:
4588:Differential geometry of surfaces
3581:{\displaystyle \pi _{*}W_{v}=v\,}
3242:{\displaystyle G^{t}(V)=\exp(tV)}
2804:) there exists a unique geodesic
2683:of the connection ∇. This is an
2401:to a continuously differentiable
4938:
4840:from the original on 2022-10-09.
4474:
4045:{\displaystyle {\bar {\nabla }}}
2394:{\displaystyle {\dot {\gamma }}}
2372:it is necessary first to extend
2365:{\displaystyle {\dot {\gamma }}}
2316:{\displaystyle {\dot {\gamma }}}
485:geodesic on a triaxial ellipsoid
73:
4749:"The Discrete Geodesic Problem"
4137:{\displaystyle f:N(\ell )\to S}
3797:Affine and projective geodesics
2942:ordinary differential equations
2190:Geodesics in general relativity
511:. The shortest path from point
287:geodesics in general relativity
270:along it. Applying this to the
5610:Differentiable/Smooth manifold
4733:Merriam-Webster.com Dictionary
4633:The spider and the fly problem
4550:Poisson surface reconstruction
4352:
4339:
4330:
4317:
4128:
4125:
4119:
4072:, but with vanishing torsion.
4036:
3971:
3942:
3930:
3901:
3820:
3767:
3688:
3674:
3527:{\displaystyle TTM=H\oplus V.}
3379:
3367:
3358:
3355:
3349:
3333:
3327:
3314:
3273:on the cotangent bundle. The
3236:
3227:
3215:
3209:
3172:
3166:
3081:
3075:
3050:
3044:
2898:
2892:
2864:{\displaystyle \gamma (0)=p\,}
2851:
2845:
2685:ordinary differential equation
2627:
2621:
2166:geodesics as Hamiltonian flows
2134:
2110:
2046:
2034:
2031:
2025:
1975:
1960:
1916:
1910:
1907:
1901:
1646:
1640:
1589:
1583:
1560:
1554:
1531:
1509:
1483:
1477:
1471:
1459:
1444:
1437:
1314:
1311:
1305:
1287:
1281:
1266:
1261:
1255:
1216:
1210:
1114:
1111:
1105:
1087:
1081:
1066:
1061:
1055:
1024:
1018:
881:
878:
865:
856:
843:
837:
758:
755:
742:
733:
720:
714:
547:shortest paths between them.
394:
308:, can be defined by using the
278:recovers the previous notion.
1:
6720:Exterior covariant derivative
6652:Tensor (intrinsic definition)
5036:, London: Benjamin-Cummings,
3838:{\displaystyle t\mapsto at+b}
3024:defined in the following way
2946:Picard–Lindelöf theorem
1666:of motion for the functional
1499:with equality if and only if
820:, i.e. in the above identity
617:minimizer. More precisely, a
469:geodesic (general relativity)
32:Geodesic (general relativity)
6745:Raising and lowering indices
5377:Raising and lowering indices
5259:Totally geodesic submanifold
4677:rather than a local minimum.
4260:{\displaystyle \varepsilon }
3863:) are called geodesics with
1873:Techniques of the classical
1857:of the metric. This is the
266:remain parallel if they are
6983:Gluon field strength tensor
6316:Classification of manifolds
5192:Encyclopedia of Mathematics
3859:
3803:
3744:along the scalar homothety
3423:{\displaystyle \gamma _{V}}
3133:{\displaystyle \gamma _{V}}
2411:
956:, the geodesic is called a
7112:
6794:Cartan formalism (physics)
6614:Penrose graphical notation
5398:Pseudo-Riemannian manifold
5168:Cambridge University Press
5110:Classical Theory of Fields
5002:(2nd ed.), New York:
4658:pseudo-Riemannian manifold
4187:in a plane into a surface
3616:pushforward (differential)
3445:
2187:
2174:(pseudo-)Riemannian metric
1652:{\displaystyle L(\gamma )}
1610:is a "convex function" of
1595:{\displaystyle L(\gamma )}
1566:{\displaystyle E(\gamma )}
558:
465:pseudo-Riemannian manifold
410:{\displaystyle t\to t^{2}}
354:) along the curve equals |
162:) between two points in a
29:
6466:Glossary of tensor theory
6462:
6392:over commutative algebras
5527:Geometrization conjecture
4858:Communications of the ACM
4754:SIAM Journal on Computing
2818:{\displaystyle \gamma \,}
1422:Cauchy–Schwarz inequality
1387:curve (more generally, a
549:Geodesics on an ellipsoid
166:, or more generally in a
44:Geodesic (disambiguation)
7050:Gregorio Ricci-Curbastro
6922:Riemann curvature tensor
6629:Van der Waerden notation
6108:Riemann curvature tensor
5034:Foundations of mechanics
4598:Hopf–Rinow theorem
4510:geodesic airframes; see
3870:An affine connection is
2756:Existence and uniqueness
1664:Euler–Lagrange equations
1360:cannot. For a piecewise
818:natural parameterization
669:there is a neighborhood
7020:Elwin Bruno Christoffel
6953:Angular momentum tensor
6624:Tetrad (index notation)
6594:Abstract index notation
5185:Volkov, Yu.A. (2001) ,
5140:Wheeler, John Archibald
4968:more precise citations.
4812:10.1073/pnas.95.15.8431
4710:Oxford University Press
4065:{\displaystyle \nabla }
4016:{\displaystyle \nabla }
1623:{\displaystyle \gamma }
1413:{\displaystyle W^{1,2}}
450:sub-Riemannian geometry
436:describe the motion of
289:describe the motion of
235:, one might consider a
172:differentiable manifold
7096:Geodesic (mathematics)
6834:Levi-Civita connection
5900:Manifold with boundary
5615:Differential structure
5537:Uniformization theorem
5470:Nash embedding theorem
5403:Riemannian volume form
5362:Levi-Civita connection
4457:
4437:
4413:
4386:
4359:
4281:
4261:
4241:
4221:
4201:
4181:
4161:
4138:
4066:
4046:
4017:
3990:
3911:
3839:
3783:
3712:
3582:
3528:
3424:
3390:
3295:
3243:
3185:
3134:
3088:
2914:
2865:
2819:
2741:
2673:
2634:
2576:
2395:
2366:
2337:
2317:
2272:
2164:, one can also regard
2144:
1985:
1875:calculus of variations
1869:Calculus of variations
1847:
1809:
1653:
1624:
1596:
1567:
1538:
1490:
1414:
1381:
1331:
1133:
927:
807:
597:
574:
496:
488:
461:Levi-Civita connection
411:
336:calculus of variations
272:Levi-Civita connection
60:
42:. For other uses, see
7091:Differential geometry
7060:Jan Arnoldus Schouten
7015:Augustin-Louis Cauchy
6495:Differential geometry
5261:at the Manifold Atlas
5217:John Wiley & Sons
5162:Ortín, Tomás (2004),
5055:Jost, Jürgen (2002),
4706:UK English Dictionary
4544:Shortest path problem
4458:
4438:
4414:
4412:{\displaystyle g_{S}}
4387:
4385:{\displaystyle g_{N}}
4360:
4282:
4262:
4242:
4240:{\displaystyle \ell }
4222:
4202:
4182:
4180:{\displaystyle \ell }
4162:
4139:
4083:Computational methods
4077:projective connection
4067:
4047:
4018:
3991:
3912:
3840:
3784:
3713:
3618:along the projection
3583:
3529:
3476:double tangent bundle
3446:Further information:
3425:
3391:
3296:
3244:
3186:
3135:
3089:
2989:geodesically complete
2915:
2866:
2820:
2742:
2674:
2635:
2577:
2396:
2367:
2338:
2318:
2273:
2145:
1986:
1848:
1810:
1654:
1625:
1597:
1568:
1539:
1491:
1415:
1382:
1380:{\displaystyle C^{1}}
1332:
1154:) between two points
1134:
928:
808:
595:
572:
494:
482:
412:
229:great-circle distance
52:
7035:Carl Friedrich Gauss
6968:stress–energy tensor
6963:Cauchy stress tensor
6715:Covariant derivative
6677:Antisymmetric tensor
6609:Multi-index notation
6047:Covariant derivative
5598:Topological manifold
5460:Gauss–Bonnet theorem
5367:Covariant derivative
5112:, Oxford: Pergamon,
5059:, Berlin, New York:
4582:Differentiable curve
4447:
4427:
4396:
4369:
4291:
4271:
4251:
4231:
4211:
4207:so that the mapping
4191:
4171:
4151:
4107:
4056:
4027:
4007:
3924:
3886:
3814:
3748:
3652:
3635:Ehresmann connection
3548:
3497:
3407:
3399:In particular, when
3308:
3285:
3196:
3144:
3117:
3031:
2877:
2839:
2808:
2699:
2648:
2589:
2441:
2432:summation convention
2376:
2347:
2327:
2298:
2230:
2009:
1892:
1822:
1677:
1634:
1614:
1577:
1548:
1503:
1431:
1391:
1364:
1204:
1012:
831:
708:
638:of the reals to the
457:Riemannian manifolds
444:, or the shape of a
388:
6912:Nonmetricity tensor
6767:(2nd-order tensors)
6735:Hodge star operator
6725:Exterior derivative
6574:Transport phenomena
6559:Continuum mechanics
6515:Multilinear algebra
6081:Exterior derivative
5683:Atiyah–Singer index
5632:Riemannian manifold
5532:Poincaré conjecture
5393:Riemannian manifold
5381:Musical isomorphism
5296:Riemannian geometry
5245:Geodesics Revisited
5164:Gravity and strings
5030:Marsden, Jerrold E.
4803:1998PNAS...95.8431K
4662:Lorentzian manifold
4573:Clairaut's relation
3436:Liouville's theorem
3432:unit tangent bundle
2780:and for any vector
2749:acceleration vector
2689:classical mechanics
2681:Christoffel symbols
2668:
2505:
2426:, we can write the
2162:classical mechanics
1855:Christoffel symbols
1842:
1741:
1344:are also minima of
1246:
1044:
985:Riemannian manifold
979:Riemannian geometry
969:length metric space
958:minimizing geodesic
824: = 1 and
565:Toponogov's theorem
424:and more generally
422:Riemannian geometry
306:Riemannian manifold
252:Riemannian manifold
168:Riemannian manifold
7045:Tullio Levi-Civita
6988:Metric tensor (GR)
6902:Levi-Civita symbol
6755:Tensor contraction
6569:General relativity
6505:Euclidean geometry
6387:Secondary calculus
6341:Singularity theory
6296:Parallel transport
6064:De Rham cohomology
5703:Generalized Stokes
5522:General relativity
5465:Hopf–Rinow theorem
5412:Types of manifolds
5388:Parallel transport
5132:Misner, Charles W.
4736:. Merriam-Webster.
4486:. You can help by
4453:
4433:
4409:
4382:
4355:
4277:
4257:
4237:
4217:
4197:
4177:
4157:
4134:
4062:
4042:
4013:
3986:
3907:
3835:
3779:
3708:
3578:
3524:
3420:
3403:is a unit vector,
3386:
3291:
3279:canonical one-form
3239:
3181:
3130:
3084:
2979:extends to all of
2967:may not be all of
2910:
2861:
2815:
2737:
2669:
2651:
2630:
2572:
2488:
2391:
2362:
2333:
2313:
2268:
2219:parallel transport
2209:∇ is defined as a
2170:Hamilton equations
2140:
1981:
1843:
1825:
1805:
1724:
1649:
1620:
1592:
1563:
1534:
1486:
1410:
1377:
1327:
1232:
1166:is defined as the
1129:
1030:
923:
803:
681:such that for any
659:such that for any
598:
587:spherical triangle
575:
501:Euclidean geometry
497:
489:
430:general relativity
407:
283:general relativity
256:geodesic curvature
191:and the adjective
61:
56:with 28 geodesics
7078:
7077:
7040:Hermann Grassmann
6996:
6995:
6948:Moment of inertia
6809:Differential form
6784:Affine connection
6599:Einstein notation
6582:
6581:
6510:Exterior calculus
6490:Coordinate system
6422:
6421:
6304:
6303:
6069:Differential form
5723:Whitney embedding
5657:Differential form
5545:
5544:
5226:978-0-471-92567-5
5177:978-0-521-82475-0
5155:978-0-7167-0344-0
5148:, W. H. Freeman,
5119:978-0-08-018176-9
5070:978-3-540-42627-1
5043:978-0-8053-0102-1
5026:Abraham, Ralph H.
5013:978-0-07-000423-8
4994:
4993:
4986:
4922:978-0-914098-71-3
4797:(15): 8431–8435.
4554:digital dentistry
4516:geodetic airframe
4512:geodesic airframe
4504:
4503:
4456:{\displaystyle S}
4436:{\displaystyle N}
4280:{\displaystyle l}
4220:{\displaystyle f}
4200:{\displaystyle S}
4160:{\displaystyle N}
4039:
3974:
3904:
3294:{\displaystyle g}
3157:
3066:
2889:
2766:affine connection
2728:
2715:
2568:
2558:
2531:
2483:
2428:geodesic equation
2420:local coordinates
2388:
2359:
2336:{\displaystyle t}
2310:
2292:
2291:
2259:
2246:
2207:affine connection
2083:
1939:
1859:geodesic equation
1794:
1767:
1719:
1302:
1278:
1230:
1196:energy functional
1117:
1102:
1078:
973:rectifiable paths
634:from an interval
579:geodesic triangle
543:, then there are
276:Riemannian metric
260:affine connection
59:
18:Geodesic equation
16:(Redirected from
7103:
7055:Bernhard Riemann
6887:
6730:Exterior product
6697:Two-point tensor
6682:Symmetric tensor
6564:Electromagnetism
6478:
6449:
6442:
6435:
6426:
6414:Stratified space
6372:Fréchet manifold
6086:Interior product
5979:
5676:
5572:
5565:
5558:
5549:
5289:
5282:
5275:
5266:
5229:
5214:
5205:Weinberg, Steven
5199:
5180:
5158:
5122:
5096:
5073:
5046:
5016:
4989:
4982:
4978:
4975:
4969:
4964:this article by
4955:inline citations
4942:
4941:
4934:
4925:
4900:
4889:
4883:
4882:
4848:
4842:
4841:
4839:
4824:
4814:
4786:
4777:
4771:
4770:
4744:
4738:
4737:
4724:
4718:
4717:
4712:. Archived from
4695:
4678:
4671:
4665:
4654:
4604:Intrinsic metric
4578:
4499:
4496:
4478:
4471:
4462:
4460:
4459:
4454:
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4434:
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3995:
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3992:
3987:
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3981:
3976:
3975:
3967:
3957:
3956:
3916:
3914:
3913:
3908:
3906:
3905:
3897:
3865:affine parameter
3844:
3842:
3841:
3836:
3791:Finsler manifold
3788:
3786:
3785:
3780:
3760:
3759:
3717:
3715:
3714:
3709:
3706:
3705:
3696:
3695:
3686:
3685:
3667:
3666:
3621:
3602:
3587:
3585:
3584:
3579:
3570:
3569:
3560:
3559:
3533:
3531:
3530:
3525:
3488:vertical bundles
3429:
3427:
3426:
3421:
3419:
3418:
3395:
3393:
3392:
3387:
3348:
3347:
3326:
3325:
3300:
3298:
3297:
3292:
3271:Hamiltonian flow
3248:
3246:
3245:
3240:
3208:
3207:
3190:
3188:
3187:
3182:
3165:
3164:
3159:
3158:
3150:
3139:
3137:
3136:
3131:
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3128:
3093:
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3090:
3085:
3074:
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2986:
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2978:
2919:
2917:
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2911:
2891:
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2870:
2868:
2867:
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2822:
2821:
2816:
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2721:
2718:
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2716:
2708:
2678:
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2670:
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2614:
2613:
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2600:
2581:
2579:
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2573:
2566:
2559:
2557:
2549:
2548:
2547:
2534:
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2507:
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2400:
2398:
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2286:
2277:
2275:
2274:
2269:
2261:
2260:
2252:
2249:
2248:
2247:
2239:
2224:
2184:Affine geodesics
2149:
2147:
2146:
2141:
2106:
2105:
2088:
2084:
2082:
2067:
2066:
2057:
2021:
2020:
2000:second variation
1990:
1988:
1987:
1982:
1956:
1955:
1944:
1940:
1938:
1927:
1852:
1850:
1849:
1844:
1841:
1836:
1814:
1812:
1811:
1806:
1795:
1793:
1785:
1784:
1783:
1770:
1768:
1766:
1758:
1757:
1756:
1743:
1740:
1735:
1720:
1718:
1717:
1716:
1703:
1702:
1701:
1692:
1691:
1681:
1658:
1656:
1655:
1650:
1629:
1627:
1626:
1621:
1601:
1599:
1598:
1593:
1572:
1570:
1569:
1564:
1543:
1541:
1540:
1535:
1530:
1519:
1495:
1493:
1492:
1487:
1452:
1451:
1419:
1417:
1416:
1411:
1409:
1408:
1386:
1384:
1383:
1378:
1376:
1375:
1336:
1334:
1333:
1328:
1304:
1303:
1295:
1280:
1279:
1271:
1265:
1264:
1245:
1240:
1231:
1223:
1138:
1136:
1135:
1130:
1118:
1104:
1103:
1095:
1080:
1079:
1071:
1065:
1064:
1046:
1043:
1038:
955:
932:
930:
929:
924:
919:
915:
914:
913:
901:
900:
877:
876:
855:
854:
812:
810:
809:
804:
799:
795:
794:
793:
781:
780:
754:
753:
732:
731:
700:
668:
658:
633:
585:arcs, forming a
541:antipodal points
416:
414:
413:
408:
406:
405:
153:
152:
149:
148:
145:
142:
137:
136:
133:
130:
127:
124:
121:
116:
115:
110:
109:
106:
103:
100:
97:
94:
91:
88:
85:
82:
79:
57:
21:
7111:
7110:
7106:
7105:
7104:
7102:
7101:
7100:
7081:
7080:
7079:
7074:
7025:Albert Einstein
6992:
6973:Einstein tensor
6936:
6917:Ricci curvature
6897:Kronecker delta
6883:Notable tensors
6878:
6799:Connection form
6776:
6770:
6701:
6687:Tensor operator
6644:
6638:
6578:
6554:Computer vision
6547:
6529:
6525:Tensor calculus
6469:
6458:
6453:
6423:
6418:
6357:Banach manifold
6350:Generalizations
6345:
6300:
6237:
6134:
6096:Ricci curvature
6052:Cotangent space
6030:
5968:
5810:
5804:
5763:Exponential map
5727:
5672:
5666:
5586:
5576:
5546:
5541:
5510:
5489:Generalizations
5484:
5438:
5407:
5342:Exponential map
5303:
5293:
5253:brachistochrone
5241:
5227:
5203:
5187:"Geodesic line"
5184:
5178:
5161:
5156:
5130:
5120:
5106:Lifshitz, E. M.
5100:
5094:
5081:
5076:See section 1.4
5071:
5061:Springer-Verlag
5054:
5049:See section 2.7
5044:
5024:
5014:
4997:
4990:
4979:
4973:
4970:
4960:Please help to
4959:
4943:
4939:
4932:
4930:Further reading
4923:
4909:Spivak, Michael
4907:
4904:
4903:
4894:(Nov 2, 2017),
4892:Michael Stevens
4890:
4886:
4871:10.1145/3131280
4850:
4849:
4845:
4837:
4784:
4779:
4778:
4774:
4767:10.1137/0216045
4746:
4745:
4741:
4726:
4725:
4721:
4697:
4696:
4692:
4687:
4682:
4681:
4672:
4668:
4655:
4648:
4643:
4638:
4593:Geodesic circle
4576:
4563:
4540:motion planning
4528:Earth geodesics
4500:
4494:
4491:
4484:needs expansion
4469:
4445:
4444:
4425:
4424:
4399:
4394:
4393:
4372:
4367:
4366:
4342:
4320:
4307:
4294:
4289:
4288:
4269:
4268:
4249:
4248:
4229:
4228:
4209:
4208:
4189:
4188:
4169:
4168:
4149:
4148:
4105:
4104:
4093:
4085:
4054:
4053:
4025:
4024:
4005:
4004:
3964:
3948:
3922:
3921:
3884:
3883:
3812:
3811:
3799:
3751:
3746:
3745:
3739:
3697:
3687:
3677:
3655:
3650:
3649:
3619:
3606: : TT
3605:
3600:
3561:
3551:
3546:
3545:
3495:
3494:
3450:
3444:
3410:
3405:
3404:
3339:
3317:
3306:
3305:
3283:
3282:
3259:closed geodesic
3251:exponential map
3199:
3194:
3193:
3147:
3142:
3141:
3120:
3115:
3114:
3056:
3034:
3029:
3028:
2997:
2984:
2983:if and only if
2980:
2976:
2875:
2874:
2837:
2836:
2806:
2805:
2789:
2758:
2747:means that the
2702:
2697:
2696:
2646:
2645:
2605:
2592:
2587:
2586:
2550:
2539:
2535:
2523:
2512:
2508:
2472:
2468:
2457:
2447:
2446:
2439:
2438:
2374:
2373:
2345:
2344:
2325:
2324:
2296:
2295:
2284:
2233:
2228:
2227:
2200:smooth manifold
2192:
2186:
2068:
2058:
2053:
2052:
2012:
2007:
2006:
1996:critical points
1931:
1923:
1922:
1890:
1889:
1883:first variation
1871:
1820:
1819:
1786:
1775:
1771:
1759:
1748:
1744:
1708:
1704:
1693:
1683:
1682:
1675:
1674:
1632:
1631:
1612:
1611:
1575:
1574:
1546:
1545:
1523:
1512:
1501:
1500:
1443:
1429:
1428:
1394:
1389:
1388:
1367:
1362:
1361:
1247:
1202:
1201:
1047:
1010:
1009:
981:
950:
943:
937:
905:
892:
891:
887:
868:
846:
829:
828:
785:
772:
771:
767:
745:
723:
706:
705:
695:
688:
682:
660:
653:
621:
607:metric geometry
603:
601:Metric geometry
567:
557:
545:infinitely many
477:
446:planetary orbit
438:point particles
432:, geodesics in
426:metric geometry
397:
386:
385:
302:
264:tangent vectors
217:spherical Earth
139:
118:
112:
76:
72:
47:
28:
23:
22:
15:
12:
11:
5:
7109:
7107:
7099:
7098:
7093:
7083:
7082:
7076:
7075:
7073:
7072:
7067:
7065:Woldemar Voigt
7062:
7057:
7052:
7047:
7042:
7037:
7032:
7030:Leonhard Euler
7027:
7022:
7017:
7012:
7006:
7004:
7002:Mathematicians
6998:
6997:
6994:
6993:
6991:
6990:
6985:
6980:
6975:
6970:
6965:
6960:
6955:
6950:
6944:
6942:
6938:
6937:
6935:
6934:
6929:
6927:Torsion tensor
6924:
6919:
6914:
6909:
6904:
6899:
6893:
6891:
6884:
6880:
6879:
6877:
6876:
6871:
6866:
6861:
6856:
6851:
6846:
6841:
6836:
6831:
6826:
6821:
6816:
6811:
6806:
6801:
6796:
6791:
6786:
6780:
6778:
6772:
6771:
6769:
6768:
6762:
6760:Tensor product
6757:
6752:
6750:Symmetrization
6747:
6742:
6740:Lie derivative
6737:
6732:
6727:
6722:
6717:
6711:
6709:
6703:
6702:
6700:
6699:
6694:
6689:
6684:
6679:
6674:
6669:
6664:
6662:Tensor density
6659:
6654:
6648:
6646:
6640:
6639:
6637:
6636:
6634:Voigt notation
6631:
6626:
6621:
6619:Ricci calculus
6616:
6611:
6606:
6604:Index notation
6601:
6596:
6590:
6588:
6584:
6583:
6580:
6579:
6577:
6576:
6571:
6566:
6561:
6556:
6550:
6548:
6546:
6545:
6540:
6534:
6531:
6530:
6528:
6527:
6522:
6520:Tensor algebra
6517:
6512:
6507:
6502:
6500:Dyadic algebra
6497:
6492:
6486:
6484:
6475:
6471:
6470:
6463:
6460:
6459:
6454:
6452:
6451:
6444:
6437:
6429:
6420:
6419:
6417:
6416:
6411:
6406:
6401:
6396:
6395:
6394:
6384:
6379:
6374:
6369:
6364:
6359:
6353:
6351:
6347:
6346:
6344:
6343:
6338:
6333:
6328:
6323:
6318:
6312:
6310:
6306:
6305:
6302:
6301:
6299:
6298:
6293:
6288:
6283:
6278:
6273:
6268:
6263:
6258:
6253:
6247:
6245:
6239:
6238:
6236:
6235:
6230:
6225:
6220:
6215:
6210:
6205:
6195:
6190:
6185:
6175:
6170:
6165:
6160:
6155:
6150:
6144:
6142:
6136:
6135:
6133:
6132:
6127:
6122:
6121:
6120:
6110:
6105:
6104:
6103:
6093:
6088:
6083:
6078:
6077:
6076:
6066:
6061:
6060:
6059:
6049:
6044:
6038:
6036:
6032:
6031:
6029:
6028:
6023:
6018:
6013:
6012:
6011:
6001:
5996:
5991:
5985:
5983:
5976:
5970:
5969:
5967:
5966:
5961:
5951:
5946:
5932:
5927:
5922:
5917:
5912:
5910:Parallelizable
5907:
5902:
5897:
5896:
5895:
5885:
5880:
5875:
5870:
5865:
5860:
5855:
5850:
5845:
5840:
5830:
5820:
5814:
5812:
5806:
5805:
5803:
5802:
5797:
5792:
5790:Lie derivative
5787:
5785:Integral curve
5782:
5777:
5772:
5771:
5770:
5760:
5755:
5754:
5753:
5746:Diffeomorphism
5743:
5737:
5735:
5729:
5728:
5726:
5725:
5720:
5715:
5710:
5705:
5700:
5695:
5690:
5685:
5679:
5677:
5668:
5667:
5665:
5664:
5659:
5654:
5649:
5644:
5639:
5634:
5629:
5624:
5623:
5622:
5617:
5607:
5606:
5605:
5594:
5592:
5591:Basic concepts
5588:
5587:
5577:
5575:
5574:
5567:
5560:
5552:
5543:
5542:
5540:
5539:
5534:
5529:
5524:
5518:
5516:
5512:
5511:
5509:
5508:
5506:Sub-Riemannian
5503:
5498:
5492:
5490:
5486:
5485:
5483:
5482:
5477:
5472:
5467:
5462:
5457:
5452:
5446:
5444:
5440:
5439:
5437:
5436:
5431:
5426:
5421:
5415:
5413:
5409:
5408:
5406:
5405:
5400:
5395:
5390:
5385:
5384:
5383:
5374:
5369:
5364:
5354:
5349:
5344:
5339:
5338:
5337:
5332:
5327:
5322:
5311:
5309:
5308:Basic concepts
5305:
5304:
5294:
5292:
5291:
5284:
5277:
5269:
5263:
5262:
5256:
5251:), mechanics (
5240:
5239:External links
5237:
5236:
5235:
5225:
5201:
5182:
5176:
5159:
5154:
5128:
5125:See section 87
5118:
5098:
5092:
5079:
5069:
5052:
5042:
5022:
5012:
4992:
4991:
4946:
4944:
4937:
4931:
4928:
4927:
4926:
4921:
4902:
4901:
4884:
4843:
4772:
4761:(4): 647–668.
4739:
4719:
4716:on 2020-03-16.
4689:
4688:
4686:
4683:
4680:
4679:
4666:
4645:
4644:
4642:
4639:
4637:
4636:
4630:
4624:
4618:
4613:
4610:Isotropic line
4607:
4601:
4595:
4590:
4585:
4579:
4570:
4564:
4562:
4559:
4558:
4557:
4546:
4536:
4530:
4524:
4522:geodesic domes
4518:
4502:
4501:
4481:
4479:
4468:
4465:
4452:
4432:
4406:
4402:
4379:
4375:
4354:
4349:
4345:
4341:
4338:
4335:
4332:
4327:
4323:
4319:
4314:
4310:
4306:
4301:
4297:
4276:
4256:
4236:
4216:
4196:
4176:
4156:
4133:
4130:
4127:
4124:
4121:
4118:
4115:
4112:
4092:
4089:
4084:
4081:
4061:
4038:
4035:
4012:
4001:skew-symmetric
3997:
3996:
3985:
3980:
3973:
3970:
3963:
3960:
3955:
3951:
3947:
3944:
3941:
3938:
3935:
3932:
3929:
3903:
3900:
3894:
3891:
3846:
3845:
3834:
3831:
3828:
3825:
3822:
3819:
3798:
3795:
3778:
3775:
3772:
3769:
3766:
3763:
3758:
3754:
3737:
3725: ∈ T
3719:
3718:
3704:
3700:
3694:
3690:
3684:
3680:
3676:
3673:
3670:
3665:
3662:
3658:
3622: : T
3610: → T
3603:
3595: ∈ T
3591:at each point
3589:
3588:
3576:
3573:
3568:
3564:
3558:
3554:
3535:
3534:
3523:
3520:
3517:
3514:
3511:
3508:
3505:
3502:
3454:tangent bundle
3443:
3442:Geodesic spray
3440:
3417:
3413:
3397:
3396:
3384:
3381:
3378:
3375:
3372:
3369:
3366:
3363:
3360:
3357:
3354:
3351:
3346:
3342:
3338:
3335:
3332:
3329:
3324:
3320:
3316:
3313:
3290:
3253:of the vector
3238:
3235:
3232:
3229:
3226:
3223:
3220:
3217:
3214:
3211:
3206:
3202:
3180:
3177:
3174:
3171:
3168:
3163:
3156:
3153:
3127:
3123:
3095:
3094:
3083:
3080:
3077:
3072:
3065:
3062:
3055:
3052:
3049:
3046:
3041:
3037:
3020:of a manifold
3015:tangent bundle
2996:
2993:
2938:
2937:
2922:
2921:
2920:
2909:
2906:
2903:
2900:
2897:
2894:
2888:
2885:
2872:
2859:
2856:
2853:
2850:
2847:
2844:
2813:
2787:
2772:For any point
2757:
2754:
2736:
2733:
2727:
2724:
2714:
2711:
2705:
2693:free particles
2666:
2661:
2658:
2654:
2629:
2626:
2623:
2620:
2617:
2612:
2608:
2604:
2599:
2595:
2583:
2582:
2571:
2565:
2562:
2556:
2553:
2546:
2542:
2538:
2529:
2526:
2519:
2515:
2511:
2503:
2498:
2495:
2491:
2487:
2479:
2475:
2471:
2464:
2460:
2454:
2450:
2387:
2384:
2358:
2355:
2332:
2309:
2306:
2290:
2289:
2280:
2278:
2267:
2264:
2258:
2255:
2245:
2242:
2236:
2185:
2182:
2151:
2150:
2139:
2136:
2133:
2130:
2127:
2124:
2121:
2118:
2115:
2112:
2109:
2104:
2101:
2098:
2095:
2092:
2087:
2081:
2078:
2074:
2071:
2065:
2061:
2055:
2051:
2048:
2045:
2042:
2039:
2036:
2033:
2030:
2027:
2024:
2019:
2015:
2002:is defined by
1992:
1991:
1980:
1977:
1974:
1971:
1968:
1965:
1962:
1959:
1954:
1951:
1948:
1943:
1937:
1934:
1930:
1925:
1921:
1918:
1915:
1912:
1909:
1906:
1903:
1900:
1897:
1870:
1867:
1840:
1835:
1832:
1828:
1816:
1815:
1804:
1801:
1798:
1792:
1789:
1782:
1778:
1774:
1765:
1762:
1755:
1751:
1747:
1739:
1734:
1731:
1727:
1723:
1715:
1711:
1707:
1700:
1696:
1690:
1686:
1648:
1645:
1642:
1639:
1619:
1591:
1588:
1585:
1582:
1573:also minimize
1562:
1559:
1556:
1553:
1533:
1529:
1526:
1522:
1518:
1515:
1511:
1508:
1497:
1496:
1485:
1482:
1479:
1476:
1473:
1470:
1467:
1464:
1461:
1458:
1455:
1450:
1446:
1442:
1439:
1436:
1407:
1404:
1401:
1397:
1374:
1370:
1340:All minima of
1338:
1337:
1326:
1323:
1320:
1316:
1313:
1310:
1307:
1301:
1298:
1292:
1289:
1286:
1283:
1277:
1274:
1268:
1263:
1260:
1257:
1254:
1250:
1244:
1239:
1235:
1229:
1226:
1221:
1218:
1215:
1212:
1209:
1186:) =
1178:) =
1140:
1139:
1128:
1125:
1122:
1116:
1113:
1110:
1107:
1101:
1098:
1092:
1089:
1086:
1083:
1077:
1074:
1068:
1063:
1060:
1057:
1054:
1050:
1042:
1037:
1033:
1029:
1026:
1023:
1020:
1017:
1005:is defined by
980:
977:
948:
941:
934:
933:
922:
918:
912:
908:
904:
899:
895:
890:
886:
883:
880:
875:
871:
867:
864:
861:
858:
853:
849:
845:
842:
839:
836:
814:
813:
802:
798:
792:
788:
784:
779:
775:
770:
766:
763:
760:
757:
752:
748:
744:
741:
738:
735:
730:
726:
722:
719:
716:
713:
693:
686:
649:if there is a
602:
599:
556:
553:
476:
473:
459:. The article
404:
400:
396:
393:
301:
298:
294:test particles
40:Earth geodesic
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
7108:
7097:
7094:
7092:
7089:
7088:
7086:
7071:
7068:
7066:
7063:
7061:
7058:
7056:
7053:
7051:
7048:
7046:
7043:
7041:
7038:
7036:
7033:
7031:
7028:
7026:
7023:
7021:
7018:
7016:
7013:
7011:
7008:
7007:
7005:
7003:
6999:
6989:
6986:
6984:
6981:
6979:
6976:
6974:
6971:
6969:
6966:
6964:
6961:
6959:
6956:
6954:
6951:
6949:
6946:
6945:
6943:
6939:
6933:
6930:
6928:
6925:
6923:
6920:
6918:
6915:
6913:
6910:
6908:
6907:Metric tensor
6905:
6903:
6900:
6898:
6895:
6894:
6892:
6888:
6885:
6881:
6875:
6872:
6870:
6867:
6865:
6862:
6860:
6857:
6855:
6852:
6850:
6847:
6845:
6842:
6840:
6837:
6835:
6832:
6830:
6827:
6825:
6822:
6820:
6819:Exterior form
6817:
6815:
6812:
6810:
6807:
6805:
6802:
6800:
6797:
6795:
6792:
6790:
6787:
6785:
6782:
6781:
6779:
6773:
6766:
6763:
6761:
6758:
6756:
6753:
6751:
6748:
6746:
6743:
6741:
6738:
6736:
6733:
6731:
6728:
6726:
6723:
6721:
6718:
6716:
6713:
6712:
6710:
6708:
6704:
6698:
6695:
6693:
6692:Tensor bundle
6690:
6688:
6685:
6683:
6680:
6678:
6675:
6673:
6670:
6668:
6665:
6663:
6660:
6658:
6655:
6653:
6650:
6649:
6647:
6641:
6635:
6632:
6630:
6627:
6625:
6622:
6620:
6617:
6615:
6612:
6610:
6607:
6605:
6602:
6600:
6597:
6595:
6592:
6591:
6589:
6585:
6575:
6572:
6570:
6567:
6565:
6562:
6560:
6557:
6555:
6552:
6551:
6549:
6544:
6541:
6539:
6536:
6535:
6532:
6526:
6523:
6521:
6518:
6516:
6513:
6511:
6508:
6506:
6503:
6501:
6498:
6496:
6493:
6491:
6488:
6487:
6485:
6483:
6479:
6476:
6472:
6468:
6467:
6461:
6457:
6450:
6445:
6443:
6438:
6436:
6431:
6430:
6427:
6415:
6412:
6410:
6409:Supermanifold
6407:
6405:
6402:
6400:
6397:
6393:
6390:
6389:
6388:
6385:
6383:
6380:
6378:
6375:
6373:
6370:
6368:
6365:
6363:
6360:
6358:
6355:
6354:
6352:
6348:
6342:
6339:
6337:
6334:
6332:
6329:
6327:
6324:
6322:
6319:
6317:
6314:
6313:
6311:
6307:
6297:
6294:
6292:
6289:
6287:
6284:
6282:
6279:
6277:
6274:
6272:
6269:
6267:
6264:
6262:
6259:
6257:
6254:
6252:
6249:
6248:
6246:
6244:
6240:
6234:
6231:
6229:
6226:
6224:
6221:
6219:
6216:
6214:
6211:
6209:
6206:
6204:
6200:
6196:
6194:
6191:
6189:
6186:
6184:
6180:
6176:
6174:
6171:
6169:
6166:
6164:
6161:
6159:
6156:
6154:
6151:
6149:
6146:
6145:
6143:
6141:
6137:
6131:
6130:Wedge product
6128:
6126:
6123:
6119:
6116:
6115:
6114:
6111:
6109:
6106:
6102:
6099:
6098:
6097:
6094:
6092:
6089:
6087:
6084:
6082:
6079:
6075:
6074:Vector-valued
6072:
6071:
6070:
6067:
6065:
6062:
6058:
6055:
6054:
6053:
6050:
6048:
6045:
6043:
6040:
6039:
6037:
6033:
6027:
6024:
6022:
6019:
6017:
6014:
6010:
6007:
6006:
6005:
6004:Tangent space
6002:
6000:
5997:
5995:
5992:
5990:
5987:
5986:
5984:
5980:
5977:
5975:
5971:
5965:
5962:
5960:
5956:
5952:
5950:
5947:
5945:
5941:
5937:
5933:
5931:
5928:
5926:
5923:
5921:
5918:
5916:
5913:
5911:
5908:
5906:
5903:
5901:
5898:
5894:
5891:
5890:
5889:
5886:
5884:
5881:
5879:
5876:
5874:
5871:
5869:
5866:
5864:
5861:
5859:
5856:
5854:
5851:
5849:
5846:
5844:
5841:
5839:
5835:
5831:
5829:
5825:
5821:
5819:
5816:
5815:
5813:
5807:
5801:
5798:
5796:
5793:
5791:
5788:
5786:
5783:
5781:
5778:
5776:
5773:
5769:
5768:in Lie theory
5766:
5765:
5764:
5761:
5759:
5756:
5752:
5749:
5748:
5747:
5744:
5742:
5739:
5738:
5736:
5734:
5730:
5724:
5721:
5719:
5716:
5714:
5711:
5709:
5706:
5704:
5701:
5699:
5696:
5694:
5691:
5689:
5686:
5684:
5681:
5680:
5678:
5675:
5671:Main results
5669:
5663:
5660:
5658:
5655:
5653:
5652:Tangent space
5650:
5648:
5645:
5643:
5640:
5638:
5635:
5633:
5630:
5628:
5625:
5621:
5618:
5616:
5613:
5612:
5611:
5608:
5604:
5601:
5600:
5599:
5596:
5595:
5593:
5589:
5584:
5580:
5573:
5568:
5566:
5561:
5559:
5554:
5553:
5550:
5538:
5535:
5533:
5530:
5528:
5525:
5523:
5520:
5519:
5517:
5513:
5507:
5504:
5502:
5499:
5497:
5494:
5493:
5491:
5487:
5481:
5480:Schur's lemma
5478:
5476:
5473:
5471:
5468:
5466:
5463:
5461:
5458:
5456:
5455:Gauss's lemma
5453:
5451:
5448:
5447:
5445:
5441:
5435:
5432:
5430:
5427:
5425:
5422:
5420:
5417:
5416:
5414:
5410:
5404:
5401:
5399:
5396:
5394:
5391:
5389:
5386:
5382:
5378:
5375:
5373:
5370:
5368:
5365:
5363:
5360:
5359:
5358:
5357:Metric tensor
5355:
5353:
5352:Inner product
5350:
5348:
5345:
5343:
5340:
5336:
5333:
5331:
5328:
5326:
5323:
5321:
5318:
5317:
5316:
5313:
5312:
5310:
5306:
5301:
5297:
5290:
5285:
5283:
5278:
5276:
5271:
5270:
5267:
5260:
5257:
5254:
5250:
5246:
5243:
5242:
5238:
5233:
5232:See chapter 3
5228:
5222:
5218:
5213:
5212:
5206:
5202:
5198:
5194:
5193:
5188:
5183:
5179:
5173:
5169:
5165:
5160:
5157:
5151:
5147:
5146:
5141:
5137:
5133:
5129:
5126:
5121:
5115:
5111:
5107:
5103:
5102:Landau, L. D.
5099:
5095:
5093:0-471-15733-3
5089:
5085:
5080:
5077:
5072:
5066:
5062:
5058:
5053:
5050:
5045:
5039:
5035:
5031:
5027:
5023:
5020:
5019:See chapter 2
5015:
5009:
5005:
5001:
4996:
4995:
4988:
4985:
4977:
4967:
4963:
4957:
4956:
4950:
4945:
4936:
4935:
4929:
4924:
4918:
4914:
4910:
4906:
4905:
4898:
4897:
4893:
4888:
4885:
4880:
4876:
4872:
4868:
4865:(11): 90–99.
4864:
4860:
4859:
4854:
4847:
4844:
4836:
4832:
4828:
4823:
4818:
4813:
4808:
4804:
4800:
4796:
4792:
4791:
4783:
4776:
4773:
4768:
4764:
4760:
4756:
4755:
4750:
4743:
4740:
4735:
4734:
4729:
4723:
4720:
4715:
4711:
4707:
4705:
4700:
4694:
4691:
4684:
4676:
4670:
4667:
4663:
4659:
4653:
4651:
4647:
4640:
4634:
4631:
4628:
4625:
4622:
4619:
4617:
4614:
4611:
4608:
4605:
4602:
4599:
4596:
4594:
4591:
4589:
4586:
4583:
4580:
4574:
4571:
4569:
4566:
4565:
4560:
4555:
4551:
4547:
4545:
4541:
4537:
4535:
4531:
4529:
4525:
4523:
4519:
4517:
4513:
4509:
4508:
4507:
4498:
4489:
4485:
4482:This section
4480:
4477:
4473:
4472:
4466:
4464:
4450:
4430:
4422:
4404:
4400:
4377:
4373:
4347:
4343:
4336:
4333:
4325:
4321:
4312:
4308:
4304:
4299:
4295:
4274:
4254:
4234:
4214:
4194:
4174:
4154:
4147:
4131:
4122:
4116:
4113:
4110:
4101:
4097:
4090:
4088:
4082:
4080:
4078:
4073:
4002:
3983:
3978:
3961:
3958:
3953:
3945:
3939:
3936:
3933:
3927:
3920:
3919:
3918:
3892:
3881:
3877:
3873:
3872:determined by
3868:
3866:
3862:
3861:
3855:
3851:
3832:
3829:
3826:
3823:
3817:
3810:
3809:
3808:
3806:
3805:
3796:
3794:
3792:
3776:
3773:
3770:
3764:
3761:
3756:
3752:
3743:
3736:
3732:
3728:
3724:
3702:
3698:
3692:
3682:
3678:
3671:
3668:
3663:
3660:
3656:
3648:
3647:
3646:
3644:
3640:
3636:
3631:
3629:
3626: →
3625:
3617:
3613:
3609:
3598:
3594:
3574:
3571:
3566:
3562:
3556:
3552:
3544:
3543:
3542:
3540:
3521:
3518:
3515:
3512:
3509:
3506:
3503:
3500:
3493:
3492:
3491:
3489:
3485:
3481:
3477:
3472:
3470:
3469:
3463:
3459:
3455:
3449:
3441:
3439:
3437:
3433:
3415:
3411:
3402:
3382:
3376:
3373:
3370:
3364:
3361:
3352:
3344:
3340:
3336:
3330:
3322:
3318:
3311:
3304:
3303:
3302:
3288:
3280:
3276:
3272:
3267:
3266:
3262:
3260:
3254:
3252:
3233:
3230:
3224:
3221:
3218:
3212:
3204:
3200:
3178:
3175:
3169:
3161:
3154:
3151:
3125:
3121:
3112:
3109: ∈
3108:
3104:
3101: ∈
3100:
3078:
3070:
3063:
3060:
3053:
3047:
3039:
3035:
3027:
3026:
3025:
3023:
3019:
3016:
3012:
3008:
3004:
3003:
2995:Geodesic flow
2994:
2992:
2990:
2974:
2970:
2966:
2961:
2959:
2955:
2951:
2947:
2943:
2936:containing 0.
2935:
2931:
2930:open interval
2928:is a maximal
2927:
2923:
2907:
2904:
2901:
2895:
2886:
2883:
2873:
2857:
2854:
2848:
2842:
2835:
2834:
2832:
2828:
2811:
2803:
2799:
2795:
2794:tangent space
2791:
2783:
2779:
2775:
2771:
2770:
2769:
2767:
2763:
2755:
2753:
2750:
2734:
2731:
2725:
2722:
2712:
2709:
2694:
2690:
2686:
2682:
2664:
2659:
2656:
2643:
2624:
2618:
2615:
2610:
2606:
2602:
2597:
2593:
2569:
2563:
2560:
2554:
2551:
2544:
2540:
2536:
2527:
2524:
2517:
2513:
2509:
2501:
2496:
2493:
2485:
2477:
2473:
2469:
2462:
2458:
2452:
2448:
2437:
2436:
2435:
2433:
2429:
2425:
2421:
2416:
2414:
2413:
2408:
2404:
2385:
2382:
2356:
2353:
2330:
2307:
2304:
2288:
2281:
2279:
2265:
2262:
2256:
2253:
2243:
2240:
2226:
2225:
2222:
2220:
2216:
2212:
2208:
2204:
2201:
2197:
2191:
2183:
2181:
2179:
2175:
2171:
2167:
2163:
2158:
2156:
2155:Jacobi fields
2137:
2131:
2128:
2125:
2122:
2119:
2116:
2113:
2107:
2102:
2099:
2096:
2093:
2090:
2085:
2079:
2072:
2063:
2049:
2043:
2040:
2037:
2028:
2022:
2017:
2013:
2005:
2004:
2003:
2001:
1997:
1978:
1972:
1969:
1966:
1963:
1957:
1952:
1949:
1946:
1941:
1935:
1919:
1913:
1904:
1898:
1895:
1888:
1887:
1886:
1884:
1880:
1876:
1868:
1866:
1864:
1860:
1856:
1838:
1833:
1830:
1802:
1799:
1796:
1790:
1787:
1780:
1776:
1772:
1763:
1760:
1753:
1749:
1745:
1737:
1732:
1729:
1721:
1713:
1709:
1705:
1698:
1694:
1688:
1684:
1673:
1672:
1671:
1669:
1665:
1660:
1643:
1637:
1617:
1609:
1605:
1586:
1580:
1557:
1551:
1527:
1524:
1520:
1516:
1513:
1506:
1480:
1474:
1468:
1465:
1462:
1456:
1453:
1448:
1440:
1434:
1427:
1426:
1425:
1423:
1405:
1402:
1399:
1395:
1372:
1368:
1359:
1355:
1351:
1347:
1343:
1324:
1321:
1318:
1308:
1299:
1296:
1290:
1284:
1275:
1272:
1258:
1252:
1248:
1242:
1237:
1233:
1227:
1224:
1219:
1213:
1207:
1200:
1199:
1198:
1197:
1193:
1189:
1185:
1181:
1177:
1173:
1169:
1165:
1161:
1157:
1153:
1149:
1145:
1142:The distance
1126:
1123:
1120:
1108:
1099:
1096:
1090:
1084:
1075:
1072:
1058:
1052:
1048:
1040:
1035:
1031:
1027:
1021:
1015:
1008:
1007:
1006:
1004:
1000:
997:, the length
996:
993:
992:metric tensor
989:
986:
978:
976:
974:
970:
965:
963:
962:shortest path
959:
954:
947:
940:
920:
916:
910:
906:
902:
897:
893:
888:
884:
873:
869:
862:
859:
851:
847:
840:
834:
827:
826:
825:
823:
819:
800:
796:
790:
786:
782:
777:
773:
768:
764:
761:
750:
746:
739:
736:
728:
724:
717:
711:
704:
703:
702:
699:
692:
685:
680:
676:
672:
667:
663:
656:
652:
648:
644:
641:
637:
632:
628:
624:
620:
616:
612:
608:
600:
594:
590:
588:
584:
580:
571:
566:
562:
554:
552:
550:
546:
542:
538:
534:
530:
526:
522:
518:
514:
510:
509:great circles
506:
502:
493:
486:
481:
474:
472:
470:
466:
462:
458:
453:
451:
447:
443:
439:
435:
431:
427:
423:
418:
402:
398:
391:
383:
379:
374:
371:
367:
365:
361:
357:
353:
349:
345:
341:
337:
333:
332:
327:
326:open interval
323:
319:
315:
311:
307:
299:
297:
295:
292:
288:
284:
279:
277:
273:
269:
265:
261:
257:
253:
248:
246:
242:
238:
234:
230:
226:
222:
218:
214:
210:
206:
202:
201:
196:
195:
190:
189:
183:
181:
180:straight line
177:
173:
169:
165:
161:
157:
151:
70:
66:
55:
54:Klein quartic
51:
45:
41:
37:
33:
19:
7070:Hermann Weyl
6874:Vector space
6859:Pseudotensor
6828:
6824:Fiber bundle
6777:abstractions
6672:Mixed tensor
6657:Tensor field
6464:
6336:Moving frame
6331:Morse theory
6321:Gauge theory
6113:Tensor field
6042:Closed/Exact
6021:Vector field
5989:Distribution
5930:Hypercomplex
5925:Quaternionic
5757:
5662:Vector field
5620:Smooth atlas
5515:Applications
5443:Main results
5346:
5231:
5215:, New York:
5210:
5190:
5163:
5144:
5124:
5109:
5083:
5075:
5056:
5048:
5033:
5018:
4999:
4980:
4971:
4952:
4912:
4895:
4887:
4862:
4856:
4846:
4794:
4788:
4775:
4758:
4752:
4742:
4731:
4722:
4714:the original
4702:
4693:
4674:
4669:
4627:Zoll surface
4621:Morse theory
4616:Jacobi field
4505:
4492:
4488:adding to it
4483:
4467:Applications
4146:neighborhood
4102:
4098:
4094:
4086:
4074:
3998:
3871:
3869:
3864:
3858:
3853:
3849:
3847:
3802:
3800:
3734:
3730:
3726:
3722:
3720:
3638:
3632:
3627:
3623:
3614:denotes the
3611:
3607:
3596:
3592:
3590:
3538:
3536:
3479:
3473:
3465:
3458:vector field
3451:
3400:
3398:
3268:
3264:
3256:
3192:
3110:
3106:
3102:
3098:
3096:
3021:
3017:
3006:
2999:
2998:
2972:
2968:
2964:
2963:In general,
2962:
2957:
2953:
2939:
2933:
2925:
2830:
2826:
2801:
2797:
2785:
2781:
2777:
2773:
2761:
2759:
2641:
2584:
2427:
2423:
2417:
2410:
2403:vector field
2293:
2282:
2217:) such that
2214:
2202:
2195:
2193:
2159:
2152:
1993:
1878:
1872:
1861:, discussed
1858:
1817:
1667:
1661:
1607:
1603:
1498:
1420:curve), the
1357:
1353:
1349:
1345:
1341:
1339:
1187:
1183:
1179:
1175:
1174:such that γ(
1171:
1163:
1159:
1155:
1151:
1147:
1143:
1141:
1002:
998:
994:
987:
982:
966:
961:
957:
952:
945:
938:
935:
821:
815:
697:
690:
683:
678:
674:
670:
665:
661:
654:
646:
642:
640:metric space
635:
630:
626:
622:
604:
583:great circle
578:
576:
544:
536:
532:
528:
524:
516:
512:
498:
454:
419:
382:great circle
377:
375:
372:
368:
364:elastic band
359:
355:
351:
347:
343:
339:
329:
321:
320:(a function
303:
300:Introduction
291:free falling
285:. Timelike
280:
249:
243:/nodes of a
239:between two
233:graph theory
225:great circle
198:
192:
186:
184:
68:
62:
7010:Élie Cartan
6958:Spin tensor
6932:Weyl tensor
6890:Mathematics
6854:Multivector
6645:definitions
6543:Engineering
6482:Mathematics
6281:Levi-Civita
6271:Generalized
6243:Connections
6193:Lie algebra
6125:Volume form
6026:Vector flow
5999:Pushforward
5994:Lie bracket
5893:Lie algebra
5858:G-structure
5647:Pushforward
5627:Submanifold
5145:Gravitation
5136:Thorne, Kip
5004:McGraw-Hill
4966:introducing
4091:Ribbon test
3880:Spivak 1999
3742:pushforward
3541:satisfying
3462:total space
3275:Hamiltonian
3005:is a local
2430:(using the
2178:Hamiltonian
268:transported
209:ellipsoidal
7085:Categories
6839:Linear map
6707:Operations
6404:Stratifold
6362:Diffeology
6158:Associated
5959:Symplectic
5944:Riemannian
5873:Hyperbolic
5800:Submersion
5708:Hopf–Rinow
5642:Submersion
5637:Smooth map
5475:Ricci flow
5424:Hyperbolic
4949:references
4728:"geodesic"
4699:"geodesic"
4685:References
4660:, e.g., a
4534:UV mapping
4167:of a line
3801:Equation (
3721:for every
3484:horizontal
2833:such that
2188:See also:
559:See also:
227:(see also
219:, it is a
197:come from
176:connection
6978:EM tensor
6814:Dimension
6765:Transpose
6286:Principal
6261:Ehresmann
6218:Subbundle
6208:Principal
6183:Fibration
6163:Cotangent
6035:Covectors
5888:Lie group
5868:Hermitian
5811:manifolds
5780:Immersion
5775:Foliation
5713:Noether's
5698:Frobenius
5693:De Rham's
5688:Darboux's
5579:Manifolds
5419:Hermitian
5372:Signature
5335:Sectional
5315:Curvature
5197:EMS Press
4974:July 2014
4552:(e.g. in
4495:June 2014
4344:ε
4313:∗
4305:−
4255:ε
4235:ℓ
4175:ℓ
4129:→
4123:ℓ
4060:∇
4037:¯
4034:∇
4011:∇
3972:¯
3969:∇
3962:−
3950:∇
3902:¯
3899:∇
3890:∇
3821:↦
3771:λ
3768:↦
3757:λ
3740:) is the
3683:λ
3661:λ
3557:∗
3553:π
3516:⊕
3466:geodesic
3412:γ
3225:
3191:. Thus,
3155:˙
3152:γ
3122:γ
3064:˙
3061:γ
3000:Geodesic
2956:and
2887:˙
2884:γ
2843:γ
2812:γ
2726:˙
2723:γ
2713:˙
2710:γ
2704:∇
2665:λ
2660:ν
2657:μ
2653:Γ
2619:γ
2616:∘
2611:μ
2598:μ
2594:γ
2545:ν
2541:γ
2518:μ
2514:γ
2502:λ
2497:ν
2494:μ
2490:Γ
2463:λ
2459:γ
2386:˙
2383:γ
2357:˙
2354:γ
2308:˙
2305:γ
2257:˙
2254:γ
2244:˙
2241:γ
2235:∇
2176:taken as
2132:ψ
2123:φ
2114:γ
2077:∂
2070:∂
2060:∂
2044:ψ
2038:φ
2029:γ
2014:δ
1973:φ
1964:γ
1933:∂
1929:∂
1914:φ
1905:γ
1896:δ
1839:λ
1834:ν
1831:μ
1827:Γ
1781:ν
1754:μ
1738:λ
1733:ν
1730:μ
1726:Γ
1699:λ
1644:γ
1618:γ
1587:γ
1558:γ
1525:γ
1514:γ
1481:γ
1466:−
1454:≤
1441:γ
1300:˙
1297:γ
1276:˙
1273:γ
1253:γ
1234:∫
1214:γ
1100:˙
1097:γ
1076:˙
1073:γ
1053:γ
1032:∫
1022:γ
903:−
863:γ
841:γ
783:−
740:γ
718:γ
555:Triangles
515:to point
442:satellite
434:spacetime
395:→
185:The noun
6844:Manifold
6829:Geodesic
6587:Notation
6382:Orbifold
6377:K-theory
6367:Diffiety
6091:Pullback
5905:Oriented
5883:Kenmotsu
5863:Hadamard
5809:Types of
5758:Geodesic
5583:Glossary
5434:Kenmotsu
5347:Geodesic
5300:Glossary
5207:(1972),
5142:(1973),
5108:(1975),
5032:(1978),
4911:(1999),
4835:Archived
4561:See also
4287:we have
3261:on
2952:on both
2950:smoothly
2829:→
2825: :
2679:are the
2407:open set
2205:with an
2196:geodesic
1853:are the
1528:′
1517:′
701:we have
651:constant
647:geodesic
625: :
615:distance
503:. On a
475:Examples
324:from an
312:for the
310:equation
241:vertices
237:geodesic
215:. For a
194:geodetic
188:geodesic
69:geodesic
65:geometry
6941:Physics
6775:Related
6538:Physics
6456:Tensors
6326:History
6309:Related
6223:Tangent
6201:)
6181:)
6148:Adjoint
6140:Bundles
6118:density
6016:Torsion
5982:Vectors
5974:Tensors
5957:)
5942:)
5938:,
5936:Pseudo−
5915:Poisson
5848:Finsler
5843:Fibered
5838:Contact
5836:)
5828:Complex
5826:)
5795:Section
5501:Hilbert
5496:Finsler
4962:improve
4879:7078650
4831:9671694
4799:Bibcode
4421:metrics
4003:, then
3876:torsion
3599:; here
3460:on the
3301:, i.e.
3249:is the
3013:on the
2172:, with
1881:. The
1168:infimum
611:locally
378:locally
358:−
221:segment
213:surface
200:geodesy
174:with a
164:surface
154:) is a
36:Geodesy
6869:Vector
6864:Spinor
6849:Matrix
6643:Tensor
6291:Vector
6276:Koszul
6256:Cartan
6251:Affine
6233:Vector
6228:Tensor
6213:Spinor
6203:Normal
6199:Stable
6153:Affine
6057:bundle
6009:bundle
5955:Almost
5878:Kähler
5834:Almost
5824:Almost
5818:Closed
5718:Sard's
5674:(list)
5429:Kähler
5325:Scalar
5320:tensor
5223:
5174:
5152:
5116:
5090:
5067:
5040:
5010:
4951:, but
4919:
4877:
4829:
4819:
4704:Lexico
4656:For a
4538:robot
4365:where
3848:where
3097:where
3011:action
2975:. Any
2924:where
2644:) and
2585:where
2567:
2418:Using
2405:in an
1818:where
1424:gives
1348:, but
1192:action
1182:and γ(
563:, and
505:sphere
314:length
6789:Basis
6474:Scope
6399:Sheaf
6173:Fiber
5949:Rizza
5920:Prime
5751:Local
5741:Curve
5603:Atlas
5330:Ricci
5249:torus
4875:S2CID
4838:(PDF)
4822:21092
4785:(PDF)
4641:Notes
4267:from
4144:of a
3482:into
3468:spray
2792:(the
2434:) as
2211:curve
2198:on a
1863:below
990:with
983:In a
645:is a
619:curve
531:. If
428:. In
346:) to
318:curve
316:of a
274:of a
250:In a
245:graph
223:of a
205:Earth
156:curve
6266:Form
6168:Dual
6101:flow
5964:Tame
5940:Sub−
5853:Flat
5733:Maps
5221:ISBN
5172:ISBN
5150:ISBN
5114:ISBN
5088:ISBN
5065:ISBN
5038:ISBN
5008:ISBN
4917:ISBN
4827:PMID
4443:and
4419:are
4392:and
4023:and
3852:and
3486:and
3113:and
3002:flow
2760:The
1994:The
1662:The
1158:and
539:are
535:and
527:and
467:and
67:, a
6188:Jet
4867:doi
4817:PMC
4807:doi
4763:doi
4514:or
4490:.
4423:on
3999:is
3434:.
3222:exp
2987:is
2932:in
2871:and
2800:at
2796:to
2784:in
2776:in
2422:on
1194:or
1162:of
960:or
677:in
673:of
657:≥ 0
605:In
521:arc
328:of
182:".
160:arc
117:-,-
63:In
7087::
6179:Co
5230:.
5219:,
5195:,
5189:,
5170:,
5166:,
5138:;
5134:;
5123:.
5104:;
5074:.
5063:,
5047:.
5028:;
5017:.
5006:,
4873:.
4863:60
4861:.
4855:.
4833:.
4825:.
4815:.
4805:.
4795:95
4793:.
4787:.
4759:16
4757:.
4751:.
4730:.
4708:.
4701:.
4649:^
4463:.
4079:.
3867:.
3793:.
3490::
3478:TT
3471:.
3255:tV
3111:TM
3105:,
3018:TM
2991:.
2960:.
2213:γ(
2194:A
2180:.
1865:.
1150:,
964:.
951:∈
944:,
696:∈
689:,
664:∈
629:→
613:a
589:.
577:A
483:A
296:.
247:.
138:,-
126:iː
114:oʊ
111:,-
84:iː
81:dʒ
6448:e
6441:t
6434:v
6197:(
6177:(
5953:(
5934:(
5832:(
5822:(
5585:)
5581:(
5571:e
5564:t
5557:v
5379:/
5302:)
5298:(
5288:e
5281:t
5274:v
5234:.
5200:.
5127:.
5097:.
5078:.
5051:.
5021:.
4987:)
4981:(
4976:)
4972:(
4958:.
4899:.
4881:.
4869::
4809::
4801::
4769:.
4765::
4675:k
4497:)
4493:(
4451:S
4431:N
4405:S
4401:g
4378:N
4374:g
4353:)
4348:2
4340:(
4337:O
4334:=
4331:)
4326:S
4322:g
4318:(
4309:f
4300:N
4296:g
4275:l
4215:f
4195:S
4155:N
4132:S
4126:)
4120:(
4117:N
4114::
4111:f
3984:Y
3979:X
3959:Y
3954:X
3946:=
3943:)
3940:Y
3937:,
3934:X
3931:(
3928:D
3893:,
3878:(
3860:1
3857:(
3854:b
3850:a
3833:b
3830:+
3827:t
3824:a
3818:t
3804:1
3777:.
3774:X
3765:X
3762::
3753:S
3738:λ
3735:S
3733:(
3731:d
3727:M
3723:X
3703:X
3699:H
3693:X
3689:)
3679:S
3675:(
3672:d
3669:=
3664:X
3657:H
3639:M
3628:M
3624:M
3620:π
3612:M
3608:M
3604:∗
3601:π
3597:M
3593:v
3575:v
3572:=
3567:v
3563:W
3539:W
3522:.
3519:V
3513:H
3510:=
3507:M
3504:T
3501:T
3480:M
3416:V
3401:V
3383:.
3380:)
3377:V
3374:,
3371:V
3368:(
3365:g
3362:=
3359:)
3356:)
3353:V
3350:(
3345:t
3341:G
3337:,
3334:)
3331:V
3328:(
3323:t
3319:G
3315:(
3312:g
3289:g
3265:.
3263:M
3237:)
3234:V
3231:t
3228:(
3219:=
3216:)
3213:V
3210:(
3205:t
3201:G
3179:V
3176:=
3173:)
3170:0
3167:(
3162:V
3126:V
3107:V
3103:R
3099:t
3082:)
3079:t
3076:(
3071:V
3054:=
3051:)
3048:V
3045:(
3040:t
3036:G
3022:M
3009:-
3007:R
2985:M
2981:ℝ
2977:γ
2973:R
2969:R
2965:I
2958:V
2954:p
2934:R
2926:I
2908:,
2905:V
2902:=
2899:)
2896:0
2893:(
2858:p
2855:=
2852:)
2849:0
2846:(
2831:M
2827:I
2802:p
2798:M
2790:M
2788:p
2786:T
2782:V
2778:M
2774:p
2735:0
2732:=
2642:t
2628:)
2625:t
2622:(
2607:x
2603:=
2570:,
2564:0
2561:=
2555:t
2552:d
2537:d
2528:t
2525:d
2510:d
2486:+
2478:2
2474:t
2470:d
2453:2
2449:d
2424:M
2412:1
2331:t
2287:)
2285:1
2283:(
2266:0
2263:=
2215:t
2203:M
2138:.
2135:)
2129:s
2126:+
2120:t
2117:+
2111:(
2108:E
2103:0
2100:=
2097:t
2094:=
2091:s
2086:|
2080:t
2073:s
2064:2
2050:=
2047:)
2041:,
2035:(
2032:)
2026:(
2023:E
2018:2
1979:.
1976:)
1970:t
1967:+
1961:(
1958:E
1953:0
1950:=
1947:t
1942:|
1936:t
1920:=
1917:)
1911:(
1908:)
1902:(
1899:E
1879:E
1803:,
1800:0
1797:=
1791:t
1788:d
1777:x
1773:d
1764:t
1761:d
1750:x
1746:d
1722:+
1714:2
1710:t
1706:d
1695:x
1689:2
1685:d
1668:E
1647:)
1641:(
1638:L
1608:E
1604:E
1590:)
1584:(
1581:L
1561:)
1555:(
1552:E
1532:)
1521:,
1510:(
1507:g
1484:)
1478:(
1475:E
1472:)
1469:a
1463:b
1460:(
1457:2
1449:2
1445:)
1438:(
1435:L
1406:2
1403:,
1400:1
1396:W
1373:1
1369:C
1358:E
1354:L
1350:L
1346:L
1342:E
1325:.
1322:t
1319:d
1315:)
1312:)
1309:t
1306:(
1291:,
1288:)
1285:t
1282:(
1267:(
1262:)
1259:t
1256:(
1249:g
1243:b
1238:a
1228:2
1225:1
1220:=
1217:)
1211:(
1208:E
1188:q
1184:b
1180:p
1176:a
1172:M
1164:M
1160:q
1156:p
1152:q
1148:p
1146:(
1144:d
1127:.
1124:t
1121:d
1115:)
1112:)
1109:t
1106:(
1091:,
1088:)
1085:t
1082:(
1067:(
1062:)
1059:t
1056:(
1049:g
1041:b
1036:a
1028:=
1025:)
1019:(
1016:L
1003:M
999:L
995:g
988:M
953:I
949:2
946:t
942:1
939:t
921:.
917:|
911:2
907:t
898:1
894:t
889:|
885:=
882:)
879:)
874:2
870:t
866:(
860:,
857:)
852:1
848:t
844:(
838:(
835:d
822:v
801:.
797:|
791:2
787:t
778:1
774:t
769:|
765:v
762:=
759:)
756:)
751:2
747:t
743:(
737:,
734:)
729:1
725:t
721:(
715:(
712:d
698:J
694:2
691:t
687:1
684:t
679:I
675:t
671:J
666:I
662:t
655:v
643:M
636:I
631:M
627:I
623:γ
537:B
533:A
529:B
525:A
517:B
513:A
487:.
403:2
399:t
392:t
360:t
356:s
352:t
350:(
348:f
344:s
342:(
340:f
331:R
322:f
150:/
147:k
144:ɪ
141:z
135:k
132:ɪ
129:s
123:d
120:ˈ
108:k
105:ɪ
102:s
99:ɛ
96:d
93:ˈ
90:ə
87:.
78:ˌ
75:/
71:(
46:.
20:)
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