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A vector flow can be thought of as a solution to the system of differential equations induced by a vector field. That is, if a (conservative) vector field is a map to the tangent space, it represents the tangent vectors to some function at each point. Splitting the tangent vectors into directional
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From a point, the rate of change of the i-th component with respect to the parametrization of the flow (“how much the flow has acted”) is described by the i-th component of the field. That is, if one parametrizes with
245:
derivatives, one can solve the resulting system of differential equations to find the function. In this sense, the function is the flow and both induces and is induced by the vector field.
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the first position component changes as described by the first component of the vector field at the point one starts from, and likewise for all other components.
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if it generates a global flow. Every smooth vector field on a compact manifold without boundary is complete.
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The image of the exponential map always lies in the connected component of the identity in
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The exponential map restricts to a diffeomorphism from some neighborhood of 0 in
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theory. These related concepts are explored in a spectrum of articles:
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253:‘length along the path of the flow,’ as one proceeds along the flow by
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123:(flow, infinitesimal generator, integral curve, complete vector field)
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515:) = γ(1) where γ is the integral curve starting at the identity in
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Every left-invariant vector field on a Lie group is complete. The
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443:(exponential map, infinitesimal generator, one-parameter group)
32:. These appear in a number of different contexts, including
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be a pseudo-Riemannian manifold (or any manifold with an
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24:refers to a set of closely related concepts of the
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400:
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240:(geodesic, exponential map, injectivity radius)
131:be a smooth vector field on a smooth manifold
609:
1505:. Unsourced material may be challenged and
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602:
1569:Learn how and when to remove this message
623:
349:there exists a unique geodesic γ :
218:. Global flows define smooth actions of
459:. There are one-to-one correspondences
298:is the unique geodesic passing through
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401:{\displaystyle {\dot {\gamma }}(0)=V.}
302:at 0 and whose tangent vector at 0 is
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467:} ⇔ {left-invariant vector fields on
49:exponential map (Riemannian geometry)
1503:adding citations to reliable sources
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117:Vector flow in differential topology
542:) is the one-parameter subgroup of
314:for which the geodesic is defined.
210:is one whose flow domain is all of
13:
234:Vector flow in Riemannian geometry
14:
1612:
585: – Computer vision framework
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310:is the maximal open interval of
437:Vector flow in Lie group theory
656:Differentiable/Smooth manifold
527:The exponential map is smooth.
451:starting at the identity is a
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1:
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463:{one-parameter subgroups of
135:. There is a unique maximal
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1362:Classification of manifolds
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1438:over commutative algebras
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1154:Riemann curvature tensor
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148:infinitesimal generator
66:infinitesimal generator
1591:Geodesic (mathematics)
946:Manifold with boundary
661:Differential structure
453:one-parameter subgroup
402:
191:is the unique maximal
78:one-parameter subgroup
1596:Differential topology
557:to a neighborhood of
499:its Lie algebra. The
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34:differential topology
1499:improve this article
1093:Covariant derivative
644:Topological manifold
583:Gradient vector flow
503:is a map exp :
429:for which 1 lies in
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226:. A vector field is
59:exponential function
1127:Exterior derivative
729:Atiyah–Singer index
678:Riemannian manifold
495:be a Lie group and
441:Relevant concepts:
238:Relevant concepts:
121:Relevant concepts:
38:Riemannian geometry
1433:Secondary calculus
1387:Singularity theory
1342:Parallel transport
1110:De Rham cohomology
749:Generalized Stokes
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286:is defined as exp(
110:injectivity radius
54:matrix exponential
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1115:Differential form
769:Whitney embedding
703:Differential form
417:be the subset of
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357:for which γ(0) =
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323:affine connection
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1418:Fréchet manifold
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93:Hamiltonian flow
74:(→ vector field)
28:determined by a
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1403:Banach manifold
1396:Generalizations
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1142:Ricci curvature
1098:Cotangent space
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809:Exponential map
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83:flow (geometry)
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956:Parallelizable
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836:Lie derivative
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792:Diffeomorphism
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637:Basic concepts
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329:be a point in
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193:integral curve
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1520: –
1519:
1518:"Vector flow"
1515:
1514:Find sources:
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1489:
1484:This article
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1455:Supermanifold
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1176:Wedge product
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1120:Vector-valued
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1050:Tangent space
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814:in Lie theory
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717:Main results
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698:Tangent space
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546:generated by
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519:generated by
518:
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511:given by exp(
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88:geodesic flow
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68:(→ Lie group)
67:
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35:
31:
27:
23:
19:
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1532:
1525:
1513:
1497:Please help
1485:
1382:Moving frame
1377:Morse theory
1367:Gauge theory
1159:Tensor field
1088:Closed/Exact
1071:
1067:Vector field
1035:Distribution
976:Hypercomplex
971:Quaternionic
708:Vector field
666:Smooth atlas
569:
562:
558:
554:
547:
543:
539:
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531:
530:For a fixed
520:
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267:exp :
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207:
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200:
199:starting at
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139:
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128:
126:
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112:(→ glossary)
30:vector field
21:
15:
1327:Levi-Civita
1317:Generalized
1289:Connections
1239:Lie algebra
1171:Volume form
1072:Vector flow
1045:Pushforward
1040:Lie bracket
939:Lie algebra
904:G-structure
693:Pushforward
673:Submanifold
208:global flow
170:. For each
168:flow domain
103:Anosov flow
22:vector flow
18:mathematics
1601:Lie groups
1585:Categories
1559:March 2009
1529:newspapers
1450:Stratifold
1408:Diffeology
1204:Associated
1005:Symplectic
990:Riemannian
919:Hyperbolic
846:Submersion
754:Hopf–Rinow
688:Submersion
683:Smooth map
590:References
534:, the map
325:) and let
98:Ricci flow
1486:does not
1332:Principal
1307:Ehresmann
1264:Subbundle
1254:Principal
1229:Fibration
1209:Cotangent
1081:Covectors
934:Lie group
914:Hermitian
857:manifolds
826:Immersion
821:Foliation
759:Noether's
744:Frobenius
739:De Rham's
734:Darboux's
625:Manifolds
375:˙
372:γ
42:Lie group
1428:Orbifold
1423:K-theory
1413:Diffiety
1137:Pullback
951:Oriented
929:Kenmotsu
909:Hadamard
855:Types of
804:Geodesic
629:Glossary
577:See also
228:complete
178:the map
1543:scholar
1507:removed
1492:sources
1372:History
1355:Related
1269:Tangent
1247:)
1227:)
1194:Adjoint
1186:Bundles
1164:density
1062:Torsion
1028:Vectors
1020:Tensors
1003:)
988:)
984:,
982:Pseudo−
961:Poisson
894:Finsler
889:Fibered
884:Contact
882:)
874:Complex
872:)
841:Section
306:. Here
166:is the
154:. Here
1545:
1538:
1531:
1524:
1516:
1337:Vector
1322:Koszul
1302:Cartan
1297:Affine
1279:Vector
1274:Tensor
1259:Spinor
1249:Normal
1245:Stable
1199:Affine
1103:bundle
1055:bundle
1001:Almost
924:Kähler
880:Almost
870:Almost
864:Closed
764:Sard's
720:(list)
538:↦ exp(
146:whose
20:, the
1550:JSTOR
1536:books
1445:Sheaf
1219:Fiber
995:Rizza
966:Prime
797:Local
787:Curve
649:Atlas
1522:news
1490:any
1488:cite
1312:Form
1214:Dual
1147:flow
1010:Tame
986:Sub−
899:Flat
779:Maps
491:Let
471:} ⇔
408:Let
361:and
317:Let
260:The
137:flow
127:Let
40:and
26:flow
1501:by
1234:Jet
561:in
455:of
337:in
222:on
195:of
150:is
16:In
1587::
1225:Co
540:tX
523:.
507:→
475:=
433:.
353:→
294:→
279:→
255:dL
214:×
206:A
203:.
187:→
174:∈
162:×
158:⊆
142:→
36:,
1572:)
1566:(
1561:)
1557:(
1547:·
1540:·
1533:·
1526:·
1509:.
1495:.
1243:(
1223:(
999:(
980:(
878:(
868:(
631:)
627:(
617:e
610:t
603:v
572:.
570:G
565:.
563:G
559:e
555:g
550:.
548:X
544:G
536:t
532:X
521:X
517:G
513:X
509:G
505:g
497:g
493:G
487:.
485:G
481:e
477:T
473:g
469:G
465:G
457:G
431:I
427:M
423:p
419:T
414:p
410:D
396:.
393:V
390:=
387:)
384:0
381:(
359:p
355:M
351:I
347:M
343:p
339:T
335:V
331:M
327:p
319:M
312:R
308:I
304:X
300:p
296:M
292:I
288:X
281:M
277:M
273:p
269:T
251:L
224:M
220:R
216:M
212:R
201:p
197:V
189:M
184:p
180:D
176:M
172:p
164:M
160:R
156:D
152:V
144:M
140:D
133:M
129:V
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