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to the curve. Unless the curve is straight, these lines will intersect among themselves in a rather complicated fashion. However, if one looks only in a narrow band around the curve, the portions of the lines in that band will not intersect, and will cover the entire band without gaps. This band is a
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The normal bundle is a tubular neighborhood and because of the diffeomorphism condition in the second point, all tubular neighborhood have the same dimension, namely (the dimension of the vector bundle considered as a manifold is) that of
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A curve, in blue, and some lines perpendicular to it, in green. Small portions of those lines around the curve are in red.
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Generalizations of smooth manifolds yield generalizations of tubular neighborhoods, such as regular neighborhoods, or
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These generalizations are used to produce analogs to the normal bundle, or rather to the
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The idea behind a tubular neighborhood can be explained in a simple example. Consider a
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A close up of the figure above. The curve is in blue, and its tubular neighborhood
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the role of the plane containing the curve. Consider the natural map
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curve in the plane without self-intersections. On each point on the
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to the curve where the curve passes through that disc's center.
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is in red. With the notation in the article, the curve is
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Pages displaying wikidata descriptions as a fallback
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54:but its sources remain unclear because it lacks
621:be smooth manifolds. A tubular neighborhood of
166:A schematic illustration of the normal bundle
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572:all the discs have the same fixed radius;
85:Learn how and when to remove this message
1096:Differential forms in algebraic topology
401:of this map to the entire normal bundle
161:
104:
96:
14:
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590:
575:the center of each disc lies on the
117:, the space containing the curve is
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528:itself, a tubular neighbourhood of
483:is called a tubular neighbourhood.
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1042:
1001:{\displaystyle J\vert _{U}:U\to V}
809:{\displaystyle S\hookrightarrow M}
25:
1186:
1094:Raoul Bott, Loring W. Tu (1982).
933:{\displaystyle 0_{E}\subseteq U}
302:plays the role of the curve and
216:to the tubular neighbourhood of
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763:{\displaystyle J\circ 0_{E}=i}
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486:Often one calls the open set
959:{\displaystyle S\subseteq V}
891:{\displaystyle V\subseteq M}
865:{\displaystyle U\subseteq E}
614:{\displaystyle S\subseteq M}
341:{\displaystyle i:N_{0}\to S}
197:in blue. The transformation
7:
1140:. Berlin: Springer-Verlag.
1136:Waldyr Muniz Oliva (2002).
1119:. Berlin: Springer-Verlag.
1098:. Berlin: Springer-Verlag.
1065:
696:together with a smooth map
689:{\displaystyle \pi :E\to S}
355:correspondence between the
10:
1191:
582:each disc lies in a plane
1115:Morris W. Hirsch (1976).
1080: – proof in topology
244:around it resembling the
212:in the figure above, and
721:{\displaystyle J:E\to M}
170:, with the zero section
40:This article includes a
568:of all discs such that
517:{\displaystyle T=j(N),}
264:tubular neighborhood.
152:{\displaystyle T=j(N).}
69:more precise citations.
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1117:Differential Topology
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836:{\displaystyle 0_{E}}
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378:{\displaystyle N_{0}}
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190:{\displaystyle N_{0}}
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18:Tubular neighbourhood
1060:stable normal bundle
1049:spherical fibrations
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476:{\displaystyle j(N)}
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431:{\displaystyle j(N)}
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389:and the submanifold
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351:which establishes a
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230:tubular neighborhood
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1138:Geometric Mechanics
1170:Geometric topology
1033:{\displaystyle M.}
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846:there exists some
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42:list of references
790:is the embedding
783:{\displaystyle i}
654:{\displaystyle M}
634:{\displaystyle S}
591:Formal definition
95:
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16:(Redirected from
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1175:Smooth manifolds
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61:Please help
53:
560:curve is a
554:normal tube
548:Normal tube
273:submanifold
234:submanifold
226:mathematics
75:August 2014
67:introducing
1159:Categories
1088:References
1078:Tube lemma
966:such that
728:such that
409:such that
282:, and let
1165:Manifolds
993:→
951:⊆
925:⊆
883:⊆
872:and some
857:⊆
801:↪
742:∘
713:→
681:→
672:π
606:⊆
353:bijective
333:→
1066:See also
562:manifold
544:exists.
536:mapping
450:between
277:manifold
242:open set
298:. Here
286:be the
63:improve
1144:
1123:
1102:
770:where
584:normal
558:smooth
253:smooth
240:is an
121:, and
1008:is a
898:with
661:is a
579:; and
577:curve
566:union
556:to a
446:is a
275:of a
271:be a
257:curve
236:of a
232:of a
201:maps
48:, or
1142:ISBN
1121:ISBN
1100:ISBN
1051:for
940:and
816:and
595:Let
454:and
442:and
228:, a
641:in
540:to
393:of
385:of
294:in
290:of
248:.
224:In
1161::
1055:.
552:A
52:,
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1150:.
1129:.
1108:.
1028:.
1025:M
1012:.
996:V
990:U
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982:U
978:|
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916:[
911:E
907:0
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880:V
860:E
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778:i
758:i
755:=
750:E
746:0
739:J
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707::
704:J
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678:E
675::
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538:N
534:j
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526:j
512:,
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506:N
503:(
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497:=
494:T
471:)
468:N
465:(
462:j
452:N
444:j
440:M
426:)
423:N
420:(
417:j
407:M
403:N
399:j
395:M
391:S
387:N
371:0
367:N
336:S
328:0
324:N
320::
317:i
304:M
300:S
296:M
292:S
284:N
280:M
269:S
220:.
218:S
214:N
210:S
206:0
203:N
199:j
183:0
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141:N
138:(
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132:=
129:T
119:M
115:S
111:T
88:)
82:(
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73:(
59:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
