180:) onto which it retracts, and whose thick part is compact (note that all manifolds have a convex core, but in general it is not compact). The simplest case is when the manifold does not have "cusps" (i.e. the fundamental group does not contain parabolic elements), in which case the manifold is geometrically finite if and only if it is the quotient of a closed, convex subset of hyperbolic space by a group acting cocompactly on this subset.
369:(i.e. vertices which lie on the sphere at infinity). In this setting the gluing construction does not always yield a complete manifold. Completeness is detected by a system of equations involving the dihedral angles around the edges adjacent to an ideal vertex, which are commonly called Thurston's gluing equations. In case the gluing is complete the ideal vertices become
95:). After the proof of the Geometrisation conjecture, understanding the topological properties of hyperbolic 3-manifolds is thus a major goal of 3-dimensional topology. Recent breakthroughs of Kahn–Markovic, Wise, Agol and others have answered most long-standing open questions on the topic but there are still many less prominent ones which have not been solved.
221:. Suppose that there is a side-pairing between the 2-dimensional faces of these polyhedra (i.e. each such face is paired with another, distinct, one so that they are isometric to each other as 2-dimensional hyperbolic polygons), and consider the space obtained by gluing the paired faces together (formally this is obtained as a
488:
properties of hyperbolic 3-manifolds are the objects of a series of conjectures by
Waldhausen and Thurston, which were recently all proven by Ian Agol following work of Jeremy Kahn, Vlad Markovic, Frédéric Haglund, Dani Wise and others. The first part of the conjectures were logically related to the
423:
In contrast to the explicit constructions above it is possible to deduce the existence of a complete hyperbolic structure on a 3-manifold purely from topological information. This is a consequence of the
Geometrisation conjecture and can be stated as follows (a statement sometimes referred to as the
384:
It is also possible to construct a finite-volume, complete hyperbolic manifold when the gluing is not complete. In this case the completion of the metric space obtained is a manifold with a torus boundary and under some (not generic) conditions it is possible to glue a hyperbolic solid torus on each
98:
In dimension 2 almost all closed surfaces are hyperbolic (all but the sphere, projective plane, torus and Klein bottle). In dimension 3 this is far from true: there are many ways to construct infinitely many non-hyperbolic closed manifolds. On the other hand, the heuristic statement that "a generic
487:
The topological properties of 3-manifolds are sufficiently intricate that in many cases it is interesting to know that a property holds virtually for a class of manifolds, that is for any manifold in the class there exists a finite covering space of the manifold with the property. The virtual
681:, so that the isolated points are volumes of compact manifolds, the manifolds with exactly one cusp are limits of compact manifolds, and so on. Together with results of Jørgensen the theorem also proves that any convergent sequence must be obtained by Dehn surgeries on the limit manifold.
347:, which is a discrete subgroup of isometries of hyperbolic space. Taking a torsion-free finite-index subgroup one obtains a hyperbolic manifold (which can be recovered by the previous construction, gluing copies of the original Coxeter polytope in a manner prescribed by an appropriate
90:
Hyperbolic geometry is the most rich and least understood of the eight geometries in dimension 3 (for example, for all other geometries it is not hard to give an explicit enumeration of the finite-volume manifolds with this geometry, while this is far from being the case for
111:
theorem), and since all 3-manifolds are obtained as surgeries on a link in the 3-sphere this gives a more precise sense to the informal statement. Another sense in which "almost all" manifolds are hyperbolic in dimension 3 is that of random models. For example, random
478:
Another consequence of the
Geometrisation conjecture is that any closed 3-manifold which admits a Riemannian metric with negative sectional curvatures admits in fact a Riemannian metric with constant sectional curvature -1. This is not true in higher dimensions.
414:
gives rise to particularly interesting hyperbolic manifolds. On the other hand, they are in some sense "rare" among hyperbolic 3-manifolds (for example hyperbolic Dehn surgery on a fixed manifold results in a non-arithmetic manifold for almost all parameters).
385:
boundary component so that the resulting space has a complete hyperbolic metric. Topologically, the manifold is obtained by hyperbolic Dehn surgery on the complete hyperbolic manifold which would result from a complete gluing.
225:). It carries a hyperbolic metric which is well-defined outside of the image of the 1-skeletons of the polyhedra. This metric extends to a hyperbolic metric on the whole space if the two following conditions are satisfied:
511:) Any hyperbolic 3-manifold of finite volume is virtually Haken; that is, it contains an embedded closed surface such that the embedding induces an injective map between fundamental groups.
341:
123:, which states that the hyperbolic structure of a hyperbolic 3-manifold of finite volume is uniquely determined by its homotopy type. In particular, geometric invariants such as the
575:
The hyperbolic volume can be used to order the space of all hyperbolic manifold. The set of manifolds corresponding to a given volume is at most finite, and the set of volumes is
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679:
217:
The oldest construction of hyperbolic manifolds, which dates back at least to
Poincaré, goes as follows: start with a finite collection of 3-dimensional hyperbolic finite
461:
172:
The thick-thin decomposition is valid for all hyperbolic 3-manifolds, though in general the thin part is not as described above. A hyperbolic 3-manifold is said to be
1992:
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It is not known whether all hyperbolic 3-manifolds of finite volume can be constructed in this way. In practice however this is how computational software (such as
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is hyperbolic. Moreover, almost all Dehn surgeries on a hyperbolic knot yield a hyperbolic manifold. A similar result is true of links (Thurston's
883:
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1010:
Petronio, Carlo; Porti, Joan (2000). "Negatively oriented ideal triangulations and a proof of
Thurston's hyperbolic Dehn filling theorem".
365:
In the previous construction the manifolds obtained are always compact. To obtain manifolds with cusps one has to use polytopes which have
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463:-injectively immersed torus is homotopic to a boundary component) then its interior carries a complete hyperbolic metric of finite volume.
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500:) The fundamental group of any hyperbolic manifold of finite volume contains a (non-free) surface group (the fundamental group of a
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405:
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This is the larger class of hyperbolic 3-manifolds for which there is a satisfying structure theory. It rests on two theorems:
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533:
Another conjecture (also proven by Agol) which implies 1-3 above but a priori has no relation to 4 is the following :
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A variation on this construction is by using hyperbolic
Coxeter polytopes (polytopes whose dihedral angles are of the form
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471:: such manifolds are always irreducible, and they carry a complete hyperbolic metric if and only if the monodromy is a
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which provides a classification of hyperbolic structure on the interior of a compact manifold by its "end invariants".
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Any hyperbolic 3-manifold of finite volume has a finite cover whose fundamental group surjects onto a non-abelian
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3-manifold tends to be hyperbolic" is verified in many contexts. For example, any knot which is not either a
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in the manifold. An example of a noncompact, finite volume hyperbolic manifold obtained in this way is the
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of genus at least 2 are almost surely hyperbolic (when the complexity of the gluing map goes to infinity).
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541:) Any hyperbolic 3-manifold of finite volume has a finite cover which is a surface bundle over the circle.
79:
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28:
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which states that such a manifold is homeomorphic to the interior of a compact manifold with boundary;
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424:"hyperbolisation theorem", which was proven by Thurston in the special case of Haken manifolds):
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563:. For the manifolds obtained as quotients this amounts to them being convergent in the pointed
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144:. It states that a hyperbolic 3-manifold of finite volume has a decomposition into two parts:
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40:
119:
The relevance of the hyperbolic geometry of a 3-manifold to its topology also comes from the
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surface groups of given genus can converge to a doubly degenerate surface group, as in the
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Gromov, Mikhail; Thurston, William (1987). "Pinching constants for hyperbolic manifolds".
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870:
609:. More precisely, Thurston's hyperbolic Dehn surgery theorem implies that a manifold with
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Any hyperbolic 3-manifold of finite volume has a finite cover with a nonzero first
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1045:. Graduate Texts in Mathematics. Vol. 149 (2nd ed.). Berlin, New York:
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230:
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In this case one important tool to understand the geometry of a manifold is the
20:
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proved by
Perelman. The study of Kleinian groups is also an important topic in
55:: in this case the manifold can be realised as a quotient of the 3-dimensional
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1982:
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890:. Lecture Notes in Mathematics. Vol. 842. Springer. pp. 40–53.
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36:
24:
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Hyperbolic 3-manifolds of finite volume have a particular importance in
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928:
847:
Callahan, Patrick J.; Hildebrand, Martin V.; Weeks, Jeffrey R. (1999).
389:
152:
part, where the injectivity radius is larger than an absolute constant;
1099:"Three-dimensional manifolds, Kleinian groups and hyperbolic geometry"
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1024:
51:
equal to −1. It is generally required that this metric be also
377:
which is constructed by gluing faces of a regular ideal hyperbolic
958:
1152:
836:
Aschenbrenner, Matthias; Friedl, Stefan; Wilton, Henry (2015).
842:. EMS Series of Lectures in Mathematics. European Math. Soc.
884:"Hyperbolic manifolds according to Thurston and Jørgensen"
808:
712:
229:
for each (non-ideal) vertex in the gluing the sum of the
16:
Manifold of dimension 3 equipped with a hyperbolic metric
294:
which is obtained by gluing opposite faces of a regular
784:
208:
Construction of hyperbolic 3-manifolds of finite volume
655:
635:
615:
588:
442:
307:
269:
239:
944:
Maher, Joseph (2010). "Random
Heegaard splittings".
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The construction of arithmetic
Kleinian groups from
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1635:
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Gluing ideal tetrahedra and hyperbolic Dehn surgery
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Manifolds with finitely generated fundamental group
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621:
601:
455:
335:
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248:
159:part, which is a disjoint union of solid tori and
127:can be used to define new topological invariants.
629:cusps is a limit of a sequence of manifolds with
263:of the polyhedra to which it belongs is equal to
233:of the polyhedra to which it belongs is equal to
428:If a compact 3-manifold with toric boundary is
426:
796:
290:A notable example of this construction is the
1164:
1103:Bulletin of the American Mathematical Society
772:
8:
1086:The geometry and topology of three-manifolds
555:A sequence of Kleinian groups is said to be
343:). Such a polytope gives rise to a Kleinian
849:"A census of cusped hyperbolic 3-manifolds"
259:for each edge in the gluing the sum of the
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176:if it contains a convex submanifold (its
724:
336:{\displaystyle \pi /m,m\in \mathbb {N} }
985:"Volumes of hyperbolic three-manifolds"
809:Aschenbrenner, Friedl & Wilton 2015
713:Aschenbrenner, Friedl & Wilton 2015
705:
546:The space of all hyperbolic 3-manifolds
213:Hyperbolic polyhedra, reflection groups
820:
983:Neumann, Walter; Zagier, Don (1985).
785:Callahan, Hildebrand & Weeks 1999
736:
7:
1142:3-dimensional geometry and topology
1043:Foundations of hyperbolic manifolds
492:. In order of strength they are:
14:
602:{\displaystyle \omega ^{\omega }}
888:Séminaire N. Bourbaki, 1979-1980
525:(such groups are usually called
406:Arithmetic hyperbolic 3-manifold
1116:10.1090/S0273-0979-1982-15003-0
571:Jørgensen–Thurston theory
467:A particular case is that of a
396:) stores hyperbolic manifolds.
39:of dimension 3 equipped with a
1211:Differentiable/Smooth manifold
469:surface bundle over the circle
168:Geometrically finite manifolds
1:
1144:. Princeton University Press.
866:10.1090/s0025-5718-99-01036-4
763:, Theorems 10.1.2 and 10.1.3.
1041:Ratcliffe, John G. (2006) .
1002:10.1016/0040-9383(85)90004-7
674:{\displaystyle 0\leq l<m}
539:virtually fibered conjecture
1917:Classification of manifolds
498:surface subgroup conjecture
419:The hyperbolisation theorem
74:as follows from Thurston's
2067:
1140:Thurston, William (1997).
1089:. Princeton lecture notes.
797:Gromov & Thurston 1987
509:Virtually Haken conjecture
490:virtually Haken conjecture
403:
358:
136:Manifolds of finite volume
1993:over commutative algebras
1055:10.1007/978-0-387-47322-2
773:Petronio & Porti 2000
201:ending lamination theorem
76:geometrisation conjecture
1709:Riemann curvature tensor
909:Inventiones Mathematicae
882:Gromov, Michael (1981).
557:geometrically convergent
456:{\displaystyle \pi _{1}}
400:Arithmetic constructions
155:and its complement, the
142:thick-thin decomposition
565:Gromov-Hausdorff metric
559:if it converges in the
434:algebraically atoroidal
361:Hyperbolic Dehn surgery
121:Mostow rigidity theorem
109:hyperbolic Dehn surgery
1501:Manifold with boundary
1216:Differential structure
675:
643:
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603:
465:
457:
337:
280:
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86:Importance in topology
80:geometric group theory
72:3-dimensional topology
968:10.1112/jtopol/jtq031
685:Quasi-Fuchsian groups
676:
644:
624:
604:
551:Geometric convergence
458:
338:
281:
279:{\displaystyle 2\pi }
251:
249:{\displaystyle 4\pi }
33:hyperbolic 3-manifold
29:differential geometry
2051:Riemannian manifolds
1648:Covariant derivative
1199:Topological manifold
695:double limit theorem
653:
633:
613:
586:
440:
436:(meaning that every
349:Schreier coset graph
305:
267:
237:
174:geometrically finite
93:hyperbolic manifolds
49:sectional curvatures
23:, more precisely in
2041:Hyperbolic geometry
1682:Exterior derivative
1284:Atiyah–Singer index
1233:Riemannian manifold
1034:1999math......1045P
946:Journal of Topology
921:1987InMat..89....1G
412:quaternion algebras
292:Seifert–Weber space
114:Heegaard splittings
1988:Secondary calculus
1942:Singularity theory
1897:Parallel transport
1665:De Rham cohomology
1304:Generalized Stokes
929:10.1007/bf01404671
839:3-manifolds groups
671:
639:
619:
599:
483:Virtual properties
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375:Gieseking manifold
333:
276:
246:
47:which has all its
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2022:
1905:
1904:
1670:Differential form
1324:Whitney embedding
1258:Differential form
1095:Thurston, William
1081:Thurston, William
1064:978-0-387-33197-3
751:, Theorem 12.7.2.
642:{\displaystyle l}
622:{\displaystyle m}
561:Chabauty topology
473:pseudo-Anosov map
63:of isometries (a
45:Riemannian metric
41:hyperbolic metric
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2015:Stratified space
1973:Fréchet manifold
1687:Interior product
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898:. Archived from
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859:(225): 321–332.
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727:, Corollary 2.5.
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57:hyperbolic space
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1958:Banach manifold
1951:Generalizations
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1697:Ricci curvature
1653:Cotangent space
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1364:Exponential map
1328:
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1139:
1093:
1079:
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1047:Springer-Verlag
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1009:
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952:(4): 997–1025.
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1192:Basic concepts
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1109:(3): 357–381.
1105:. New Series.
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1007:
995:(3): 307–332.
980:
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904:
902:on 2016-01-10.
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761:Ratcliffe 2006
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749:Ratcliffe 2006
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691:quasi-fuchsian
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649:cusps for any
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502:closed surface
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404:Main article:
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367:ideal vertices
359:Main article:
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223:quotient space
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101:satellite knot
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65:Kleinian group
61:discrete group
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2013:
2011:
2010:Supermanifold
2008:
2006:
2003:
2001:
1998:
1994:
1991:
1990:
1989:
1986:
1984:
1981:
1979:
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1966:
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1961:
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1930:
1928:
1925:
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1912:
1908:
1898:
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1765:
1762:
1760:
1757:
1755:
1752:
1750:
1747:
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1742:
1738:
1732:
1731:Wedge product
1729:
1727:
1724:
1720:
1717:
1716:
1715:
1712:
1710:
1707:
1703:
1700:
1699:
1698:
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1690:
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1685:
1683:
1680:
1676:
1675:Vector-valued
1673:
1672:
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1668:
1666:
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1659:
1656:
1655:
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1634:
1628:
1625:
1623:
1620:
1618:
1615:
1611:
1608:
1607:
1606:
1605:Tangent space
1603:
1601:
1598:
1596:
1593:
1591:
1588:
1587:
1585:
1581:
1578:
1576:
1572:
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1397:
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1379:
1377:
1374:
1370:
1369:in Lie theory
1367:
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1362:
1360:
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1300:
1297:
1295:
1292:
1290:
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1281:
1279:
1276:
1272:Main results
1270:
1264:
1261:
1259:
1256:
1254:
1253:Tangent space
1251:
1249:
1246:
1244:
1241:
1239:
1236:
1234:
1231:
1229:
1226:
1222:
1219:
1217:
1214:
1213:
1212:
1209:
1205:
1202:
1201:
1200:
1197:
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1190:
1185:
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1169:
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1155:
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1126:
1122:
1117:
1112:
1108:
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1100:
1096:
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1074:
1070:
1066:
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1056:
1052:
1048:
1044:
1039:
1035:
1031:
1026:
1021:
1017:
1013:
1008:
1003:
998:
994:
990:
986:
981:
977:
973:
969:
965:
960:
955:
951:
947:
942:
938:
934:
930:
926:
922:
918:
914:
910:
905:
901:
897:
893:
889:
885:
880:
876:
872:
867:
862:
858:
854:
850:
845:
841:
840:
834:
833:
829:
822:
817:
814:
810:
805:
802:
798:
793:
790:
786:
781:
778:
774:
769:
766:
762:
757:
754:
750:
745:
742:
738:
733:
730:
726:
725:Thurston 1982
721:
718:
714:
709:
706:
700:
698:
696:
692:
689:Sequences of
684:
682:
668:
665:
662:
659:
656:
636:
616:
594:
590:
582:
578:
570:
568:
566:
562:
558:
550:
545:
540:
536:
535:
534:
528:
524:
520:
517:
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510:
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495:
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491:
482:
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448:
444:
435:
431:
425:
418:
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413:
407:
399:
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391:
386:
382:
380:
376:
372:
368:
362:
354:
352:
350:
346:
325:
322:
319:
316:
312:
308:
299:
297:
293:
273:
270:
262:
258:
243:
240:
232:
228:
227:
226:
224:
220:
212:
207:
202:
198:
195:
191:
190:
189:
183:
181:
179:
175:
167:
162:
158:
154:
151:
147:
146:
145:
143:
135:
130:
128:
126:
122:
117:
115:
110:
106:
102:
96:
94:
85:
83:
81:
77:
73:
68:
66:
62:
58:
54:
50:
46:
42:
38:
34:
30:
26:
22:
1937:Moving frame
1932:Morse theory
1922:Gauge theory
1714:Tensor field
1643:Closed/Exact
1622:Vector field
1590:Distribution
1531:Hypercomplex
1526:Quaternionic
1263:Vector field
1221:Smooth atlas
1141:
1106:
1102:
1084:
1042:
1025:math/9901045
1015:
1011:
992:
988:
949:
945:
912:
908:
900:the original
887:
856:
852:
838:
816:
804:
792:
780:
768:
756:
744:
732:
720:
715:, Chapter 7.
708:
688:
577:well-ordered
574:
556:
554:
532:
526:
516:Betti number
486:
477:
466:
427:
422:
409:
387:
383:
364:
300:
296:dodecahedron
289:
231:solid angles
216:
187:
177:
171:
156:
149:
139:
118:
97:
89:
69:
43:, that is a
32:
18:
2036:3-manifolds
1882:Levi-Civita
1872:Generalized
1844:Connections
1794:Lie algebra
1726:Volume form
1627:Vector flow
1600:Pushforward
1595:Lie bracket
1494:Lie algebra
1459:G-structure
1248:Pushforward
1228:Submanifold
821:Gromov 1981
430:irreducible
379:tetrahedron
178:convex core
21:mathematics
2030:Categories
2005:Stratifold
1963:Diffeology
1759:Associated
1560:Symplectic
1545:Riemannian
1474:Hyperbolic
1401:Submersion
1309:Hopf–Rinow
1243:Submersion
1238:Smooth map
1012:Expo. Math
853:Math. Comp
830:References
737:Maher 2010
581:order type
523:free group
381:together.
105:torus knot
1887:Principal
1862:Ehresmann
1819:Subbundle
1809:Principal
1784:Fibration
1764:Cotangent
1636:Covectors
1489:Lie group
1469:Hermitian
1412:manifolds
1381:Immersion
1376:Foliation
1314:Noether's
1299:Frobenius
1294:De Rham's
1289:Darboux's
1180:Manifolds
1125:0002-9904
959:0809.4881
937:119850633
660:≤
595:ω
591:ω
445:π
326:∈
309:π
274:π
244:π
219:polytopes
131:Structure
1983:Orbifold
1978:K-theory
1968:Diffiety
1692:Pullback
1506:Oriented
1484:Kenmotsu
1464:Hadamard
1410:Types of
1359:Geodesic
1184:Glossary
1097:(1982).
1083:(1980).
1018:: 1–35.
989:Topology
976:14179122
915:: 1–12.
537:5. (the
53:complete
37:manifold
25:topology
1927:History
1910:Related
1824:Tangent
1802:)
1782:)
1749:Adjoint
1741:Bundles
1719:density
1617:Torsion
1583:Vectors
1575:Tensors
1558:)
1543:)
1539:,
1537:Pseudo−
1516:Poisson
1449:Finsler
1444:Fibered
1439:Contact
1437:)
1429:Complex
1427:)
1396:Section
1133:0648524
1073:2249478
1030:Bibcode
917:Bibcode
896:0636516
875:1620219
579:and of
390:SnapPea
1892:Vector
1877:Koszul
1857:Cartan
1852:Affine
1834:Vector
1829:Tensor
1814:Spinor
1804:Normal
1800:Stable
1754:Affine
1658:bundle
1610:bundle
1556:Almost
1479:Kähler
1435:Almost
1425:Almost
1419:Closed
1319:Sard's
1275:(list)
1131:
1123:
1071:
1061:
974:
935:
894:
873:
394:Regina
125:volume
2000:Sheaf
1774:Fiber
1550:Rizza
1521:Prime
1352:Local
1342:Curve
1204:Atlas
1020:arXiv
972:S2CID
954:arXiv
933:S2CID
701:Notes
527:large
507:(the
496:(the
371:cusps
161:cusps
150:thick
103:or a
59:by a
35:is a
1867:Form
1769:Dual
1702:flow
1565:Tame
1541:Sub−
1454:Flat
1334:Maps
1121:ISSN
1059:ISBN
666:<
432:and
199:The
192:The
157:thin
148:the
31:, a
27:and
1789:Jet
1111:doi
1051:doi
997:doi
964:doi
925:doi
861:doi
392:or
351:).
67:).
19:In
2032::
1780:Co
1129:MR
1127:.
1119:.
1101:.
1069:MR
1067:.
1057:.
1049:.
1028:.
1016:18
1014:.
993:24
991:.
987:.
970:.
962:.
948:.
931:.
923:.
913:89
911:.
892:MR
886:.
871:MR
869:.
857:68
855:.
851:.
697:.
567:.
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475:.
298:.
82:.
1798:(
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1433:(
1423:(
1186:)
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1158:v
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1113::
1107:6
1075:.
1053::
1036:.
1032::
1022::
1005:.
999::
978:.
966::
956::
950:3
939:.
927::
919::
877:.
863::
823:.
811:.
799:.
787:.
775:.
739:.
669:m
663:l
657:0
637:l
617:m
518:.
449:1
330:N
323:m
320:,
317:m
313:/
286:.
271:2
256:;
241:4
163:.
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