1731:
588:
597:, then gluing the faces together in pairs using affine-linear maps. Label the vertices 0, 1, 2, 3. Glue the face with vertices 0, 1, 2 to the face with vertices 3, 1, 0 in that order. Glue the face 0, 2, 3 to the face 3, 2, 1 in that order. In the hyperbolic structure of the Gieseking manifold, this ideal tetrahedron is the canonical polyhedral decomposition of
278:
338:
75:
583:{\displaystyle K=\operatorname {Cl} _{2}\left({\frac {\pi }{2}}\right)=\sum _{n=0}^{\infty }{\frac {1}{(4n+1)^{2}}}-\sum _{n=0}^{\infty }{\frac {1}{(4n+3)^{2}}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)^{2}}}=0.91596559\dots }
273:{\displaystyle V=\operatorname {Cl} _{2}\left({\frac {\pi }{3}}\right)={\frac {3{\sqrt {3}}}{4}}\left(\sum _{n=0}^{\infty }{\frac {1}{(3n+1)^{2}}}-\sum _{n=0}^{\infty }{\frac {1}{(3n+2)^{2}}}\right)=1.0149416\dots }
322:
59:
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629:. The triangulation has one tetrahedron, two faces, one edge and no vertices, so all the edges of the original tetrahedron are glued together.
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and this gives another way to see that the
Gieseking manifold is double covered by the complement of the figure-eight knot.
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and has the smallest volume among non-compact hyperbolic manifolds, having volume approximately
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which also manifests as a volume can also be expressed in terms of the
Clausen function,
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The
Gieseking manifold can be constructed by removing the vertices from a
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The
Gieseking manifold is a fiber bundle over the circle with fiber the
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767:(1987), "The noncompact hyperbolic 3-manifold of minimal volume",
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and Robert C. Penner. Moreover, the angle made by the faces is
1270:
810:"Euclidean decompositions of noncompact hyperbolic manifolds"
317:{\displaystyle \operatorname {Cl} _{2}\left(\varphi \right)}
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61:. It was discovered by Hugo Gieseking (
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770:Proceedings of the American Mathematical Society
1766:
860:
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654:of the Gieseking manifold is the integers.
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646:. The underlying compact manifold has a
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13:
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159:
54:{\displaystyle V\approx 1.0149416}
14:
1818:
713:{\displaystyle (x,y)\to (x+y,x).}
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815:Journal of Differential Geometry
907:Differentiable/Smooth manifold
734:List of mathematical constants
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661:torus and monodromy given by
632:The Gieseking manifold has a
1745:. You can help Knowledge by
808:; Penner, Robert C. (1988).
16:Cusped hyperbolic 3-manifold
7:
1613:Classification of manifolds
727:
10:
1823:
1724:
720:The square of this map is
1689:over commutative algebras
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1605:
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1331:
1278:
1269:
1105:
1028:
967:
887:
31:of finite volume. It is
1405:Riemann curvature tensor
746:Gieseking, Hugo (1912),
650:boundary, and the first
65:). The volume is called
69:and has a closed-form,
1741:-related article is a
1197:Manifold with boundary
912:Differential structure
829:10.4310/jdg/1214441650
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623:
622:{\displaystyle \pi /3}
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1807:Metric geometry stubs
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29:hyperbolic 3-manifold
1344:Covariant derivative
895:Topological manifold
752:, Thesis, Muenster,
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605:
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287:
76:
39:
1802:Hyperbolic geometry
1739:hyperbolic geometry
1378:Exterior derivative
980:Atiyah–Singer index
929:Riemannian manifold
806:Epstein, David B.A.
599:David B. A. Epstein
1797:Geometric topology
1684:Secondary calculus
1638:Singularity theory
1593:Parallel transport
1361:De Rham cohomology
1000:Generalized Stokes
710:
619:
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330:Catalan's constant
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270:
67:Gieseking constant
51:
25:Gieseking manifold
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1366:Differential form
1020:Whitney embedding
954:Differential form
641:figure-eight knot
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1647:Generalizations
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1393:Ricci curvature
1349:Cotangent space
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1060:Exponential map
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783:10.2307/2046691
765:Adams, Colin C.
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1207:Parallelizable
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1082:Integral curve
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1043:Diffeomorphism
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888:Basic concepts
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777:(4): 601–606,
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652:homology group
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33:non-orientable
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1706:Supermanifold
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1427:Wedge product
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1371:Vector-valued
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1301:Tangent space
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1080:
1078:
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1070:
1066:
1065:in Lie theory
1063:
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972:
968:Main results
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960:
957:
955:
952:
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949:Tangent space
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328:. Similarly,
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1747:expanding it
1736:
1721:
1633:Moving frame
1628:Morse theory
1618:Gauge theory
1410:Tensor field
1339:Closed/Exact
1318:Vector field
1286:Distribution
1227:Hypercomplex
1222:Quaternionic
959:Vector field
917:Smooth atlas
822:(1): 67–80.
819:
813:
774:
768:
748:
656:
648:Klein bottle
637:homeomorphic
634:double cover
631:
592:
282:
27:is a cusped
24:
18:
1792:3-manifolds
1578:Levi-Civita
1568:Generalized
1540:Connections
1490:Lie algebra
1422:Volume form
1323:Vector flow
1296:Pushforward
1291:Lie bracket
1190:Lie algebra
1155:G-structure
944:Pushforward
924:Submanifold
595:tetrahedron
21:mathematics
1786:Categories
1701:Stratifold
1659:Diffeology
1455:Associated
1256:Symplectic
1241:Riemannian
1170:Hyperbolic
1097:Submersion
1005:Hopf–Rinow
939:Submersion
934:Smooth map
758:43.0202.03
740:References
644:complement
575:0.91596559
1583:Principal
1558:Ehresmann
1515:Subbundle
1505:Principal
1480:Fibration
1460:Cotangent
1332:Covectors
1185:Lie group
1165:Hermitian
1108:manifolds
1077:Immersion
1072:Foliation
1010:Noether's
995:Frobenius
990:De Rham's
985:Darboux's
876:Manifolds
791:0002-9939
684:→
609:π
578:…
526:−
515:∞
500:∑
457:∞
442:∑
438:−
399:∞
384:∑
368:π
359:
308:φ
301:
268:…
265:1.0149416
218:∞
203:∑
199:−
160:∞
145:∑
105:π
96:
49:1.0149416
46:≈
1679:Orbifold
1674:K-theory
1664:Diffiety
1388:Pullback
1202:Oriented
1180:Kenmotsu
1160:Hadamard
1106:Types of
1055:Geodesic
880:Glossary
728:See also
1623:History
1606:Related
1520:Tangent
1498:)
1478:)
1445:Adjoint
1437:Bundles
1415:density
1313:Torsion
1279:Vectors
1271:Tensors
1254:)
1239:)
1235:,
1233:Pseudo−
1212:Poisson
1145:Finsler
1140:Fibered
1135:Contact
1133:)
1125:Complex
1123:)
1092:Section
838:0918457
799:0894423
639:to the
324:is the
1588:Vector
1573:Koszul
1553:Cartan
1548:Affine
1530:Vector
1525:Tensor
1510:Spinor
1500:Normal
1496:Stable
1450:Affine
1354:bundle
1306:bundle
1252:Almost
1175:Kähler
1131:Almost
1121:Almost
1115:Closed
1015:Sard's
971:(list)
836:
797:
789:
756:
283:where
23:, the
1737:This
1696:Sheaf
1470:Fiber
1246:Rizza
1217:Prime
1048:Local
1038:Curve
900:Atlas
1743:stub
1563:Form
1465:Dual
1398:flow
1261:Tame
1237:Sub−
1150:Flat
1030:Maps
787:ISSN
63:1912
1485:Jet
824:doi
779:doi
775:100
754:JFM
19:In
1788::
1476:Co
834:MR
832:.
820:27
818:.
812:.
795:MR
793:,
785:,
773:,
350:Cl
292:Cl
87:Cl
1774:e
1767:t
1760:v
1749:.
1494:(
1474:(
1250:(
1231:(
1129:(
1119:(
882:)
878:(
868:e
861:t
854:v
840:.
826::
781::
708:.
705:)
702:x
699:,
696:y
693:+
690:x
687:(
681:)
678:y
675:,
672:x
669:(
617:3
613:/
572:=
564:2
560:)
556:1
553:+
550:n
547:2
544:(
537:n
533:)
529:1
523:(
510:0
507:=
504:n
496:=
488:2
484:)
480:3
477:+
474:n
471:4
468:(
464:1
452:0
449:=
446:n
430:2
426:)
422:1
419:+
416:n
413:4
410:(
406:1
394:0
391:=
388:n
380:=
376:)
371:2
363:(
354:2
346:=
343:K
311:)
305:(
296:2
262:=
258:)
249:2
245:)
241:2
238:+
235:n
232:3
229:(
225:1
213:0
210:=
207:n
191:2
187:)
183:1
180:+
177:n
174:3
171:(
167:1
155:0
152:=
149:n
140:(
134:4
128:3
123:3
117:=
113:)
108:3
100:(
91:2
83:=
80:V
43:V
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