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The cotangent bundle of a manifold is locally modeled on a space similar to the first example. It can be shown that we can glue these affine symplectic forms hence this bundle forms a symplectic manifold. A less trivial example of a
Lagrangian submanifold is the zero section of the cotangent bundle
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945:{\displaystyle {\mathcal {L}}_{V_{H}}(\omega )=0\;\Leftrightarrow \;\mathrm {d} (\iota _{V_{H}}\omega )+\iota _{V_{H}}\mathrm {d} \omega =\mathrm {d} (\mathrm {d} \,H)+\mathrm {d} \omega (V_{H})=\mathrm {d} \omega (V_{H})=0}
1924:
5741:
3816:{\displaystyle \omega (X,Y)=\omega (f(x)\partial _{x},g(x)\partial _{x})={\frac {1}{2}}f(x)g(x)(\mathrm {d} x(\partial _{x})\mathrm {d} y(\partial _{x})-\mathrm {d} y(\partial _{x})\mathrm {d} x(\partial _{x}))}
6340:
3047:
2912:
2277:. Upper and lower indexes transform contra and covariantly under a change of coordinate frames. The phrase "fibrewise coordinates with respect to the cotangent vectors" is meant to convey that the momenta
5239:{\displaystyle {\frac {\partial }{\partial u_{i}}}={\frac {\partial q_{k}}{\partial u_{i}}}{\frac {\partial }{\partial q_{k}}}+{\frac {\partial p_{k}}{\partial u_{i}}}{\frac {\partial }{\partial p_{k}}}}
5879:
5587:{\displaystyle \omega \left({\frac {\partial }{\partial q_{k}}},{\frac {\partial }{\partial p_{k}}}\right)=-\omega \left({\frac {\partial }{\partial p_{k}}},{\frac {\partial }{\partial q_{k}}}\right)=1}
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1798:{\displaystyle \omega (v_{i},v_{j})={\begin{cases}1&j-i=n{\text{ with }}1\leqslant i\leqslant n\\-1&i-j=n{\text{ with }}1\leqslant j\leqslant n\\0&{\text{otherwise}}\end{cases}}}
5604:
on a symplectic manifold take on the canonical form, this example suggests that
Lagrangian submanifolds are relatively unconstrained. The classification of symplectic manifolds is done via
5050:{\displaystyle =\sum _{k}{\frac {\partial q_{k}}{\partial u_{i}}}{\frac {\partial p_{k}}{\partial u_{j}}}-{\frac {\partial p_{k}}{\partial u_{i}}}{\frac {\partial q_{k}}{\partial u_{j}}}=0}
3865:
4189:
3055:
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for maps between
Lagrangian submanifolds. In physics, the action describes the time evolution of a physical system; here, it can be taken as the description of the dynamics of branes.
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predicts that the existence of a special
Lagrangian submanifolds on Calabi–Yau manifolds in Hamiltonian isotopy classes of Lagrangians is equivalent to stability with respect to a
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2263:
2211:
2139:
121:
of classical mechanics, which provides one of the major motivations for the field, the set of all possible configurations of a system is modeled as a manifold, and this manifold's
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of the ambient manifold's tangent space. Similarly, if each tangent subspace to a submanifold is co-isotropic (the dual of an isotropic subspace), the submanifold is called
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2338:. The soldering is an expression of the idea that velocity and momentum are colinear, in that both move in the same direction, and differ by a scale factor.
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span the tangent space at each point the argument is applied at, we see that the requirement for the vanishing Lie derivative along flows of
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6689:{\displaystyle \tau \circ i_{1}=i_{2}\circ \sigma ,\ \nu \circ \pi _{1}=\pi _{2}\circ \tau ,\ \tau ^{*}\omega _{2}=\omega _{1}\,,}
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4307:, giving us the zero section. This example can be repeated for any manifold defined by the vanishing locus of smooth functions
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of classical mechanics. The point of distinguishing upper and lower indexes is driven by the case of the manifold having a
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4145:{\displaystyle T^{*}X=\{(x,y,\mathrm {d} x,\mathrm {d} y)\in \mathbb {R} ^{4}:y^{2}-x=0,2y\mathrm {d} y-\mathrm {d} x=0\}}
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3146:{\displaystyle \omega =\mathrm {d} x_{1}\wedge \mathrm {d} y_{1}+\dotsb +\mathrm {d} x_{n}\wedge \mathrm {d} y_{n}.}
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is a symplectic manifold equipped with a compatible integrable complex structure. They form a particular class of
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5374:{\displaystyle \omega \left({\frac {\partial }{\partial u_{i}}},{\frac {\partial }{\partial u_{j}}}\right)=0}
7889:
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The
Classification of Critical Points, Caustics and Wave Fronts: Singularities of Differentiable Maps, Vol 1
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as a lower bound for the number of self intersections of a smooth
Lagrangian submanifold, rather than the
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is
Lagrangian. Their intersections display rigidity properties not possessed by smooth manifolds; the
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6919: – differentiable manifold equipped with a nondegenerate (but not necessarily closed) 2‐form
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3389:{\displaystyle X=f_{i}({\textbf {x}})\partial _{x_{i}},Y=g_{i}({\textbf {x}})\partial _{x_{i}},}
3218:{\displaystyle \mathbb {R} _{\mathbf {x} }^{n}\to \mathbb {R} _{\mathbf {x} ,\mathbf {y} }^{2n}}
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6793:. For example, the cotangent bundle of a smooth manifold is an exact symplectic manifold. The
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are submanifolds where the symplectic form restricts to zero, i.e. each tangent space is an
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6995: – canonical differential form defined on the cotangent bundle of a smooth manifold
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4765:{\displaystyle q_{i}=q_{i}(u_{1},\dotsc ,u_{n})\quad p_{i}=p_{i}(u_{1},\dotsc ,u_{n})}
17:
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539:. Since one desires the Hamiltonian to be constant along flow lines, one should have
5801:{\displaystyle M\cap \mathbb {V} (\varepsilon \cdot \mathrm {d} f)={\text{Crit}}(f)}
2481:. That is, they do not necessarily arise from a complex structure on the manifold.
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7354:. Mathematics and Its Applications. Vol. 62. Dordrecht: Springer Netherlands.
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1919:{\displaystyle \Omega ={\begin{pmatrix}0&I_{n}\\-I_{n}&0\end{pmatrix}}.}
142:
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Dunin-Barkowski, Petr (2022). "Symplectic duality for topological recursion".
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5736:{\displaystyle \mathbb {V} (\varepsilon \cdot \mathrm {d} f)\subset T^{*}M}
6938: – Formalism in classical field theory based on Hamiltonian mechanics
5743:. For a generic Morse function we have a Lagrangian intersection given by
97:, called the symplectic form. The study of symplectic manifolds is called
8331:
8326:
8316:
7707:
7528:
1070:
7043:"Why symplectic geometry is the natural setting for classical mechanics"
5685:
one can construct a
Lagrangian submanifold given by the vanishing locus
5410:. Simplify the result by making use of the canonical symplectic form on
7497:
7176:
6093:. The following examples are known as special Lagrangian submanifolds,
7483:
Hitchin, Nigel (1999). "Lectures on
Special Lagrangian Submanifolds".
7923:
7489:
6230:
or the canonical two-form. Using this set-up we can locally think of
1093:
7398:
6947: – symplectic manifold equipped with a torsion-free connection
2007:
has a natural symplectic form, called the
Poincaré two-form or the
955:
so that, on repeating this argument for different smooth functions
105:. Symplectic manifolds arise naturally in abstract formulations of
7455:
6349:
6974: – Branch of differential geometry and differential topology
5979:
if in addition to the above Lagrangian condition the restriction
2815:. Lagrangian submanifolds are the maximal isotropic submanifolds.
149:
allow one to derive the time evolution of a system from a set of
2213:
are fibrewise coordinates with respect to the cotangent vectors
7501:
6335:{\displaystyle \pi :T^{*}\mathbb {R} ^{n}\to \mathbb {R} ^{n}.}
3042:{\displaystyle \mathbb {R} _{{\textbf {x}},{\textbf {y}}}^{2n}}
2907:{\displaystyle \mathbb {R} _{{\textbf {x}},{\textbf {y}}}^{2n}}
7445:(2009). "Fibre bundles, jet manifolds and Lagrangian theory".
2682:
are submanifolds where the restriction of the symplectic form
2477:. They generalize Kähler manifolds, in that they need not be
7162:(1999). "Covariant Hamiltonian equations for field theory".
762:
643:
and hence a 2-form. Finally, one makes the requirement that
6932:—an odd-dimensional counterpart of the symplectic manifold.
6112:
deals with the study of special Lagrangian submanifolds in
5874:{\displaystyle \Omega =\Omega _{1}+\mathrm {i} \Omega _{2}}
1830:
then the matrix, Ω, of this quadratic form is given by the
1791:
1377:
has an even dimension. The closed condition means that the
3959:{\displaystyle X=\{(x,y)\in \mathbb {R} ^{2}:y^{2}-x=0\}.}
2079:{\displaystyle \omega =\sum _{i=1}^{n}dp_{i}\wedge dq^{i}}
6104:
fixed points of a real structure of Calabi–Yau manifolds.
5624:
Another useful class of Lagrangian submanifolds occur in
4524:{\displaystyle (q_{1},\dotsc ,q_{n},p_{1},\dotsc ,p_{n})}
3577:{\displaystyle \omega =\mathrm {d} x\wedge \mathrm {d} y}
2991:{\displaystyle (x_{1},\dotsc ,x_{n},y_{1},\dotsc ,y_{n})}
7321:"Symplectic manifolds and their lagrangian submanifolds"
3535:{\displaystyle X=f(x)\partial _{x},Y=g(x)\partial _{x},}
157:
describing the flow of the system from the differential
6976:
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6940:
Pages displaying short descriptions of redirect targets
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such that both sides of the diagram given on the right
6361:
be a Lagrangian submanifold of a symplectic manifold (
6279:
and the Lagrangian fibration as the trivial fibration
5269:. This is that the symplectic form must vanish on the
4238:{\displaystyle \mathbb {R} ^{4}=T^{*}\mathbb {R} ^{2}}
2785:
2399:
has a symplectic form which is the restriction of the
1865:
1194:
is non-degenerate. That is to say, if there exists an
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In this case the symplectic form reduces to a simple
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is a symplectic form. Assigning a symplectic form to
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There is a standard Lagrangian submanifold given by
2808:{\displaystyle {\text{dim }}L={\tfrac {1}{2}}\dim M}
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7348:Arnold, V. I. (1990). "Ch.1, Symplectic geometry".
6838:is a manifold equipped with a closed nondegenerate
5295:; that is, it must vanish for all tangent vectors:
4245:. We can consider the subset where the coordinates
153:, the symplectic form should allow one to obtain a
7469:"Symplectic Structures—A New Approach to Geometry"
6877:
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6163:is even-dimensional we can take local coordinates
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1115:. Here, non-degenerate means that for every point
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663:should not change under flow lines, i.e. that the
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141:; in particular, they are a generalization of the
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4775:This manifold is a Lagrangian submanifold if the
7125:Cantrijn, F.; Ibort, L. A.; de León, M. (1999).
4400:{\displaystyle \mathrm {d} f_{1},\dotsc ,df_{k}}
612:{\displaystyle \omega (V_{H},V_{H})=dH(V_{H})=0}
3274:{\displaystyle \mathbb {R} _{\mathbf {x} }^{n}}
2489:There are several natural geometric notions of
7279:. London: Benjamin-Cummings. See Section 3.2.
7127:"On the Geometry of Multisymplectic Manifolds"
6853:provided with a polysymplectic tangent-valued
5249:in the condition for a Lagrangian submanifold
2354:. A large class of examples come from complex
7513:
6557:preserves the symplectic form. Symbolically:
461:there is a unique corresponding vector field
8:
6823:Symplectic manifolds are special cases of a
6812:in the sense that the tangent bundle has an
6026:is vanishing. In other words, the real part
4583:is one that is parameterized by coordinates
4139:
4025:
3950:
3892:
3860:{\displaystyle \mathrm {d} y(\partial _{x})}
1582:
1547:
6989: – Isomorphism of symplectic manifolds
4184:{\displaystyle \mathrm {d} x,\mathrm {d} y}
7928:
7520:
7506:
7498:
6956: – Operation in Hamiltonian mechanics
3281:because given any pair of tangent vectors
2392:{\displaystyle V\subset \mathbb {CP} ^{n}}
1337:are always singular, the requirement that
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795:
7488:
7454:
7397:
7351:Singularities of Caustics and Wave Fronts
7338:
7299:Symplectic Geometry and Quantum Mechanics
7175:
7142:
7061:Symplectic Geometry and Quantum Mechanics
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4708:
4694:
4675:
4662:
4649:
4643:
4616:
4597:
4588:
4564:
4560:
4559:
4556:
4536:
4512:
4493:
4480:
4461:
4452:
4428:
4424:
4423:
4420:
4391:
4369:
4360:
4358:
4337:
4318:
4312:
4283:
4281:
4252:
4250:
4229:
4225:
4224:
4217:
4204:
4200:
4199:
4196:
4173:
4162:
4160:
4125:
4114:
4087:
4074:
4070:
4069:
4054:
4043:
4013:
4007:
3980:
3974:
3932:
3919:
3915:
3914:
3884:
3848:
3833:
3831:
3801:
3786:
3777:
3762:
3750:
3735:
3726:
3711:
3674:
3662:
3637:
3592:
3566:
3555:
3547:
3523:
3492:
3468:
3442:
3401:
3375:
3370:
3357:
3356:
3347:
3326:
3321:
3308:
3307:
3298:
3286:
3265:
3259:
3258:
3254:
3253:
3250:
3230:
3206:
3200:
3192:
3191:
3187:
3186:
3176:
3170:
3169:
3165:
3164:
3161:
3134:
3125:
3116:
3107:
3092:
3083:
3074:
3065:
3057:
3030:
3023:
3022:
3013:
3012:
3011:
3007:
3006:
3003:
2979:
2960:
2947:
2928:
2919:
2895:
2888:
2887:
2878:
2877:
2876:
2872:
2871:
2868:
2819:One major example is that the graph of a
2784:
2773:
2771:
2744:
2739:
2733:
2707:
2687:
2655:
2615:
2594:
2589:
2583:
2557:
2537:
2498:
2454:
2422:
2418:
2415:
2414:
2411:
2383:
2379:
2376:
2375:
2366:
2322:
2313:
2288:
2282:
2249:
2227:
2218:
2194:
2175:
2166:
2146:
2122:
2103:
2094:
2070:
2054:
2041:
2030:
2018:
1988:
1982:
1959:
1939:
1894:
1877:
1860:
1852:
1783:
1754:
1707:
1679:
1667:
1654:
1642:
1608:
1604:
1603:
1600:
1573:
1554:
1545:
1502:
1482:
1462:
1442:
1410:
1386:
1362:
1342:
1312:
1288:
1276:
1235:
1211:
1199:
1179:
1155:
1149:
1120:
1100:
1077:
1034:
1013:
1007:
986:
980:
960:
927:
912:
900:
885:
875:
870:
862:
851:
843:
838:
817:
812:
800:
772:
767:
761:
760:
757:
729:
723:
698:
692:
672:
648:
624:
594:
569:
556:
544:
514:
493:
472:
466:
443:
419:
395:
379:
373:
349:
325:
309:
303:
279:
273:
247:
220:
205:
185:
162:
82:
53:
27:Type of manifold in differential geometry
7084:
7082:
7080:
7063:. Basel: Birkhäuser Verlag. p. 10.
7054:
7052:
7028:"What is a symplectic manifold, really?"
7004: – inequality applicable to 2-forms
1588:{\displaystyle \{v_{1},\ldots ,v_{2n}\}}
1092:is a closed non-degenerate differential
532:{\displaystyle dH=\omega (V_{H},\cdot )}
145:of a closed system. In the same way the
7018:
6117:
6272:{\displaystyle T^{*}\mathbb {R} ^{n},}
3584:. Notice that when we expand this out
1049:is equivalent to the requirement that
7408:"How to find Lagrangian Submanifolds"
6800:A symplectic manifold endowed with a
6199:can be, at least locally, written as
4628:{\displaystyle (u_{1},\dotsc ,u_{n})}
2258:{\displaystyle dq^{1},\ldots ,dq^{n}}
2206:{\displaystyle (p_{1},\ldots ,p_{n})}
2134:{\displaystyle (q^{1},\ldots ,q^{n})}
7:
5955:imaginary. A Lagrangian submanifold
3049:with the canonical symplectic form
2265:. Cotangent bundles are the natural
1141:, the skew-symmetric pairing on the
438:ensures that for every differential
407:{\displaystyle T^{*}M\otimes T^{*}M}
337:{\displaystyle T^{*}M\otimes T^{*}M}
232:{\displaystyle TM\rightarrow T^{*}M}
7224:Introduction to Symplectic Topology
6159:are Lagrangian submanifolds. Since
6097:complex Lagrangian submanifolds of
5658:{\displaystyle f:M\to \mathbb {R} }
4346:{\displaystyle f_{1},\dotsc ,f_{k}}
3867:factor, which is 0, by definition.
3358:
3309:
3024:
3014:
2889:
2879:
2847:gives the sum of the submanifold's
2823:in the product symplectic manifold
7226:. Oxford Mathematical Monographs.
6936:Covariant Hamiltonian field theory
6034:
5987:
5936:
5909:
5862:
5857:
5844:
5837:
5771:
5707:
5557:
5553:
5532:
5528:
5491:
5487:
5466:
5462:
5344:
5340:
5319:
5315:
5220:
5216:
5198:
5183:
5161:
5157:
5139:
5124:
5102:
5098:
5025:
5010:
4991:
4976:
4954:
4939:
4920:
4905:
4361:
4284:
4253:
4174:
4163:
4155:where we are treating the symbols
4126:
4115:
4055:
4044:
3845:
3834:
3798:
3787:
3774:
3763:
3747:
3736:
3723:
3712:
3659:
3634:
3567:
3556:
3520:
3489:
3367:
3318:
3126:
3108:
3084:
3066:
1954:be a smooth manifold of dimension
1854:
1623:{\displaystyle \mathbb {R} ^{2n}.}
1029:corresponding to arbitrary smooth
913:
886:
871:
863:
852:
801:
117:of manifolds. For example, in the
25:
6728:Special cases and generalizations
5435:{\displaystyle \mathbb {R} ^{2n}}
5086:. This can be seen by expanding
4576:{\displaystyle \mathbb {R} ^{2n}}
4440:{\displaystyle \mathbb {R} ^{2n}}
3876:of a manifold. For example, let
2914:have global coordinates labelled
2485:Lagrangian and other submanifolds
2431:{\displaystyle \mathbb {CP} ^{n}}
298:, or equivalently, an element of
7158:Giachetta, G.; Mangiarotti, L.;
6900:
4850:. That is, it is Lagrangian if
3437:To elucidate, consider the case
3260:
3201:
3193:
3171:
137:Symplectic manifolds arise from
7026:Webster, Ben (9 January 2012).
6808:with the symplectic form is an
6404:give a Lagrangian fibration of
6342:This is the canonical picture.
5901:as a holomorphic n-form, where
5818:Special Lagrangian submanifolds
4703:
4411:Example: Parametric submanifold
4300:{\displaystyle \mathrm {d} y=0}
4269:{\displaystyle \mathrm {d} x=0}
3430:{\displaystyle \omega (X,Y)=0.}
7560:Differentiable/Smooth manifold
7002:Wirtinger inequality (2-forms)
6872:
6860:
6752:
6740:
6314:
5795:
5789:
5778:
5761:
5714:
5697:
5647:
4886:
4860:
4811:
4785:
4759:
4727:
4700:
4668:
4622:
4590:
4518:
4454:
4062:
4028:
3907:
3895:
3854:
3841:
3810:
3807:
3794:
3783:
3770:
3756:
3743:
3732:
3719:
3708:
3705:
3699:
3693:
3687:
3668:
3655:
3649:
3630:
3624:
3618:
3609:
3597:
3516:
3510:
3485:
3479:
3418:
3406:
3363:
3353:
3314:
3304:
3182:
2985:
2921:
2759:{\displaystyle \omega |_{L}=0}
2740:
2669:
2657:
2590:
2512:
2500:
2200:
2168:
2128:
2096:
1974:. Then the total space of the
1673:
1647:
1630:We define our symplectic form
1424:
1412:
1357:be nondegenerate implies that
1264:{\displaystyle \omega (X,Y)=0}
1252:
1240:
933:
920:
906:
893:
879:
867:
828:
805:
796:
786:
780:
600:
587:
575:
549:
526:
507:
213:
1:
4415:Consider the canonical space
2141:are any local coordinates on
200:. So we require a linear map
7340:10.1016/0001-8708(71)90020-X
7301:. Basel: Birkhäuser Verlag.
7246:"Seminar on Mirror Symmetry"
7213:General and cited references
6983: – Mathematical concept
6968: – Mathematical concept
5678:{\displaystyle \varepsilon }
2603:{\displaystyle \omega |_{S}}
975:such that the corresponding
8266:Classification of manifolds
7431:Encyclopedia of Mathematics
7194:10.1088/0305-4470/32/38/302
7059:de Gosson, Maurice (2006).
6758:{\displaystyle (M,\omega )}
6046:{\displaystyle \Omega _{1}}
5999:{\displaystyle \Omega _{2}}
5948:{\displaystyle \Omega _{2}}
5921:{\displaystyle \Omega _{1}}
5608:—this is an application of
4531:. A parametric submanifold
2675:{\displaystyle (M,\omega )}
2518:{\displaystyle (M,\omega )}
1430:{\displaystyle (M,\omega )}
1333:. Since in odd dimensions,
1300:{\displaystyle Y\in T_{p}M}
1223:{\displaystyle X\in T_{p}M}
8416:
7447:Lectures for Theoreticians
6962: – Mathematical group
6928: – Branch of geometry
6917:Almost symplectic manifold
6226:. This form is called the
5810:
5597:and all others vanishing.
2721:{\displaystyle L\subset M}
2571:{\displaystyle S\subset M}
1634:on this basis as follows:
1533:
738:{\displaystyle \iota _{X}}
180:of a Hamiltonian function
8342:over commutative algebras
7360:10.1007/978-94-011-3330-2
7144:10.1017/S1446788700036636
6885:-form; it is utilized in
6795:canonical symplectic form
6147:of a symplectic manifold
6073:leads the volume form on
3871:Example: Cotangent bundle
2650:of a symplectic manifold
2493:of a symplectic manifold
2009:canonical symplectic form
1497:is referred to as giving
1457:is a smooth manifold and
8058:Riemann curvature tensor
7277:Foundations of Mechanics
6887:Hamiltonian field theory
6832:multisymplectic manifold
6814:almost complex structure
4353:and their differentials
2610:is a symplectic form on
2475:almost-complex manifolds
2471:almost complex structure
2442:Almost-complex manifolds
1530:Symplectic vector spaces
718:, this amounts to (here
7326:Advances in Mathematics
6981:Symplectic vector space
6847:polysymplectic manifold
6816:, but this need not be
6782:{\displaystyle \omega }
6769:if the symplectic form
5830:) we can make a choice
5665:and for a small enough
3238:{\displaystyle \omega }
2695:{\displaystyle \omega }
2648:Lagrangian submanifolds
2530:Symplectic submanifolds
2462:{\displaystyle \omega }
1536:Symplectic vector space
1470:{\displaystyle \omega }
1394:{\displaystyle \omega }
1350:{\displaystyle \omega }
1335:skew-symmetric matrices
1187:{\displaystyle \omega }
1108:{\displaystyle \omega }
680:{\displaystyle \omega }
656:{\displaystyle \omega }
632:{\displaystyle \omega }
427:{\displaystyle \omega }
414:, the requirement that
357:{\displaystyle \omega }
119:Hamiltonian formulation
90:{\displaystyle \omega }
7850:Manifold with boundary
7565:Differential structure
7426:"Symplectic Structure"
7222:; Salamon, D. (1998).
6879:
6810:almost Kähler manifold
6783:
6759:
6733:A symplectic manifold
6690:
6354:
6336:
6273:
6218:, where d denotes the
6087:
6067:
6047:
6020:
6000:
5969:
5949:
5922:
5895:
5875:
5802:
5737:
5679:
5659:
5588:
5436:
5404:
5375:
5289:
5263:
5240:
5080:
5051:
4844:
4818:
4766:
4629:
4577:
4545:
4525:
4441:
4401:
4347:
4301:
4270:
4239:
4185:
4146:
3993:
3992:{\displaystyle T^{*}X}
3960:
3861:
3817:
3578:
3536:
3457:
3431:
3390:
3275:
3239:
3219:
3147:
3043:
2992:
2908:
2809:
2760:
2722:
2696:
2676:
2634:Isotropic submanifolds
2624:
2604:
2572:
2546:
2519:
2463:
2432:
2393:
2332:
2331:{\displaystyle dq^{i}}
2298:
2259:
2207:
2155:
2135:
2080:
2046:
2001:
2000:{\displaystyle T^{*}Q}
1968:
1948:
1920:
1799:
1624:
1589:
1511:
1491:
1471:
1451:
1431:
1395:
1371:
1351:
1327:
1301:
1265:
1224:
1188:
1168:
1167:{\displaystyle T_{p}M}
1135:
1134:{\displaystyle p\in M}
1109:
1086:
1043:
1023:
996:
969:
946:
739:
708:
681:
657:
633:
613:
533:
482:
455:
428:
408:
358:
338:
292:
291:{\displaystyle T^{*}M}
259:
233:
194:
174:
151:differential equations
91:
62:
18:Lagrangian submanifold
8390:Hamiltonian mechanics
8385:Differential topology
7295:de Gosson, Maurice A.
7258:"Symplectic Geometry"
7131:J. Austral. Math. Soc
6993:Tautological one-form
6880:
6878:{\displaystyle (n+2)}
6784:
6760:
6691:
6532:Lagrangian equivalent
6353:
6337:
6274:
6125:Thomas–Yau conjecture
6099:hyperkähler manifolds
6088:
6068:
6048:
6021:
6001:
5970:
5950:
5928:is the real part and
5923:
5896:
5876:
5803:
5738:
5680:
5660:
5620:Example: Morse theory
5589:
5437:
5405:
5376:
5290:
5264:
5241:
5081:
5052:
4845:
4819:
4767:
4630:
4578:
4546:
4526:
4442:
4402:
4348:
4302:
4271:
4240:
4186:
4147:
3994:
3969:Then, we can present
3961:
3862:
3826:both terms we have a
3818:
3579:
3537:
3458:
3432:
3391:
3276:
3240:
3220:
3148:
3044:
2998:. Then, we can equip
2993:
2909:
2810:
2761:
2723:
2697:
2677:
2625:
2605:
2573:
2547:
2520:
2464:
2433:
2394:
2358:. Any smooth complex
2333:
2299:
2297:{\displaystyle p_{i}}
2273:, as is the case for
2260:
2208:
2156:
2136:
2081:
2026:
2002:
1969:
1949:
1921:
1800:
1625:
1590:
1512:
1492:
1472:
1452:
1432:
1396:
1372:
1352:
1328:
1302:
1266:
1225:
1189:
1169:
1136:
1110:
1087:
1044:
1024:
1022:{\displaystyle V_{H}}
997:
995:{\displaystyle V_{H}}
970:
947:
740:
709:
707:{\displaystyle V_{H}}
682:
658:
634:
619:, which implies that
614:
534:
483:
481:{\displaystyle V_{H}}
456:
429:
409:
359:
339:
293:
260:
234:
195:
175:
92:
63:
34:differential geometry
7997:Covariant derivative
7548:Topological manifold
7416:. December 17, 2014.
6857:
6773:
6737:
6564:
6454:Two Lagrangian maps
6388:Lagrangian immersion
6283:
6241:
6195:the symplectic form
6145:Lagrangian fibration
6139:Lagrangian fibration
6077:
6057:
6030:
6010:
5983:
5959:
5932:
5905:
5885:
5834:
5828:Calabi–Yau manifolds
5747:
5689:
5669:
5635:
5449:
5414:
5388:
5302:
5276:
5253:
5093:
5064:
4857:
4828:
4782:
4642:
4587:
4555:
4535:
4451:
4419:
4357:
4311:
4280:
4249:
4195:
4159:
4006:
3973:
3883:
3830:
3591:
3546:
3467:
3441:
3400:
3285:
3249:
3229:
3160:
3056:
3002:
2918:
2867:
2855:in the smooth case.
2853:Euler characteristic
2770:
2732:
2706:
2686:
2654:
2614:
2582:
2556:
2536:
2497:
2453:
2447:Riemannian manifolds
2410:
2365:
2312:
2308:" to the velocities
2281:
2275:Riemannian manifolds
2217:
2165:
2145:
2093:
2017:
1981:
1958:
1938:
1851:
1641:
1599:
1544:
1519:symplectic structure
1501:
1481:
1461:
1441:
1409:
1385:
1361:
1341:
1311:
1275:
1234:
1198:
1178:
1148:
1119:
1099:
1076:
1033:
1006:
979:
959:
756:
722:
691:
671:
647:
623:
543:
492:
465:
442:
418:
372:
348:
302:
272:
246:
204:
184:
161:
111:analytical mechanics
81:
52:
8400:Symplectic geometry
8031:Exterior derivative
7633:Atiyah–Singer index
7582:Riemannian manifold
7273:Marsden, Jerrold E.
7254:Meinrenken, Eckhard
7186:1999JPhA...32.6629G
6972:Symplectic topology
6220:exterior derivative
6129:stability condition
5813:symplectic category
5403:{\displaystyle i,j}
5079:{\displaystyle i,j}
4843:{\displaystyle i,j}
3456:{\displaystyle n=1}
3270:
3214:
3181:
3038:
2903:
2728:is vanishing, i.e.
1403:symplectic manifold
1379:exterior derivative
1326:{\displaystyle X=0}
714:vanishes. Applying
139:classical mechanics
107:classical mechanics
103:symplectic topology
99:symplectic geometry
76:differential 2-form
42:symplectic manifold
8337:Secondary calculus
8291:Singularity theory
8246:Parallel transport
8014:De Rham cohomology
7653:Generalized Stokes
7476:Notices of the AMS
7164:Journal of Physics
7098:Gusein-Zade, S. M.
6908:Mathematics portal
6875:
6779:
6755:
6686:
6437:critical value set
6433:Lagrangian mapping
6355:
6346:Lagrangian mapping
6332:
6269:
6222:and ∧ denotes the
6083:
6063:
6043:
6016:
5996:
5965:
5945:
5918:
5891:
5871:
5798:
5733:
5675:
5655:
5584:
5432:
5400:
5371:
5288:{\displaystyle TL}
5285:
5259:
5236:
5076:
5047:
4901:
4840:
4814:
4762:
4625:
4573:
4541:
4521:
4437:
4397:
4343:
4297:
4266:
4235:
4191:as coordinates of
4181:
4142:
3989:
3956:
3857:
3813:
3574:
3532:
3453:
3427:
3386:
3271:
3252:
3235:
3215:
3185:
3163:
3143:
3039:
3005:
2988:
2904:
2870:
2805:
2794:
2756:
2718:
2692:
2672:
2638:isotropic subspace
2620:
2600:
2568:
2542:
2515:
2459:
2428:
2389:
2360:projective variety
2356:algebraic geometry
2328:
2294:
2255:
2203:
2151:
2131:
2076:
1997:
1964:
1944:
1916:
1907:
1795:
1790:
1620:
1585:
1507:
1487:
1467:
1447:
1427:
1391:
1367:
1347:
1323:
1297:
1261:
1220:
1184:
1164:
1131:
1105:
1082:
1039:
1019:
992:
965:
942:
735:
704:
677:
653:
629:
609:
529:
478:
454:{\displaystyle dH}
451:
424:
404:
354:
334:
288:
267:cotangent manifold
258:{\displaystyle TM}
255:
229:
190:
173:{\displaystyle dH}
170:
147:Hamilton equations
87:
68:, equipped with a
58:
8372:
8371:
8254:
8253:
8019:Differential form
7673:Whitney embedding
7607:Differential form
7467:(November 1998).
7443:Sardanashvily, G.
7369:978-1-4020-0333-2
7170:(38): 6629–6642.
7160:Sardanashvily, G.
6987:Symplectomorphism
6966:Symplectic matrix
6648:
6607:
6228:Poincaré two-form
6193:Darboux's theorem
6155:where all of the
6135:of the manifold.
6086:{\displaystyle L}
6066:{\displaystyle L}
6019:{\displaystyle L}
5968:{\displaystyle L}
5894:{\displaystyle M}
5787:
5614:action functional
5571:
5546:
5505:
5480:
5358:
5333:
5262:{\displaystyle L}
5234:
5212:
5175:
5153:
5116:
5039:
5005:
4968:
4934:
4892:
4824:vanishes for all
4544:{\displaystyle L}
4447:with coordinates
3682:
3360:
3311:
3026:
3016:
2891:
2881:
2845:Arnold conjecture
2821:symplectomorphism
2793:
2776:
2623:{\displaystyle S}
2545:{\displaystyle M}
2401:Fubini—Study form
2352:complex manifolds
2154:{\displaystyle Q}
1967:{\displaystyle n}
1947:{\displaystyle Q}
1930:Cotangent bundles
1786:
1757:
1710:
1510:{\displaystyle M}
1490:{\displaystyle M}
1450:{\displaystyle M}
1370:{\displaystyle M}
1085:{\displaystyle M}
1042:{\displaystyle H}
968:{\displaystyle H}
193:{\displaystyle H}
115:cotangent bundles
61:{\displaystyle M}
16:(Redirected from
8407:
8395:Smooth manifolds
8364:Stratified space
8322:Fréchet manifold
8036:Interior product
7929:
7626:
7522:
7515:
7508:
7499:
7494:
7492:
7479:
7473:
7460:
7458:
7438:
7417:
7403:
7401:
7381:
7344:
7342:
7312:
7290:
7264:
7262:
7249:
7237:
7206:
7205:
7179:
7155:
7149:
7148:
7146:
7122:
7116:
7115:
7094:Varchenko, A. N.
7086:
7075:
7074:
7056:
7047:
7046:
7038:
7032:
7031:
7023:
7007:
6998:
6977:
6960:Symplectic group
6950:
6945:Fedosov manifold
6941:
6931:
6926:Contact manifold
6922:
6910:
6905:
6904:
6884:
6882:
6881:
6876:
6825:Poisson manifold
6788:
6786:
6785:
6780:
6764:
6762:
6761:
6756:
6695:
6693:
6692:
6687:
6681:
6680:
6668:
6667:
6658:
6657:
6646:
6636:
6635:
6623:
6622:
6605:
6595:
6594:
6582:
6581:
6529:
6491:
6430:
6408:. The composite
6403:
6381:
6365:,ω) given by an
6341:
6339:
6338:
6333:
6328:
6327:
6322:
6313:
6312:
6307:
6301:
6300:
6278:
6276:
6275:
6270:
6265:
6264:
6259:
6253:
6252:
6236:cotangent bundle
6224:exterior product
6217:
6190:
6092:
6090:
6089:
6084:
6072:
6070:
6069:
6064:
6052:
6050:
6049:
6044:
6042:
6041:
6025:
6023:
6022:
6017:
6005:
6003:
6002:
5997:
5995:
5994:
5974:
5972:
5971:
5966:
5954:
5952:
5951:
5946:
5944:
5943:
5927:
5925:
5924:
5919:
5917:
5916:
5900:
5898:
5897:
5892:
5880:
5878:
5877:
5872:
5870:
5869:
5860:
5852:
5851:
5824:Kähler manifolds
5807:
5805:
5804:
5799:
5788:
5785:
5774:
5760:
5742:
5740:
5739:
5734:
5729:
5728:
5710:
5696:
5684:
5682:
5681:
5676:
5664:
5662:
5661:
5656:
5654:
5593:
5591:
5590:
5585:
5577:
5573:
5572:
5570:
5569:
5568:
5552:
5547:
5545:
5544:
5543:
5527:
5511:
5507:
5506:
5504:
5503:
5502:
5486:
5481:
5479:
5478:
5477:
5461:
5441:
5439:
5438:
5433:
5431:
5430:
5422:
5409:
5407:
5406:
5401:
5380:
5378:
5377:
5372:
5364:
5360:
5359:
5357:
5356:
5355:
5339:
5334:
5332:
5331:
5330:
5314:
5294:
5292:
5291:
5286:
5271:tangent manifold
5268:
5266:
5265:
5260:
5245:
5243:
5242:
5237:
5235:
5233:
5232:
5231:
5215:
5213:
5211:
5210:
5209:
5196:
5195:
5194:
5181:
5176:
5174:
5173:
5172:
5156:
5154:
5152:
5151:
5150:
5137:
5136:
5135:
5122:
5117:
5115:
5114:
5113:
5097:
5085:
5083:
5082:
5077:
5056:
5054:
5053:
5048:
5040:
5038:
5037:
5036:
5023:
5022:
5021:
5008:
5006:
5004:
5003:
5002:
4989:
4988:
4987:
4974:
4969:
4967:
4966:
4965:
4952:
4951:
4950:
4937:
4935:
4933:
4932:
4931:
4918:
4917:
4916:
4903:
4900:
4885:
4884:
4872:
4871:
4849:
4847:
4846:
4841:
4823:
4821:
4820:
4817:{\displaystyle }
4815:
4810:
4809:
4797:
4796:
4777:Lagrange bracket
4771:
4769:
4768:
4763:
4758:
4757:
4739:
4738:
4726:
4725:
4713:
4712:
4699:
4698:
4680:
4679:
4667:
4666:
4654:
4653:
4634:
4632:
4631:
4626:
4621:
4620:
4602:
4601:
4582:
4580:
4579:
4574:
4572:
4571:
4563:
4550:
4548:
4547:
4542:
4530:
4528:
4527:
4522:
4517:
4516:
4498:
4497:
4485:
4484:
4466:
4465:
4446:
4444:
4443:
4438:
4436:
4435:
4427:
4406:
4404:
4403:
4398:
4396:
4395:
4374:
4373:
4364:
4352:
4350:
4349:
4344:
4342:
4341:
4323:
4322:
4306:
4304:
4303:
4298:
4287:
4275:
4273:
4272:
4267:
4256:
4244:
4242:
4241:
4236:
4234:
4233:
4228:
4222:
4221:
4209:
4208:
4203:
4190:
4188:
4187:
4182:
4177:
4166:
4151:
4149:
4148:
4143:
4129:
4118:
4092:
4091:
4079:
4078:
4073:
4058:
4047:
4018:
4017:
3998:
3996:
3995:
3990:
3985:
3984:
3965:
3963:
3962:
3957:
3937:
3936:
3924:
3923:
3918:
3866:
3864:
3863:
3858:
3853:
3852:
3837:
3822:
3820:
3819:
3814:
3806:
3805:
3790:
3782:
3781:
3766:
3755:
3754:
3739:
3731:
3730:
3715:
3683:
3675:
3667:
3666:
3642:
3641:
3583:
3581:
3580:
3575:
3570:
3559:
3541:
3539:
3538:
3533:
3528:
3527:
3497:
3496:
3462:
3460:
3459:
3454:
3436:
3434:
3433:
3428:
3395:
3393:
3392:
3387:
3382:
3381:
3380:
3379:
3362:
3361:
3352:
3351:
3333:
3332:
3331:
3330:
3313:
3312:
3303:
3302:
3280:
3278:
3277:
3272:
3269:
3264:
3263:
3257:
3244:
3242:
3241:
3236:
3224:
3222:
3221:
3216:
3213:
3205:
3204:
3196:
3190:
3180:
3175:
3174:
3168:
3152:
3150:
3149:
3144:
3139:
3138:
3129:
3121:
3120:
3111:
3097:
3096:
3087:
3079:
3078:
3069:
3048:
3046:
3045:
3040:
3037:
3029:
3028:
3027:
3018:
3017:
3010:
2997:
2995:
2994:
2989:
2984:
2983:
2965:
2964:
2952:
2951:
2933:
2932:
2913:
2911:
2910:
2905:
2902:
2894:
2893:
2892:
2883:
2882:
2875:
2842:
2814:
2812:
2811:
2806:
2795:
2786:
2777:
2774:
2765:
2763:
2762:
2757:
2749:
2748:
2743:
2727:
2725:
2724:
2719:
2701:
2699:
2698:
2693:
2681:
2679:
2678:
2673:
2629:
2627:
2626:
2621:
2609:
2607:
2606:
2601:
2599:
2598:
2593:
2577:
2575:
2574:
2569:
2551:
2549:
2548:
2543:
2524:
2522:
2521:
2516:
2468:
2466:
2465:
2460:
2437:
2435:
2434:
2429:
2427:
2426:
2421:
2405:projective space
2398:
2396:
2395:
2390:
2388:
2387:
2382:
2342:Kähler manifolds
2337:
2335:
2334:
2329:
2327:
2326:
2303:
2301:
2300:
2295:
2293:
2292:
2264:
2262:
2261:
2256:
2254:
2253:
2232:
2231:
2212:
2210:
2209:
2204:
2199:
2198:
2180:
2179:
2160:
2158:
2157:
2152:
2140:
2138:
2137:
2132:
2127:
2126:
2108:
2107:
2085:
2083:
2082:
2077:
2075:
2074:
2059:
2058:
2045:
2040:
2006:
2004:
2003:
1998:
1993:
1992:
1976:cotangent bundle
1973:
1971:
1970:
1965:
1953:
1951:
1950:
1945:
1925:
1923:
1922:
1917:
1912:
1911:
1899:
1898:
1882:
1881:
1840:
1804:
1802:
1801:
1796:
1794:
1793:
1787:
1784:
1758:
1756: with
1755:
1711:
1709: with
1708:
1672:
1671:
1659:
1658:
1629:
1627:
1626:
1621:
1616:
1615:
1607:
1594:
1592:
1591:
1586:
1581:
1580:
1559:
1558:
1516:
1514:
1513:
1508:
1496:
1494:
1493:
1488:
1476:
1474:
1473:
1468:
1456:
1454:
1453:
1448:
1436:
1434:
1433:
1428:
1400:
1398:
1397:
1392:
1376:
1374:
1373:
1368:
1356:
1354:
1353:
1348:
1332:
1330:
1329:
1324:
1306:
1304:
1303:
1298:
1293:
1292:
1270:
1268:
1267:
1262:
1229:
1227:
1226:
1221:
1216:
1215:
1193:
1191:
1190:
1185:
1173:
1171:
1170:
1165:
1160:
1159:
1140:
1138:
1137:
1132:
1114:
1112:
1111:
1106:
1091:
1089:
1088:
1083:
1048:
1046:
1045:
1040:
1028:
1026:
1025:
1020:
1018:
1017:
1001:
999:
998:
993:
991:
990:
974:
972:
971:
966:
951:
949:
948:
943:
932:
931:
916:
905:
904:
889:
874:
866:
855:
850:
849:
848:
847:
824:
823:
822:
821:
804:
779:
778:
777:
776:
766:
765:
747:interior product
744:
742:
741:
736:
734:
733:
716:Cartan's formula
713:
711:
710:
705:
703:
702:
686:
684:
683:
678:
662:
660:
659:
654:
638:
636:
635:
630:
618:
616:
615:
610:
599:
598:
574:
573:
561:
560:
538:
536:
535:
530:
519:
518:
487:
485:
484:
479:
477:
476:
460:
458:
457:
452:
433:
431:
430:
425:
413:
411:
410:
405:
400:
399:
384:
383:
363:
361:
360:
355:
343:
341:
340:
335:
330:
329:
314:
313:
297:
295:
294:
289:
284:
283:
264:
262:
261:
256:
241:tangent manifold
238:
236:
235:
230:
225:
224:
199:
197:
196:
191:
179:
177:
176:
171:
123:cotangent bundle
96:
94:
93:
88:
67:
65:
64:
59:
21:
8415:
8414:
8410:
8409:
8408:
8406:
8405:
8404:
8375:
8374:
8373:
8368:
8307:Banach manifold
8300:Generalizations
8295:
8250:
8187:
8084:
8046:Ricci curvature
8002:Cotangent space
7980:
7918:
7760:
7754:
7713:Exponential map
7677:
7622:
7616:
7536:
7526:
7482:
7471:
7463:
7441:
7420:
7406:
7391:
7388:
7386:Further reading
7370:
7347:
7315:
7309:
7293:
7287:
7267:
7260:
7252:
7240:
7234:
7218:
7215:
7210:
7209:
7157:
7156:
7152:
7124:
7123:
7119:
7112:
7088:
7087:
7078:
7071:
7058:
7057:
7050:
7040:
7039:
7035:
7025:
7024:
7020:
7015:
7010:
7005:
6996:
6975:
6954:Poisson bracket
6948:
6939:
6929:
6920:
6906:
6899:
6896:
6855:
6854:
6851:Legendre bundle
6771:
6770:
6735:
6734:
6730:
6719:
6708:
6672:
6659:
6649:
6627:
6614:
6586:
6573:
6562:
6561:
6536:diffeomorphisms
6534:if there exist
6528:
6521:
6514:
6507:
6500:
6493:
6490:
6483:
6476:
6469:
6462:
6455:
6409:
6391:
6369:
6348:
6317:
6302:
6292:
6281:
6280:
6254:
6244:
6239:
6238:
6212:
6200:
6180:
6171:
6164:
6141:
6133:Fukaya category
6114:mirror symmetry
6075:
6074:
6055:
6054:
6033:
6028:
6027:
6008:
6007:
5986:
5981:
5980:
5957:
5956:
5935:
5930:
5929:
5908:
5903:
5902:
5883:
5882:
5861:
5843:
5832:
5831:
5822:In the case of
5820:
5815:
5745:
5744:
5720:
5687:
5686:
5667:
5666:
5633:
5632:
5622:
5560:
5556:
5535:
5531:
5525:
5521:
5494:
5490:
5469:
5465:
5459:
5455:
5447:
5446:
5417:
5412:
5411:
5386:
5385:
5347:
5343:
5322:
5318:
5312:
5308:
5300:
5299:
5274:
5273:
5251:
5250:
5223:
5219:
5201:
5197:
5186:
5182:
5164:
5160:
5142:
5138:
5127:
5123:
5105:
5101:
5091:
5090:
5062:
5061:
5028:
5024:
5013:
5009:
4994:
4990:
4979:
4975:
4957:
4953:
4942:
4938:
4923:
4919:
4908:
4904:
4876:
4863:
4855:
4854:
4826:
4825:
4801:
4788:
4780:
4779:
4749:
4730:
4717:
4704:
4690:
4671:
4658:
4645:
4640:
4639:
4612:
4593:
4585:
4584:
4558:
4553:
4552:
4533:
4532:
4508:
4489:
4476:
4457:
4449:
4448:
4422:
4417:
4416:
4413:
4387:
4365:
4355:
4354:
4333:
4314:
4309:
4308:
4278:
4277:
4247:
4246:
4223:
4213:
4198:
4193:
4192:
4157:
4156:
4083:
4068:
4009:
4004:
4003:
3976:
3971:
3970:
3928:
3913:
3881:
3880:
3873:
3844:
3828:
3827:
3797:
3773:
3746:
3722:
3658:
3633:
3589:
3588:
3544:
3543:
3519:
3488:
3465:
3464:
3439:
3438:
3398:
3397:
3371:
3366:
3343:
3322:
3317:
3294:
3283:
3282:
3247:
3246:
3227:
3226:
3158:
3157:
3130:
3112:
3088:
3070:
3054:
3053:
3000:
2999:
2975:
2956:
2943:
2924:
2916:
2915:
2865:
2864:
2861:
2824:
2768:
2767:
2738:
2730:
2729:
2704:
2703:
2684:
2683:
2652:
2651:
2612:
2611:
2588:
2580:
2579:
2554:
2553:
2534:
2533:
2495:
2494:
2487:
2451:
2450:
2444:
2413:
2408:
2407:
2374:
2363:
2362:
2348:Kähler manifold
2344:
2318:
2310:
2309:
2284:
2279:
2278:
2245:
2223:
2215:
2214:
2190:
2171:
2163:
2162:
2143:
2142:
2118:
2099:
2091:
2090:
2066:
2050:
2015:
2014:
1984:
1979:
1978:
1956:
1955:
1936:
1935:
1932:
1906:
1905:
1900:
1890:
1884:
1883:
1873:
1871:
1861:
1849:
1848:
1831:
1828:identity matrix
1817:
1789:
1788:
1781:
1775:
1774:
1737:
1728:
1727:
1690:
1680:
1663:
1650:
1639:
1638:
1602:
1597:
1596:
1595:be a basis for
1569:
1550:
1542:
1541:
1538:
1532:
1527:
1499:
1498:
1479:
1478:
1459:
1458:
1439:
1438:
1407:
1406:
1383:
1382:
1359:
1358:
1339:
1338:
1309:
1308:
1284:
1273:
1272:
1232:
1231:
1207:
1196:
1195:
1176:
1175:
1151:
1146:
1145:
1117:
1116:
1097:
1096:
1074:
1073:
1067:symplectic form
1063:
1031:
1030:
1009:
1004:
1003:
982:
977:
976:
957:
956:
923:
896:
839:
834:
813:
808:
768:
759:
754:
753:
725:
720:
719:
694:
689:
688:
669:
668:
645:
644:
621:
620:
590:
565:
552:
541:
540:
510:
490:
489:
468:
463:
462:
440:
439:
416:
415:
391:
375:
370:
369:
346:
345:
321:
305:
300:
299:
275:
270:
269:
244:
243:
216:
202:
201:
182:
181:
159:
158:
135:
129:of the system.
79:
78:
50:
49:
46:smooth manifold
36:, a subject of
28:
23:
22:
15:
12:
11:
5:
8413:
8411:
8403:
8402:
8397:
8392:
8387:
8377:
8376:
8370:
8369:
8367:
8366:
8361:
8356:
8351:
8346:
8345:
8344:
8334:
8329:
8324:
8319:
8314:
8309:
8303:
8301:
8297:
8296:
8294:
8293:
8288:
8283:
8278:
8273:
8268:
8262:
8260:
8256:
8255:
8252:
8251:
8249:
8248:
8243:
8238:
8233:
8228:
8223:
8218:
8213:
8208:
8203:
8197:
8195:
8189:
8188:
8186:
8185:
8180:
8175:
8170:
8165:
8160:
8155:
8145:
8140:
8135:
8125:
8120:
8115:
8110:
8105:
8100:
8094:
8092:
8086:
8085:
8083:
8082:
8077:
8072:
8071:
8070:
8060:
8055:
8054:
8053:
8043:
8038:
8033:
8028:
8027:
8026:
8016:
8011:
8010:
8009:
7999:
7994:
7988:
7986:
7982:
7981:
7979:
7978:
7973:
7968:
7963:
7962:
7961:
7951:
7946:
7941:
7935:
7933:
7926:
7920:
7919:
7917:
7916:
7911:
7901:
7896:
7882:
7877:
7872:
7867:
7862:
7860:Parallelizable
7857:
7852:
7847:
7846:
7845:
7835:
7830:
7825:
7820:
7815:
7810:
7805:
7800:
7795:
7790:
7780:
7770:
7764:
7762:
7756:
7755:
7753:
7752:
7747:
7742:
7740:Lie derivative
7737:
7735:Integral curve
7732:
7727:
7722:
7721:
7720:
7710:
7705:
7704:
7703:
7696:Diffeomorphism
7693:
7687:
7685:
7679:
7678:
7676:
7675:
7670:
7665:
7660:
7655:
7650:
7645:
7640:
7635:
7629:
7627:
7618:
7617:
7615:
7614:
7609:
7604:
7599:
7594:
7589:
7584:
7579:
7574:
7573:
7572:
7567:
7557:
7556:
7555:
7544:
7542:
7541:Basic concepts
7538:
7537:
7527:
7525:
7524:
7517:
7510:
7502:
7496:
7495:
7480:
7461:
7439:
7418:
7413:Stack Exchange
7404:
7387:
7384:
7383:
7382:
7368:
7345:
7317:Alan Weinstein
7313:
7307:
7291:
7285:
7269:Abraham, Ralph
7265:
7250:
7238:
7232:
7214:
7211:
7208:
7207:
7177:hep-th/9904062
7150:
7137:(3): 303–330.
7117:
7110:
7104:. Birkhäuser.
7076:
7069:
7048:
7033:
7017:
7016:
7014:
7011:
7009:
7008:
6999:
6990:
6984:
6978:
6969:
6963:
6957:
6951:
6942:
6933:
6923:
6913:
6912:
6911:
6895:
6892:
6891:
6890:
6874:
6871:
6868:
6865:
6862:
6843:
6828:
6821:
6798:
6778:
6754:
6751:
6748:
6745:
6742:
6729:
6726:
6717:
6706:
6697:
6696:
6685:
6679:
6675:
6671:
6666:
6662:
6656:
6652:
6645:
6642:
6639:
6634:
6630:
6626:
6621:
6617:
6613:
6610:
6604:
6601:
6598:
6593:
6589:
6585:
6580:
6576:
6572:
6569:
6526:
6519:
6512:
6505:
6498:
6488:
6481:
6474:
6467:
6460:
6347:
6344:
6331:
6326:
6321:
6316:
6311:
6306:
6299:
6295:
6291:
6288:
6268:
6263:
6258:
6251:
6247:
6208:
6176:
6169:
6140:
6137:
6110:SYZ conjecture
6106:
6105:
6102:
6082:
6062:
6053:restricted on
6040:
6036:
6015:
5993:
5989:
5964:
5942:
5938:
5915:
5911:
5890:
5868:
5864:
5859:
5855:
5850:
5846:
5842:
5839:
5819:
5816:
5797:
5794:
5791:
5783:
5780:
5777:
5773:
5769:
5766:
5763:
5759:
5755:
5752:
5732:
5727:
5723:
5719:
5716:
5713:
5709:
5705:
5702:
5699:
5695:
5674:
5653:
5649:
5646:
5643:
5640:
5630:Morse function
5621:
5618:
5606:Floer homology
5595:
5594:
5583:
5580:
5576:
5567:
5563:
5559:
5555:
5550:
5542:
5538:
5534:
5530:
5524:
5520:
5517:
5514:
5510:
5501:
5497:
5493:
5489:
5484:
5476:
5472:
5468:
5464:
5458:
5454:
5429:
5426:
5421:
5399:
5396:
5393:
5382:
5381:
5370:
5367:
5363:
5354:
5350:
5346:
5342:
5337:
5329:
5325:
5321:
5317:
5311:
5307:
5284:
5281:
5258:
5247:
5246:
5230:
5226:
5222:
5218:
5208:
5204:
5200:
5193:
5189:
5185:
5179:
5171:
5167:
5163:
5159:
5149:
5145:
5141:
5134:
5130:
5126:
5120:
5112:
5108:
5104:
5100:
5075:
5072:
5069:
5058:
5057:
5046:
5043:
5035:
5031:
5027:
5020:
5016:
5012:
5001:
4997:
4993:
4986:
4982:
4978:
4972:
4964:
4960:
4956:
4949:
4945:
4941:
4930:
4926:
4922:
4915:
4911:
4907:
4899:
4895:
4891:
4888:
4883:
4879:
4875:
4870:
4866:
4862:
4839:
4836:
4833:
4813:
4808:
4804:
4800:
4795:
4791:
4787:
4773:
4772:
4761:
4756:
4752:
4748:
4745:
4742:
4737:
4733:
4729:
4724:
4720:
4716:
4711:
4707:
4702:
4697:
4693:
4689:
4686:
4683:
4678:
4674:
4670:
4665:
4661:
4657:
4652:
4648:
4624:
4619:
4615:
4611:
4608:
4605:
4600:
4596:
4592:
4570:
4567:
4562:
4540:
4520:
4515:
4511:
4507:
4504:
4501:
4496:
4492:
4488:
4483:
4479:
4475:
4472:
4469:
4464:
4460:
4456:
4434:
4431:
4426:
4412:
4409:
4394:
4390:
4386:
4383:
4380:
4377:
4372:
4368:
4363:
4340:
4336:
4332:
4329:
4326:
4321:
4317:
4296:
4293:
4290:
4286:
4265:
4262:
4259:
4255:
4232:
4227:
4220:
4216:
4212:
4207:
4202:
4180:
4176:
4172:
4169:
4165:
4153:
4152:
4141:
4138:
4135:
4132:
4128:
4124:
4121:
4117:
4113:
4110:
4107:
4104:
4101:
4098:
4095:
4090:
4086:
4082:
4077:
4072:
4067:
4064:
4061:
4057:
4053:
4050:
4046:
4042:
4039:
4036:
4033:
4030:
4027:
4024:
4021:
4016:
4012:
3988:
3983:
3979:
3967:
3966:
3955:
3952:
3949:
3946:
3943:
3940:
3935:
3931:
3927:
3922:
3917:
3912:
3909:
3906:
3903:
3900:
3897:
3894:
3891:
3888:
3872:
3869:
3856:
3851:
3847:
3843:
3840:
3836:
3824:
3823:
3812:
3809:
3804:
3800:
3796:
3793:
3789:
3785:
3780:
3776:
3772:
3769:
3765:
3761:
3758:
3753:
3749:
3745:
3742:
3738:
3734:
3729:
3725:
3721:
3718:
3714:
3710:
3707:
3704:
3701:
3698:
3695:
3692:
3689:
3686:
3681:
3678:
3673:
3670:
3665:
3661:
3657:
3654:
3651:
3648:
3645:
3640:
3636:
3632:
3629:
3626:
3623:
3620:
3617:
3614:
3611:
3608:
3605:
3602:
3599:
3596:
3573:
3569:
3565:
3562:
3558:
3554:
3551:
3531:
3526:
3522:
3518:
3515:
3512:
3509:
3506:
3503:
3500:
3495:
3491:
3487:
3484:
3481:
3478:
3475:
3472:
3452:
3449:
3446:
3426:
3423:
3420:
3417:
3414:
3411:
3408:
3405:
3385:
3378:
3374:
3369:
3365:
3355:
3350:
3346:
3342:
3339:
3336:
3329:
3325:
3320:
3316:
3306:
3301:
3297:
3293:
3290:
3268:
3262:
3256:
3234:
3212:
3209:
3203:
3199:
3195:
3189:
3184:
3179:
3173:
3167:
3154:
3153:
3142:
3137:
3133:
3128:
3124:
3119:
3115:
3110:
3106:
3103:
3100:
3095:
3091:
3086:
3082:
3077:
3073:
3068:
3064:
3061:
3036:
3033:
3021:
3009:
2987:
2982:
2978:
2974:
2971:
2968:
2963:
2959:
2955:
2950:
2946:
2942:
2939:
2936:
2931:
2927:
2923:
2901:
2898:
2886:
2874:
2860:
2857:
2817:
2816:
2804:
2801:
2798:
2792:
2789:
2783:
2780:
2755:
2752:
2747:
2742:
2737:
2717:
2714:
2711:
2691:
2671:
2668:
2665:
2662:
2659:
2645:
2631:
2619:
2597:
2592:
2587:
2567:
2564:
2561:
2541:
2514:
2511:
2508:
2505:
2502:
2486:
2483:
2458:
2443:
2440:
2425:
2420:
2417:
2386:
2381:
2378:
2373:
2370:
2343:
2340:
2325:
2321:
2317:
2291:
2287:
2252:
2248:
2244:
2241:
2238:
2235:
2230:
2226:
2222:
2202:
2197:
2193:
2189:
2186:
2183:
2178:
2174:
2170:
2150:
2130:
2125:
2121:
2117:
2114:
2111:
2106:
2102:
2098:
2087:
2086:
2073:
2069:
2065:
2062:
2057:
2053:
2049:
2044:
2039:
2036:
2033:
2029:
2025:
2022:
1996:
1991:
1987:
1963:
1943:
1931:
1928:
1927:
1926:
1915:
1910:
1904:
1901:
1897:
1893:
1889:
1886:
1885:
1880:
1876:
1872:
1870:
1867:
1866:
1864:
1859:
1856:
1815:
1810:quadratic form
1806:
1805:
1792:
1782:
1780:
1777:
1776:
1773:
1770:
1767:
1764:
1761:
1753:
1750:
1747:
1744:
1741:
1738:
1736:
1733:
1730:
1729:
1726:
1723:
1720:
1717:
1714:
1706:
1703:
1700:
1697:
1694:
1691:
1689:
1686:
1685:
1683:
1678:
1675:
1670:
1666:
1662:
1657:
1653:
1649:
1646:
1619:
1614:
1611:
1606:
1584:
1579:
1576:
1572:
1568:
1565:
1562:
1557:
1553:
1549:
1534:Main article:
1531:
1528:
1526:
1523:
1506:
1486:
1466:
1446:
1426:
1423:
1420:
1417:
1414:
1390:
1366:
1346:
1322:
1319:
1316:
1296:
1291:
1287:
1283:
1280:
1260:
1257:
1254:
1251:
1248:
1245:
1242:
1239:
1219:
1214:
1210:
1206:
1203:
1183:
1163:
1158:
1154:
1130:
1127:
1124:
1104:
1081:
1062:
1059:
1038:
1016:
1012:
989:
985:
964:
953:
952:
941:
938:
935:
930:
926:
922:
919:
915:
911:
908:
903:
899:
895:
892:
888:
884:
881:
878:
873:
869:
865:
861:
858:
854:
846:
842:
837:
833:
830:
827:
820:
816:
811:
807:
803:
798:
794:
791:
788:
785:
782:
775:
771:
764:
732:
728:
701:
697:
676:
665:Lie derivative
652:
628:
608:
605:
602:
597:
593:
589:
586:
583:
580:
577:
572:
568:
564:
559:
555:
551:
548:
528:
525:
522:
517:
513:
509:
506:
503:
500:
497:
475:
471:
450:
447:
436:non-degenerate
423:
403:
398:
394:
390:
387:
382:
378:
353:
333:
328:
324:
320:
317:
312:
308:
287:
282:
278:
254:
251:
228:
223:
219:
215:
212:
209:
189:
169:
166:
134:
131:
125:describes the
86:
57:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
8412:
8401:
8398:
8396:
8393:
8391:
8388:
8386:
8383:
8382:
8380:
8365:
8362:
8360:
8359:Supermanifold
8357:
8355:
8352:
8350:
8347:
8343:
8340:
8339:
8338:
8335:
8333:
8330:
8328:
8325:
8323:
8320:
8318:
8315:
8313:
8310:
8308:
8305:
8304:
8302:
8298:
8292:
8289:
8287:
8284:
8282:
8279:
8277:
8274:
8272:
8269:
8267:
8264:
8263:
8261:
8257:
8247:
8244:
8242:
8239:
8237:
8234:
8232:
8229:
8227:
8224:
8222:
8219:
8217:
8214:
8212:
8209:
8207:
8204:
8202:
8199:
8198:
8196:
8194:
8190:
8184:
8181:
8179:
8176:
8174:
8171:
8169:
8166:
8164:
8161:
8159:
8156:
8154:
8150:
8146:
8144:
8141:
8139:
8136:
8134:
8130:
8126:
8124:
8121:
8119:
8116:
8114:
8111:
8109:
8106:
8104:
8101:
8099:
8096:
8095:
8093:
8091:
8087:
8081:
8080:Wedge product
8078:
8076:
8073:
8069:
8066:
8065:
8064:
8061:
8059:
8056:
8052:
8049:
8048:
8047:
8044:
8042:
8039:
8037:
8034:
8032:
8029:
8025:
8024:Vector-valued
8022:
8021:
8020:
8017:
8015:
8012:
8008:
8005:
8004:
8003:
8000:
7998:
7995:
7993:
7990:
7989:
7987:
7983:
7977:
7974:
7972:
7969:
7967:
7964:
7960:
7957:
7956:
7955:
7954:Tangent space
7952:
7950:
7947:
7945:
7942:
7940:
7937:
7936:
7934:
7930:
7927:
7925:
7921:
7915:
7912:
7910:
7906:
7902:
7900:
7897:
7895:
7891:
7887:
7883:
7881:
7878:
7876:
7873:
7871:
7868:
7866:
7863:
7861:
7858:
7856:
7853:
7851:
7848:
7844:
7841:
7840:
7839:
7836:
7834:
7831:
7829:
7826:
7824:
7821:
7819:
7816:
7814:
7811:
7809:
7806:
7804:
7801:
7799:
7796:
7794:
7791:
7789:
7785:
7781:
7779:
7775:
7771:
7769:
7766:
7765:
7763:
7757:
7751:
7748:
7746:
7743:
7741:
7738:
7736:
7733:
7731:
7728:
7726:
7723:
7719:
7718:in Lie theory
7716:
7715:
7714:
7711:
7709:
7706:
7702:
7699:
7698:
7697:
7694:
7692:
7689:
7688:
7686:
7684:
7680:
7674:
7671:
7669:
7666:
7664:
7661:
7659:
7656:
7654:
7651:
7649:
7646:
7644:
7641:
7639:
7636:
7634:
7631:
7630:
7628:
7625:
7621:Main results
7619:
7613:
7610:
7608:
7605:
7603:
7602:Tangent space
7600:
7598:
7595:
7593:
7590:
7588:
7585:
7583:
7580:
7578:
7575:
7571:
7568:
7566:
7563:
7562:
7561:
7558:
7554:
7551:
7550:
7549:
7546:
7545:
7543:
7539:
7534:
7530:
7523:
7518:
7516:
7511:
7509:
7504:
7503:
7500:
7491:
7486:
7481:
7477:
7470:
7466:
7462:
7457:
7452:
7448:
7444:
7440:
7437:
7433:
7432:
7427:
7423:
7419:
7415:
7414:
7409:
7405:
7400:
7395:
7390:
7389:
7385:
7379:
7375:
7371:
7365:
7361:
7357:
7353:
7352:
7346:
7341:
7336:
7333:(3): 329–46.
7332:
7328:
7327:
7322:
7318:
7314:
7310:
7308:3-7643-7574-4
7304:
7300:
7296:
7292:
7288:
7286:0-8053-0102-X
7282:
7278:
7274:
7270:
7266:
7259:
7255:
7251:
7247:
7243:
7242:Auroux, Denis
7239:
7235:
7233:0-19-850451-9
7229:
7225:
7221:
7217:
7216:
7212:
7203:
7199:
7195:
7191:
7187:
7183:
7178:
7173:
7169:
7165:
7161:
7154:
7151:
7145:
7140:
7136:
7132:
7128:
7121:
7118:
7113:
7111:0-8176-3187-9
7107:
7103:
7099:
7095:
7091:
7090:Arnold, V. I.
7085:
7083:
7081:
7077:
7072:
7070:3-7643-7574-4
7066:
7062:
7055:
7053:
7049:
7044:
7041:Cohn, Henry.
7037:
7034:
7029:
7022:
7019:
7012:
7003:
7000:
6994:
6991:
6988:
6985:
6982:
6979:
6973:
6970:
6967:
6964:
6961:
6958:
6955:
6952:
6946:
6943:
6937:
6934:
6927:
6924:
6918:
6915:
6914:
6909:
6903:
6898:
6893:
6888:
6869:
6866:
6863:
6852:
6848:
6844:
6841:
6837:
6833:
6829:
6826:
6822:
6819:
6815:
6811:
6807:
6803:
6799:
6796:
6792:
6776:
6768:
6749:
6746:
6743:
6732:
6731:
6727:
6725:
6723:
6716:
6712:
6705:
6702:
6683:
6677:
6673:
6669:
6664:
6660:
6654:
6650:
6643:
6640:
6637:
6632:
6628:
6624:
6619:
6615:
6611:
6608:
6602:
6599:
6596:
6591:
6587:
6583:
6578:
6574:
6570:
6567:
6560:
6559:
6558:
6556:
6552:
6548:
6544:
6540:
6537:
6533:
6525:
6518:
6511:
6504:
6497:
6487:
6480:
6473:
6466:
6459:
6452:
6450:
6446:
6442:
6438:
6434:
6429:
6425:
6421:
6417:
6413:
6407:
6402:
6398:
6394:
6389:
6385:
6380:
6376:
6372:
6368:
6364:
6360:
6352:
6345:
6343:
6329:
6324:
6309:
6297:
6293:
6289:
6286:
6266:
6261:
6249:
6245:
6237:
6234:as being the
6233:
6229:
6225:
6221:
6216:
6211:
6207:
6203:
6198:
6194:
6188:
6184:
6179:
6175:
6168:
6162:
6158:
6154:
6150:
6146:
6138:
6136:
6134:
6130:
6126:
6121:
6119:
6115:
6111:
6103:
6100:
6096:
6095:
6094:
6080:
6060:
6038:
6013:
5991:
5978:
5962:
5940:
5913:
5888:
5866:
5853:
5848:
5840:
5829:
5825:
5817:
5814:
5809:
5792:
5781:
5775:
5767:
5764:
5753:
5750:
5730:
5725:
5721:
5717:
5711:
5703:
5700:
5672:
5644:
5641:
5638:
5631:
5627:
5619:
5617:
5615:
5611:
5607:
5603:
5598:
5581:
5578:
5574:
5565:
5561:
5548:
5540:
5536:
5522:
5518:
5515:
5512:
5508:
5499:
5495:
5482:
5474:
5470:
5456:
5452:
5445:
5444:
5443:
5427:
5424:
5397:
5394:
5391:
5368:
5365:
5361:
5352:
5348:
5335:
5327:
5323:
5309:
5305:
5298:
5297:
5296:
5282:
5279:
5272:
5256:
5228:
5224:
5206:
5202:
5191:
5187:
5177:
5169:
5165:
5147:
5143:
5132:
5128:
5118:
5110:
5106:
5089:
5088:
5087:
5073:
5070:
5067:
5044:
5041:
5033:
5029:
5018:
5014:
4999:
4995:
4984:
4980:
4970:
4962:
4958:
4947:
4943:
4928:
4924:
4913:
4909:
4897:
4893:
4889:
4881:
4877:
4873:
4868:
4864:
4853:
4852:
4851:
4837:
4834:
4831:
4806:
4802:
4798:
4793:
4789:
4778:
4754:
4750:
4746:
4743:
4740:
4735:
4731:
4722:
4718:
4714:
4709:
4705:
4695:
4691:
4687:
4684:
4681:
4676:
4672:
4663:
4659:
4655:
4650:
4646:
4638:
4637:
4636:
4617:
4613:
4609:
4606:
4603:
4598:
4594:
4568:
4565:
4538:
4513:
4509:
4505:
4502:
4499:
4494:
4490:
4486:
4481:
4477:
4473:
4470:
4467:
4462:
4458:
4432:
4429:
4410:
4408:
4392:
4388:
4384:
4381:
4378:
4375:
4370:
4366:
4338:
4334:
4330:
4327:
4324:
4319:
4315:
4294:
4291:
4288:
4263:
4260:
4257:
4230:
4218:
4214:
4210:
4205:
4178:
4170:
4167:
4136:
4133:
4130:
4122:
4119:
4111:
4108:
4105:
4102:
4099:
4096:
4093:
4088:
4084:
4080:
4075:
4065:
4059:
4051:
4048:
4040:
4037:
4034:
4031:
4022:
4019:
4014:
4010:
4002:
4001:
4000:
3986:
3981:
3977:
3953:
3947:
3944:
3941:
3938:
3933:
3929:
3925:
3920:
3910:
3904:
3901:
3898:
3889:
3886:
3879:
3878:
3877:
3870:
3868:
3849:
3838:
3802:
3791:
3778:
3767:
3759:
3751:
3740:
3727:
3716:
3702:
3696:
3690:
3684:
3679:
3676:
3671:
3663:
3652:
3646:
3643:
3638:
3627:
3621:
3615:
3612:
3606:
3603:
3600:
3594:
3587:
3586:
3585:
3571:
3563:
3560:
3552:
3549:
3529:
3524:
3513:
3507:
3504:
3501:
3498:
3493:
3482:
3476:
3473:
3470:
3450:
3447:
3444:
3424:
3421:
3415:
3412:
3409:
3403:
3396:we have that
3383:
3376:
3372:
3348:
3344:
3340:
3337:
3334:
3327:
3323:
3299:
3295:
3291:
3288:
3266:
3232:
3210:
3207:
3197:
3177:
3140:
3135:
3131:
3122:
3117:
3113:
3104:
3101:
3098:
3093:
3089:
3080:
3075:
3071:
3062:
3059:
3052:
3051:
3050:
3034:
3031:
3019:
2980:
2976:
2972:
2969:
2966:
2961:
2957:
2953:
2948:
2944:
2940:
2937:
2934:
2929:
2925:
2899:
2896:
2884:
2858:
2856:
2854:
2850:
2849:Betti numbers
2846:
2840:
2836:
2832:
2828:
2822:
2802:
2799:
2796:
2790:
2787:
2781:
2778:
2753:
2750:
2745:
2735:
2715:
2712:
2709:
2689:
2666:
2663:
2660:
2649:
2646:
2643:
2639:
2635:
2632:
2617:
2595:
2585:
2565:
2562:
2559:
2539:
2531:
2528:
2527:
2526:
2509:
2506:
2503:
2492:
2484:
2482:
2480:
2476:
2472:
2456:
2448:
2441:
2439:
2423:
2406:
2402:
2384:
2371:
2368:
2361:
2357:
2353:
2349:
2341:
2339:
2323:
2319:
2315:
2307:
2289:
2285:
2276:
2272:
2271:metric tensor
2268:
2250:
2246:
2242:
2239:
2236:
2233:
2228:
2224:
2220:
2195:
2191:
2187:
2184:
2181:
2176:
2172:
2148:
2123:
2119:
2115:
2112:
2109:
2104:
2100:
2071:
2067:
2063:
2060:
2055:
2051:
2047:
2042:
2037:
2034:
2031:
2027:
2023:
2020:
2013:
2012:
2011:
2010:
1994:
1989:
1985:
1977:
1961:
1941:
1929:
1913:
1908:
1902:
1895:
1891:
1887:
1878:
1874:
1868:
1862:
1857:
1847:
1846:
1845:
1843:
1839:
1835:
1829:
1826:
1822:
1818:
1811:
1778:
1771:
1768:
1765:
1762:
1759:
1751:
1748:
1745:
1742:
1739:
1734:
1731:
1724:
1721:
1718:
1715:
1712:
1704:
1701:
1698:
1695:
1692:
1687:
1681:
1676:
1668:
1664:
1660:
1655:
1651:
1644:
1637:
1636:
1635:
1633:
1617:
1612:
1609:
1577:
1574:
1570:
1566:
1563:
1560:
1555:
1551:
1537:
1529:
1524:
1522:
1520:
1504:
1484:
1464:
1444:
1421:
1418:
1415:
1404:
1388:
1380:
1364:
1344:
1336:
1320:
1317:
1314:
1294:
1289:
1285:
1281:
1278:
1258:
1255:
1249:
1246:
1243:
1237:
1217:
1212:
1208:
1204:
1201:
1181:
1161:
1156:
1152:
1144:
1143:tangent space
1128:
1125:
1122:
1102:
1095:
1079:
1072:
1068:
1060:
1058:
1056:
1052:
1036:
1014:
1010:
987:
983:
962:
939:
936:
928:
924:
917:
909:
901:
897:
890:
882:
876:
859:
856:
844:
840:
835:
831:
825:
818:
814:
809:
792:
789:
783:
773:
769:
752:
751:
750:
748:
730:
726:
717:
699:
695:
674:
666:
650:
642:
626:
606:
603:
595:
591:
584:
581:
578:
570:
566:
562:
557:
553:
546:
523:
520:
515:
511:
504:
501:
498:
495:
473:
469:
448:
445:
437:
421:
401:
396:
392:
388:
385:
380:
376:
367:
351:
331:
326:
322:
318:
315:
310:
306:
285:
280:
276:
268:
252:
249:
242:
226:
221:
217:
210:
207:
187:
167:
164:
156:
152:
148:
144:
140:
132:
130:
128:
124:
120:
116:
112:
108:
104:
100:
84:
77:
74:
73:nondegenerate
71:
55:
47:
43:
39:
35:
30:
19:
8286:Moving frame
8281:Morse theory
8271:Gauge theory
8063:Tensor field
7992:Closed/Exact
7971:Vector field
7939:Distribution
7908:
7880:Hypercomplex
7875:Quaternionic
7612:Vector field
7570:Smooth atlas
7490:math/9907034
7475:
7446:
7429:
7411:
7350:
7330:
7324:
7298:
7276:
7223:
7220:McDuff, Dusa
7167:
7163:
7153:
7134:
7130:
7120:
7101:
7060:
7036:
7021:
6846:
6839:
6835:
6831:
6766:
6721:
6714:
6709:denotes the
6703:
6700:
6698:
6554:
6546:
6542:
6538:
6531:
6523:
6516:
6509:
6502:
6495:
6485:
6478:
6471:
6464:
6457:
6453:
6447:is called a
6444:
6440:
6436:
6432:
6427:
6423:
6419:
6415:
6411:
6405:
6400:
6396:
6392:
6387:
6386:is called a
6383:
6378:
6374:
6370:
6362:
6358:
6356:
6231:
6214:
6209:
6205:
6201:
6196:
6186:
6182:
6177:
6173:
6166:
6160:
6148:
6144:
6142:
6122:
6118:Hitchin 1999
6107:
5976:
5821:
5626:Morse theory
5623:
5610:Morse theory
5602:local charts
5599:
5596:
5383:
5248:
5059:
4774:
4414:
4154:
3968:
3874:
3825:
3245:vanishes on
3155:
2862:
2838:
2834:
2830:
2826:
2818:
2647:
2642:co-isotropic
2641:
2633:
2529:
2488:
2469:-compatible
2445:
2345:
2267:phase spaces
2088:
1933:
1842:block matrix
1837:
1833:
1824:
1820:
1819:denotes the
1813:
1807:
1631:
1539:
1518:
1402:
1401:vanishes. A
1069:on a smooth
1066:
1064:
1050:
954:
155:vector field
136:
41:
31:
29:
8231:Levi-Civita
8221:Generalized
8193:Connections
8143:Lie algebra
8075:Volume form
7976:Vector flow
7949:Pushforward
7944:Lie bracket
7843:Lie algebra
7808:G-structure
7597:Pushforward
7577:Submanifold
6530:are called
6204:= ∑ d
4635:such that
3225:. The form
2491:submanifold
2473:are termed
1174:defined by
641:alternating
143:phase space
127:phase space
38:mathematics
8379:Categories
8354:Stratifold
8312:Diffeology
8108:Associated
7909:Symplectic
7894:Riemannian
7823:Hyperbolic
7750:Submersion
7658:Hopf–Rinow
7592:Submersion
7587:Smooth map
7465:McDuff, D.
7422:Lumist, Ü.
7399:2206.14792
7133:. Ser. A.
6834:of degree
6818:integrable
6806:compatible
5975:is called
5811:See also:
5628:. Given a
2578:such that
2479:integrable
1405:is a pair
1230:such that
1061:Definition
1053:should be
488:such that
344:. Letting
133:Motivation
8236:Principal
8211:Ehresmann
8168:Subbundle
8158:Principal
8133:Fibration
8113:Cotangent
7985:Covectors
7838:Lie group
7818:Hermitian
7761:manifolds
7730:Immersion
7725:Foliation
7663:Noether's
7648:Frobenius
7643:De Rham's
7638:Darboux's
7529:Manifolds
7456:0908.1886
7436:EMS Press
7424:(2001) ,
7202:204899025
7013:Citations
6797:is exact.
6777:ω
6750:ω
6711:pull back
6674:ω
6661:ω
6655:∗
6651:τ
6641:τ
6638:∘
6629:π
6616:π
6612:∘
6609:ν
6600:σ
6597:∘
6571:∘
6568:τ
6508:) :
6470:) :
6418:) :
6367:immersion
6315:→
6298:∗
6287:π
6250:∗
6213:∧ d
6153:fibration
6035:Ω
5988:Ω
5937:Ω
5910:Ω
5863:Ω
5845:Ω
5838:Ω
5768:⋅
5765:ε
5754:∩
5726:∗
5718:⊂
5704:⋅
5701:ε
5673:ε
5648:→
5558:∂
5554:∂
5533:∂
5529:∂
5519:ω
5516:−
5492:∂
5488:∂
5467:∂
5463:∂
5453:ω
5345:∂
5341:∂
5320:∂
5316:∂
5306:ω
5221:∂
5217:∂
5199:∂
5184:∂
5162:∂
5158:∂
5140:∂
5125:∂
5103:∂
5099:∂
5026:∂
5011:∂
4992:∂
4977:∂
4971:−
4955:∂
4940:∂
4921:∂
4906:∂
4894:∑
4744:…
4685:…
4607:…
4503:…
4471:…
4379:…
4328:…
4219:∗
4123:−
4094:−
4066:∈
4015:∗
3982:∗
3939:−
3911:∈
3846:∂
3799:∂
3775:∂
3760:−
3748:∂
3724:∂
3660:∂
3635:∂
3616:ω
3595:ω
3564:∧
3550:ω
3521:∂
3490:∂
3404:ω
3368:∂
3319:∂
3233:ω
3183:→
3123:∧
3102:⋯
3081:∧
3060:ω
2970:…
2938:…
2800:
2775:dim
2736:ω
2713:⊂
2690:ω
2667:ω
2586:ω
2563:⊂
2510:ω
2457:ω
2372:⊂
2237:…
2185:…
2113:…
2061:∧
2028:∑
2021:ω
1990:∗
1888:−
1855:Ω
1785:otherwise
1769:⩽
1763:⩽
1743:−
1732:−
1722:⩽
1716:⩽
1696:−
1645:ω
1564:…
1465:ω
1422:ω
1389:ω
1345:ω
1282:∈
1238:ω
1205:∈
1182:ω
1126:∈
1103:ω
918:ω
891:ω
857:ω
836:ι
826:ω
810:ι
797:⇔
784:ω
727:ι
675:ω
651:ω
627:ω
547:ω
524:⋅
505:ω
422:ω
397:∗
389:⊗
381:∗
364:denote a
352:ω
327:∗
319:⊗
311:∗
281:∗
239:from the
222:∗
214:→
85:ω
8332:Orbifold
8327:K-theory
8317:Diffiety
8041:Pullback
7855:Oriented
7833:Kenmotsu
7813:Hadamard
7759:Types of
7708:Geodesic
7533:Glossary
7378:22509804
7319:(1971).
7297:(2006).
7275:(1978).
7100:(1985).
6894:See also
6804:that is
6395: :
6373: :
5384:for all
5060:for all
3463:. Then,
2859:Examples
2449:with an
2306:soldered
1525:Examples
1271:for all
1071:manifold
8276:History
8259:Related
8173:Tangent
8151:)
8131:)
8098:Adjoint
8090:Bundles
8068:density
7966:Torsion
7932:Vectors
7924:Tensors
7907:)
7892:)
7888:,
7886:Pseudo−
7865:Poisson
7798:Finsler
7793:Fibered
7788:Contact
7786:)
7778:Complex
7776:)
7745:Section
7182:Bibcode
6551:commute
6449:caustic
6390:). Let
6191:and by
6131:on the
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5977:special
5612:to the
2403:on the
1307:, then
745:is the
366:section
265:to the
113:as the
8241:Vector
8226:Koszul
8206:Cartan
8201:Affine
8183:Vector
8178:Tensor
8163:Spinor
8153:Normal
8149:Stable
8103:Affine
8007:bundle
7959:bundle
7905:Almost
7828:Kähler
7784:Almost
7774:Almost
7768:Closed
7668:Sard's
7624:(list)
7376:
7366:
7305:
7283:
7230:
7200:
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7067:
6842:-form.
6802:metric
6699:where
6647:
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6553:, and
6496:π
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6435:. The
6412:π
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6157:fibres
1437:where
1094:2-form
1055:closed
687:along
70:closed
8349:Sheaf
8123:Fiber
7899:Rizza
7870:Prime
7701:Local
7691:Curve
7553:Atlas
7485:arXiv
7472:(PDF)
7451:arXiv
7394:arXiv
7261:(PDF)
7198:S2CID
7172:arXiv
6849:is a
6791:exact
6767:exact
6431:is a
6185:,...,
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2304:are "
2089:Here
1812:. If
44:is a
8216:Form
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7803:Flat
7683:Maps
7374:OCLC
7364:ISBN
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7281:ISBN
7228:ISBN
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7065:ISBN
6545:and
6492:and
6357:Let
6123:The
6120:).
6108:The
5826:(or
5786:Crit
4276:and
3542:and
2863:Let
2766:and
2161:and
1934:Let
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109:and
40:, a
8138:Jet
7356:doi
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6522:↠
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4887:]
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