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Moerdijk I., Mrcun J. — Introduction to Foliations and Lie Groupoids
Moerdijk I., Mrcun J. — Introduction to Foliations and Lie Groupoids

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Название: Introduction to Foliations and Lie Groupoids

Авторы: Moerdijk I., Mrcun J.

Аннотация:

The relations between foliations, Lie groupoids, and Lie algebroids are a main emphasis of this quick introduction for students who are familiar with the basic concepts of differential geometry. Lie groupoids are the main tool for studying the transversal structure of a foliation, by means of its holonomy groupoid; foliations are also a special kind of Lie algebroids; and the elementary theory of foliations is a useful tool for studying Lie groupoids and Lie algebroids. And that is only the surface.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 1st edition

Год издания: 2003

Количество страниц: 173

Добавлена в каталог: 27.05.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$(K, \epsilon)$-cocycle      50
$(K, \epsilon)$-cocycle, normalized      50
$G_x$      112
$hol(\alpha)$      21
$\Gamma (M )$      134
$\Gamma^q$      134
$\mathfrak{X}(F )$      10 83
$\mathfrak{X}^s_{inv} (G)$      150
$\mathfrak{X}_{bas} (F )$      84
$\Omega ^0_{bas}(M, F )$      83
$\phi^*(G)$      121
Action of a Lie groupoid on a Lie groupoid      126
Action of a Lie groupoid on a manifold      125
Anchor of a Lie algebroid      151 153
Arrows of a groupoid      111
Atiyah sequence      156
Atlas      1
Atlas, foliation      5
Atlas, orbifold      38
Aut(M, F )      82
Automorphism of a foliated manifold      82
B(M, F )      161
Base manifold of a Lie algebroid      153
Base manifold of a Lie groupoid      112
Basic function      83
Bis(G)      115
Bisection, global      114
Bisection, local      114
Bundle map      93
Bundle, complex frame, of an orbifold      44
Bundle, flat      17
Bundle, foliated principal G-      98
Bundle, frame      93
Bundle, frame, of a manifold      42
Bundle, frame, of an orbifold      43
Bundle, normal, of a foliation      12
Bundle, orientable      3
Bundle, orthogonal frame      43
Bundle, orthogonal frame, of an orbifold      43
Bundle, positive frame, of an orbifold      43
Bundle, positive orthogonal frame, of an orbifold      43
Bundle, principal G-      93 145
Bundle, principal G-, trivial      93
Bundle, tangent, of a foliation      10
Bundle, tangent, of a manifold      1
Bundle, transverse frame,      99
Bundle, transverse orthogonal frame      100
Bundle, transverse principal G-      98
Bundle, trivial principal G-      145
Bundle, unit principal G-      145
Bundle, unitary frame, of an orbifold      44
Canonical form      94
Canonical form on OF (M )      94
Canonical form, transverse      100
Canonical local form      2
Canonical local form of a submersion      2
Canonical local form of an immersion      2
Canonical parallelism      98
Change-of-charts homeomorphism      1
Chart      1
Chart, foliation —      5
Chart, orbifold —      37
Codimension of a foliation      5
Compatible orbifold charts      38
Connection      94
Connection, F -partial      148
Connection, flat      95
Connection, flat F -partial      148
Connection, Levi-Civita      97
Connection, torsion free      97
Connection, transverse Levi-Civita      101
Connection,projectable      99
Critical point      58
Critical value      58
Curvature form      95
Darboux cover      104
De Rham cohomology      2
Derivative      2
Diff$^+_x$ (M )      21
Diff$_x$ (M )      21
Diff(M )      35
Diffeomorphism      2
Diffeomorphism, local      2 134
Differential form      2
Differential form, integrable      12
Differential form, Maurer — Cartan      102
Differential ideal      9
EFF      136
Effect of an etale groupoid      137
Embedding      3
Embedding between orbifold charts      37
Equivalence of categories      127
Equivalence of pseudogroups      139
Equivalence, Morita, of Lie groupoids      130
Equivalence, strong, of Lie groupoids      128
Equivalence, weak, of Lie groupoids      128 130
Equivariant map of principal bundles      145
Essentially surjective functor      127
Etale map      2 134
Exterior differentiation      2
Exterior product      2
F$_{bas}$      84
Fibred product of Lie groupoids, strong      123
Fibred product of Lie groupoids, weak      124
Flow      16
Foliated action of a Lie group      16
Foliation      5
Foliation of the Moebius band      6
Foliation, associated to a Lie group action      16
Foliation, associated to a submersion      6
Foliation, basic      84
Foliation, developable      161
Foliation, homogeneous      82
Foliation, invariant      15
Foliation, Kronecker      6
Foliation, Lie      102
Foliation, lifted      99
Foliation, orientable      12
Foliation, quotient      15
Foliation, Reeb      7
Foliation, Reeb, of $S^3$      8
Foliation, Riemannian      26
Foliation, simple      6
Foliation, strictly simple      6
Foliation, transversely orientable      12
Foliation, transversely parallelizable      88
Foliation, trivial      6
Full and faithful functor      127
G(x, y)      112
Gauge group      114
Germ      20
Germ, one-sided      57
Godbillon — Vey class      13
Groupoid      111
Groupoid, action      113
Groupoid, effective      137
Groupoid, etale      134
Groupoid, etale holonomy      135
Groupoid, etale monodromy      135
Groupoid, fundamental      113
Groupoid, gauge      114
Groupoid, Haefliger      134
Groupoid, Hausdorff Lie      112
Groupoid, holonomy      118
Groupoid, induced      121
Groupoid, kernel      113
Groupoid, Lie      112
Groupoid, monodromy —      117
Groupoid, of an orbifold      142
Groupoid, over $G_0$      112
Groupoid, pair      113
Groupoid, proper effective      141
Groupoid, source-connected Lie      157
Groupoid, source-simply-connected Lie      157
Groupoid, transitive Lie      129 132
Groupoid, translation      113 125
Groupoid, unit      113
Groupoid, weakly equivalent to a discrete group      133
Haefliger cocycle      9
Haefliger’s theorem      57
Hessian matrix      58
Hol(L, x)      23
Hol(M, F )      118
Holonomy      21
Holonomy class      23
Holonomy cover      31
Holonomy group      23
Holonomy group of a flat connection      96
Holonomy group of a Lie foliation      103
Holonomy group, linear      23
Holonomy groupoid      118
Holonomy groupoid, etale      135
Holonomy homomorphism      23
Holonomy homomorphism of a Lie foliation      104
Holonomy homomorphism, linear      23
Holonomy pseudogroup      139
Holonomy, one-sided      57
Homomorphism of groupoids      112
Homomorphism of Lie groupoids      112
Horizontal vectors      95
Immersion      2
Infinitesimal action      155
Integrable subbundle      9
Integral of a Lie algebroid      154
Inverse map      111
Iso$_z$(Q)      40
Isomorphism of Lie groupoids      127
Isotropy groups, discrete      136
Isotropy in a groupoid      112
Isotropy of a leaf      16
Isotropy of a point      16
Isotropy of a point of an orbifold      40
Isotropy of a subset      35
L(M, F )      86
Leaf      5
Leibniz identity      153
Lie algebroid      153
Lie algebroid of a Lie groupoid      152
Lie algebroid, basic, of a transversely parallelizable foliation      161
Lie algebroid, integrable      154
Lie algebroid, regular      156
Lie algebroid, transformation      155
Lie algebroid, transitive      156
Lie subalgebroid      158
Line field      12
Manifold      1
Manifold, foliated      6
Manifold, homogeneous      82
Manifold, non-Hausdorff      1
Manifold, non-second-countable      1
Metric, bundle-like      29
Metric, Riemannian      3
Metric, Riemannian, on an orbifold      41
Metric, transverse      25
Molino’s structure theorem      108
Mon(M, F )      117
Morphism of Lie algebroids      153
Morse function      58 60
Morse index      59
Morse lemma      58
Morse theory      58
Multiplication map      111
Natural transformation      122
Novikov’s theorem      66
Objects of a groupoid      111
Orbifold      39 141
Orbifold map      39
Orbifold, orientable      43
Orbit in a Lie groupoid      115
Orbit of a Lie group action      16
Orientation cover of a foliation      15
Orientation cover of a foliation, transverse      15
Orientation of a bundle      3
Orientation of a foliation      12
Orientation of an orbifold      43
Orientation, transverse, of a foliation      12
Plaque      5
Poincare — Hopf theorem      61
Poisson manifold      156
Product of foliations      14
Product of Lie groupoids      123
Proper groupoid      140
Proper groupoid, effective      141
Proper map      140
Pseudogroup      138
Pseudogroup, holonomy      139
Pull-back of a foliation      14
Pull-back of a principal bundle      145
Reeb component      66
Reeb stability theorem, global      45
Reeb stability theorem, local      31
Reeb — Thurston stability theorem, global      50
Reeb — Thurston stability theorem, local      50
Saturated subset      31
Semi-direct product of Lie groupoids      125 126
Singular locus      40
Singularity      58
Singularity, Morse      58
Singularity, non-degenerate      58
Source of an arrow      111
Space of leaves      6
Space of orbits of a Lie groupoid      143
Stabilizer subgroup      16
Stable subset      35
Structure maps of a groupoid      111
Submanifold      3
Submersion      2
Sum of Lie groupoids      123
Suspension      16
Tangent space      1
Target of an arrow      111
Torsion form      97
Transition      138 139
Transversal section      21
Transversal section, complete      135
Transversality      3
Transverse parallelism      88
Tubular neighbourhood      3
Unit map      111
Units of a groupoid      112
Vanishing cycle      66
Vanishing cycle. negative      66
Vanishing cycle. positive      66
Vanishing cycle. simple      70
Vector field      2
Vector field, basic      29
Vector field, foliated      29
Vector field, G-invariant      150
Vector field, normal      29
Vector field, projectable      29 86
Vector field, tangent      10
Vector field, transverse      86
Vertical vectors      95
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