Kumpera A., Spencer D.C. — Lie Equations: General Theory. Vol. 1 :: Электронная библиотека попечительского совета мехмата МГУ

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Kumpera A., Spencer D.C. — Lie Equations: General Theory. Vol. 1

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Название: Lie Equations: General Theory. Vol. 1

Авторы: Kumpera A., Spencer D.C.

Аннотация:

As the title indicates, the content of these notes is a lengthy construction of techniques devised to study specific differential geometric problems. In this introduction we state our main objectives and illustrate by examples some of their geometric implications.

Язык:

Рубрика: Математика/Симметрия и группы/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1972

Количество страниц: 293

Добавлена в каталог: 30.04.2005

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Предметный указатель

$\delta$ -cohomology      34 83 85—86
$\delta$ -complex      24 83
$\delta$ -Poincare lemma      85—86
$\Gamma_{G}$ -structure      42
$\Gamma_{G}$ -structures, almost      7 9
$\Gamma_{G}$ -structures, deformation of      30 33—35
$\Gamma_{G}$ -structures, differentiable family of      30 31
$\Gamma_{G}$ -structures, locally trivial family of      31 32
$\hat{D}$ (linear operator), definition of      233
$\hat{\mathscr{D}_{1}}$ (non-linear operator), definition of      247
$\hat{\mathscr{D}}$ (non-linear operator), definition of      247
$\mathscr{D}$ (non-linear operator), definition of      159
$\mathscr{D}$ (non-linear operator), formula for      161
$\mathscr{D}_{1}$ (non-linear operator), definition of      162
$\mathscr{O}$ -algebra (left, right)      51 58 60 61
$\mathscr{O}$ -module (left, right)      57 59 61
$\tilde{D}$ (linear operator), definition of      114
$\tilde{D}^{'}$ (linear operator), definition of      203
$\tilde{\mathscr{D}}$ (non-linear operator), Buttin formula for      176
$\tilde{\mathscr{D}}$ (non-linear operator), definition of      174
$\tilde{\mathscr{D}}^{'}$ (non-linear operator), definition of      204
$\tilde{\mathscr{D}}^{'}_{1}$ (non-linear operator), definition of      207
$\tilde{\mathscr{D}}_{1}$ (non-linear operator), definition of      183
Actions of $\Gamma_{k}X$       156
AD      see also "Adjoint representation"
Ad (finite form of ad) in projective limit      255—256
Ad (finite form of ad), commutation with ad      178—179
Ad (finite form of ad), commutation with bracket      182
Ad (finite form of ad), definition of      170—171
ad as $\mathscr{O}$ -linear sheaf map of Lie derivative      118—119 239—240
ad in projective limit      253
ad, commutation with Ad      178—179
ad, first definition of      117
ad, morphism of graded Lie algebras      241—242
ad, properties of      123
Adjoint representation      97
Adjoint representation, ad      116—117 239
Admissible derivations      215—216 217—218 220—222 224 225 226 227 228 232
Admissible derivations, strongly admissible      236—237 240
Admissible elements of $\underline{\Pi_{k}X}$       137
Admissible local maps      139 141
Admissible transformations      169 192 193
Admissible vector fields      90 140
Arens (R.)      13*
Ba (B.)      19*
Basis (left, right)      63
Beltrami equation      23
Bernard (D.)      24
Bers (L.)      23*
Bkouche (R.)      83
Bracket      104 see
Bracket in $\mathscr{I}_{k}T$       88
Bracket, $[ , ]_{\infty}$       253
Bracket, commutation of [ , ] with Ad      182
Bracket, invariance of [ , ]      177
Bracket, [ , ]      107
Buttin (C.)      vii 43 176
Buttin formula      176
Campbell — Hausdorff formula      268
Canonical transformations      40
Cartan (E.)      26 230
Cartan (H.)      22
Cartan formula (for Lie derivative)      230
Cartan — Kaehler theory      26
Characteristic ideal      44
Characteristic variety      44 46
Chern (S.S.)      23 24
Cohen — Macaulay equation (structure)      43—44
Complex analytic structure      46
Complexes, $\bar{\partial}$ -complex      24 83
Complexes, $\hat{D}$ -complex      233 235
Complexes, $\hat{D}$ -complex in the projective limit      254
Complexes, $\hat{\mathscr{D}}$ -complex      246
Complexes, $\mathscr{D}$ -complex in the projective limit      255
Complexes, first linear complex      4 64 66
Complexes, first non-linear complex      4 159 162
Complexes, fourth non-linear complex ( $\hat{\mathscr{D}}$ -complex)      247
Complexes, non-linear complex for groupoids      275—276
Complexes, non-linear complex in deformation theory      34
Complexes, principal symbol complex      44
Complexes, restricted $\hat{D}$ -complex      245—246
Complexes, second linear complex      4 75 76 86
Complexes, second non-linear complex      4 177 183 188 195
Complexes, third non-linear complex      4 7 202 207
Connections, Levi-Civita connections      11 25
Connections, linear connections      11
Contact elements      9
coordinates      61 98 142
Courant (R.)      23*
D (linear operator), characterization of      75
D (linear operator), definition of      66
D' (linear operator), definition of      76
Darboux (G.)      12 13 41
de Barros (C.M.)      viii 19 28
de Rham (G.)      3 4 15 20 28 29
Deformation of $\Gamma$ -structure      39
Deformation of $\Gamma_{G}$ -structure      42
Deformation of complex analytic structure      46
Deformation of Lie structure      33—35
Deformation of multifoliate structure      42
Deformation of symplectic structure      40
Deformation of volume structure      38
Deformation theory of Lie structure      29
Deformation theory of Lie structure, general mechanism for      33—35
Deformation, germs of      34
Derivations of $\mathbf{R}$ -Lie algebra ( $\mathscr{I}_{k}T$ , [ , ])      97
Derivations of jet forms      212
Derivations of scalar differential forms      104
Derivations, admissible      215—218 220 222 224—228 232
Derivations, decomposition of      106 228
Derivations, formulas for brackets of      232
Derivations, interior      105 216 222—224 233
Derivations, Lie      106 117 216 225—227
Derivations, strongly admissible      236—237 240
Diagonal      52 272
Diagonal approach      139
Diagonal module (submodule)      89—90
Diagonal structure      88—92 108—109 111
Diagonal, ideal of      53
Diagonal, reflection on      54 59
Differential equations (homogeneous linear partial)      77 79
Differential forms (canonical)      11
Differential operators (linear)      87
Dolbeault (P.)      21—22
Dolbeault complex      21 22 46—47
Eckmann (B.)      20 22*
Ehresmann (C.)      vi 16 18 20 22 24 26* 52 138 257 270
Eliopoulos (H.A.)      28
Exponential map (Exp) for Lie groupoid      261
Exponential map (Exp), bijectivity of      201
Exponential map (Exp), definition of      144—145
Exponential map (Exp), differentiability of      147
Finite type      25 27
Formally integrable (differential) equation      83—84 86
Forms, horizontal (vertical)      64 109—110 126
Frames, admissible      6
Frames, fields of      10
Frechet derivative      70 158
Frobenius integrability condition      14 25
Frobenius integrability condition, complex      21
Frobenius theorem      9—11 13
Frobenius theorem, complex      20 22—24
Froelicher (A.)      14* 15* 20 22* 104* 223*
Froelicher — Nijenhuis resolution      246
Froelicher — Nijenhuis theory      104 212 216 233
Fundamental identities for $\hat{\mathscr{D}}$ , $\hat{\mathscr{D}}_{1}$       252
Fundamental identities for $\mathscr{D}$ , $\mathscr{D}_{1}$       164—165
Fundamental identities for $\tilde{\mathscr{D}^{'}}$ , $\tilde{\mathscr{D}^{'}_{1}}$       210—211
Fundamental identities for $\tilde{\mathscr{D}}$ , $\tilde{\mathscr{D}_{1}}$       184—186
Fundamental identities for brackets of derivations      232

Fundamental identities for D, $\tilde{D}$ (and brackets)      124
Generalized differentiable manifold      54
Godbillon (C.)      13*
Goldschmidt (H.)      80 83—85
Goursat (E.)      13*
Graded Lie algebra      112 114 121 221 223 255
Gray (J.W.)      13*
Griffiths (P.A.)      29
Grothendieck (A.)      v
Groupoid      8 29 33 257 see
Groupoid of invertible elements      136 137 141
Groupoid, differentiable      258
Gugenheim (V.K.A.M.)      13* 15* 21*
Guillemin (V.W.)      4 25 26 44 45* 170
Haantjes (J.)      14
Hilbert (D.)      23*
Hoermander (L.)      22 24* 45
Holonomic elements (left, right)      63 77 80 87 95
Homological length      45
Homotopy operator      73
Horizontal (vertical) projection      126
Horizontal differentiation      68
Infinite type      26
Infinitesimal deformation      32—33
Infinitesimal neighborhood (of diagonal)      53 56
Infinitesimal transformations      6 40 42
Integrability condition      9—10 14—15 21 22 24
Integrability of $\Gamma(\mathscr{R}_{k})$ -structure      7
Integrability of almost $\Gamma$ -structure      7 9
Integrability of field of contact elements      9
Integrability of Lie structure      7
Integrability, formal      80 83—84 86
Integrable almost $\Gamma$ -structure      7 9
Integrable equation      80
Integrable G-structure      6
Integrable structure      8
Integral submanifold      9
Interior derivations      105 216 222—224 233
Involutive equation      26
Involutive systems      26 83
Isothermal parameters      23
Jacobi identity      105 116 120 121 131 134 183 233
Jet bundles      50
Jet forms      212
Jet sheaves, linear      49
Jet sheaves, non-linear      136
Klemola (T.)      viii
Kobayashi (E.T.)      27 104* 223*
Kodaira (K.)      32* 33* 36 42* 43* 48
Kohn(J.J.)      22
Korn (A.)      23
Kumpera (A.)      3* 271*
Kuranishi (M.)      48
Lehmann (D.)      80
Lehmann-Lejeune (J.)      27 28*
Levi-Civita (T.)      11 25
Libermann (P.)      5 22* 27
Lichnerowicz (A.)      13*
Lichtenstein (L.)      23
Lie (S.)      5
Lie algebra of groupoid      144 260
Lie algebra, graded      112 114 121 144 221 223 255
Lie derivations      106 117 216 225—227
Lie derivative      10 37 88 118—119 217 230 239—240
Lie equations      33 37 43 46
Lie equations, elliptic      v 3 7
Lie equations, linear      5 6 7 10 14 30 88 100 102
Lie equations, non-linear      6 7 8 30
Lie groupoid      6 257 259
Lie groupoid, prolongation of      269
Lie pseudogroup      6
Lie structures      7 33 43
Lie structures, deformation of      29 33—35
Lie structures, elliptic      22
Local rings (sheaf of)      49
Local solution (of differential equation)      80
Locally universal family      41
Malgrange (B.)      v vi vii 4 7* 22 24* 26 27 36 53 83 102 138 170 251
Matsushima (Y.)      24
Model category      30
Model category, $M(\psi)$       37
Multifoliate group (structure)      42—43
Neumann problem      45
Newlander (A.)      22 27 36
Ngo Van Que      4 271 275
Nickerson (H.K.)      13 15* 21*
Nijenhuis (A.)      14* 15* 22 24 104* 223*
Nijenhuis bracket      4 14 20 48 107 112 114 115 117 131 134 159 169 216 227 230 238 239 246 253
Nijenhuis torsion tensor      24 27
Nilpotent elements      54
Nirenberg (L.)      22 23 24* 27 36 48*
Parallelism      10
Partial exactness of second non-linear complex      188 195
Partial exactness of third non-linear complex      208
Poincare complex      45
Poincare lemma      15 21 28 40 69 73 85
Pradines (J.)      261
Projective limits      252
Prolongation of differential equations      80
Prolongation of Lie equations      103
Prolongation of Lie groupoids      269
Prolongation of maps      79
Prolongation of sheaves      55
Prolongation space      52 54
Pseudodifferential operators      45
Pseudogroup, continuous      7 8
Pseudogroup, primitive      36
Pseudogroup, simple      36
Pseudogroup, transitive (continuous)      35 42
QUE      see "Ngo Van Que"
Quillen (D.G.)      83
Regular functions      42
Regular maps      29
Restriction map      32
Samelson (H.)      23*
Schouten (J.A.)      13
Semi-holonomic jets      27
Singer (I.M.)      24 25 26
Solutions (sheaf of)      87
Source (target) maps      136—137 272
Spencer (D.C.)      v vi vii 3 7* 13 15* 21* 26 27 32* 33* 36* 38* 42* 43* 47* 48* 80 242 246 248 251
Split exact      50
Split injective      60
Stable cohomology (range)      86
Sternberg (S.)      4 13* 24 25 26 44 45* 170
Structive tensor      24
Structive tensor of higher order      25—27
Structure equations      3 162 177 183 207 208 247 256
Structures, $M(\Gamma)$ -structure      30
Structures, $M(\psi)$ -structure      37
Structures, $\Gamma$ -structures      7 9 30—35
Structures, $\Gamma_{G}$ -structures      42
Structures, almost $\Gamma$ -structures      7 9
Structures, almost-complex      7 15 19 23
Structures, almost-product      13—14
Structures, almost-symplectic      18—19
Structures, Cohen — Macaulay      43
Structures, complex analytic      42 46
Structures, conformal      26
Structures, contact      12 36
Structures, defined by differential forms      37
Structures, defined by fields of endomorphisms      27
Structures, defined by fields of p-frames      10
Structures, defined by linear connections      11
Structures, diagonal      88—92 108—109 111
Structures, elliptic almost-structures      8
Structures, foliate      42
Structures, G-structures      5 6 8 9 10 24
Structures, left (right)      51—53 56 59 77
Structures, Lie      7 22 29 33—35 43

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