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Ivey Th.A., Landsberg J.M. — Cartan for Beginners: Differential Geometry Via Moving Frames and Exterior Differential Systems
Ivey Th.A., Landsberg J.M. — Cartan for Beginners: Differential Geometry Via Moving Frames and Exterior Differential Systems

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Название: Cartan for Beginners: Differential Geometry Via Moving Frames and Exterior Differential Systems

Авторы: Ivey Th.A., Landsberg J.M.

Аннотация:

This book is an introduction to Cartan's approach to differential geometry. Two central methods in Cartan's geometry are the theory of exterior, differential systems and the method of moving frames. The book presents thorough and modern treatments of both subjects, including their applications to classic and contemporary problems.

The book begins with the classical geometry of surfaces and basic Riemannian geometry in the language of moving frames, along with an elementary introduction to exterior differential systems. Key concepts are developed incrementally, with motivating examples leading to definitions, theorems and proofs.

Once the basics of the methods are established, applications and advanced topics are developed. One particularly notable application is to complex algebraic geometry, where important results from projective differential geometry are expanded and updated. The book features an introduction to G-structures and a treatment of the theory of connections. The Cartan machinery is also applied to obtain explicit solutions of PDEs, via Darboux's method, the method of characteristics, and Cartan's method of equivalence.

This text is suitable for a one-year graduate course in differential geometry. It has numerous exercises and examples throughout. The book will also be of use to experts in such areas as PDEs and algebraic geometry who want to learn how moving frames and exterior differential systems apply to their fields.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2003

Количество страниц: 378

Добавлена в каталог: 11.06.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
#      53
$A^{(1)}$      147
$A^{(l)}$      147
$C^{\infty}{M)$      335
$c_k$, codimension of polar space      256
$E_6$, exceptional Lie group      102
$F_4$, differential invariant      107
$F_4$, exceptional Lie group      102
$F_k$      108
$G(n,T\Sigma)$      177
$G_2$, exceptional Lie group      323
$Hoi^{\theta}_u$      287
$H^{0,2}(A)$      175
$H^{i,j}(A)$      180
$I II_{M,x} I$      80
$III^v$      129
$I^{(1)}$, derived system      216
$k_g$      59
$K_n$      60
$m_x$, functions vanishing at x      335
$Sp(V, \omega)$, symplectic group      317
$S^2V$      313
$S^kV$      314
$s_k$, characters of a tableau      154
$s_k$, characters of an EDS      258
$T^*_xM$, cotangent space      335
$T_xM$, tangent space      335
$V_{\mathbb{C}}$, complexification of V      343
$X_{smooth}$      82
$[R_{\theta}]$      282
$\Delta r^{\mu}_{\alpha \beta \gamma}$      95
$\delta_*$, dual defect      120
$\delta_{\sigma}(X)$, secant defect      129
$\delta_{\tau}(X)$, tangential defect      129
$\Gamma(E)$, smooth sections of E      335
$\Gamma^{\beta}_{\alpha, i}$      277
$\Lamdba^2 V$      313
$\Lamdba^k V$      314
$\mathbb{E}^3$, Euclidean three-space      2
$\mathbb{F}\mathbb{F}^k$      97
$\mathbb{S}_m$, spinor variety      106
$\mathbb{T}(V)$      273
$\mathcal{F}^1$, Euclidean      37
$\mathcal{F}^1$, projective      78
$\mathcal{H}^{i, j}(\mathfrak{G})$      283
$\mathcal{I}$, differential ideal      340
$\mathcal{I}^k$, k-th homogeneous component of $\mathcal{I}$      340
$\mathcal{K}(V)$      330
$\mathcal{L}_X$, Lie derivative      339
$\mathfrak{g}$, Lie algebra of Lie group G      17
$\nabla$      277
$\Omega^k(M)$, $\Omega^*(M)$      336
$\Omega^k(M,V)$      338
$\Omega_{(p, q)}(M)$      345
$\otimes$, tensor product      312
$\phi^*$, pullback by $\phi$      337
$\phi_*$, pushforward by $\phi$      337
$\rfloor$, interior product      315
$\tau(X)$, tangential variety      86
$\tau(Y,X)$      131
$\tau_g$      60
$\underline{d}^k$      97
$\Xi_A$, characteristic variety of a tableau      157
$\{\}_{alg}$      340
${}_{diff}$      340
$| \mathbb{F}\mathbb{F}^k |$      97
(I, J), linear Pfafflan system      164
(p, g)-forms      345
Abuse of notation      29 72 170
Adjoint representation      321
Affine connection      285
Affine tangent space      76
Algebraic variety      82
Algebraic variety, degree of      82
Algebraic variety, dimension of      82
Algebraic variety, general point of      83
Algebraic variety, ideal of      82
Almost complex manifold      274 282 344
Almost complex structure      344
Almost symplectic manifold      274
Ambrose — Singer Theorem      290
Ann(v)      129
Apparent torsion      165
Arclength parameter      14
ASO(2)      12
ASO(3)      23
ASO(3) as space of frames      24
Associated hypersurface      124
Associated varieties      123
Associative submanifolds      201 265
Associator      325
Asymptotic directions      80
Asymptotic line      60 226 238
Baecklund transformations      235—241
Baecklund's Theorem      237
Baseloc $| II_{M,x} |$      80
Basic differential form      339
Bertini Theorem      112
Bertini Theorem, higher-order      112
Bertrand curve      26
Bezout's theorem      82
Bianchi identities      53—54
Bonnet surface      44 231
Burger's equation      208 232
Calibrated submanifold      198
Calibration      197
Calibration, associative      201
Calibration, Cayley form      202
Calibration, coassociative      201
Calibration, special Lagrangian      200
Canonical system on Grassmann bundle      177
Canonical system on space of jets      28
Cartan geometry      296
Cartan integer      156 179
Cartan lemma      314
Cartan system      209
Cartan — Dieudonne Theorem      331
Cartan — Janet theorem      192
Cartan — Kahler Theorem      254—256
Cartan — Kahler Theorem for linear Pfaffian systems      176
Cartan — Kahler Theorem for tableaux      156
Cartan — Kahler Theorem, Goldschmidt version      181
Cartan's algorithm for linear Pfaffian systems      178
Cartan's five variables paper      217
Cartan's test      256
Catenoid      43
Cauchy problem      349
Cauchy — Kowalevski form      350
Cauchy — Kowalevski Theorem      243 351
Cauchy — Riemann equations      347
Cauchy — Riemann equations, tableau      144 156
Cayley submanifold      202
Character of a tableau      156
Characteristic hyperplane      181
Characteristic systems (Monge)      213
Characteristic variety      157
Characteristic variety, dimension and degree of      159
Characteristics, Cauchy      205 259
Characteristics, Cauchy, quotient by      210
Characteristics, confounded      213
Characteristics, first-order      214
Characteristics, method of      207—208
Characteristics, Monge      213
Characters      258
Characters of linear Pfaffian system      179
Characters of tableau      154
Chebyshev net      227
Christoffel symbols      277
Cl(V,Q)      331
Clifford algebras      331
Clifford algebras, fundamental lemma of      332
Clifford torus      58
Co-roots      329
Coassociative submanifold      201
Codazzi equation for Darboux frames      43
Codazzi equation, matrix form      49
Codimension      245
Coisotropic hypersurface      124
Complete intersection      140
Complex characteristic variety      158
Complex contact structure      348
Complex manifold      343 344
Complex structure      318 344
Complexification of a real vector space      318
Cone      44
Cone over a variety      86
Cone, characterization of      125
Connection affine      285
Connection form      279
Connection on coframe bundle      278—283
Connection on induced vector bundles      284
Connection on vector bundle      277
Connection, symmetric      285
Conormal space, of submanifold in $\mathbb{P}^N$      77
Contact manifold      33
Contact system on space of jets      28
Contact, order of      83
Cotangent bundle      335
Cotangent space      335
Covariant differential operator      54 277
Cubic form      94
Curvature of curve in $\mathbb{E}^2$      14
Curvature of curve in $\mathbb{E}^3$      25
Curvature of G-structure      280
Curvature, Gauss      38
Curvature, Gauss, geometric interpretation of      47
Curvature, Gauss, in coordinates      4
Curvature, mean      38
Curvature, mean, geometric interpretation of      68
Curvature, mean, in coordinates      4
Curvature, Ricci      53 262
Curvature, scalar      53 262 266 330
Curvature, sectional      53
Curvature, traceless Ricci      330
Curvature, Weyl      330
Curvature-line coordinates      188
Curve in $\mathbb{E}^2$, curvature      14
Curve in $\mathbb{E}^2$, osculating circle      14
Curve in $\mathbb{E}^3$, curvature      25
Curve in $\mathbb{E}^3$, differential invariants      25—26
Curve in $\mathbb{E}^3$, torsion      25
Curve, arclength parameter      14
Curve, Bertrand      26
Curve, regular      13
Curve, speed of      14
Cylinder      44
d, exterior derivative      337
Darboux frame      42
Darboux's theorem      32
Darboux, integrable      218 239
Darboux, method of      217—222
Darboux, semi-integrable      222
de Rham Splitting Theorem      289
Decomposable tensor      312
Derived flag      216
Derived system      216
DET      102 315
Determinant of linear endomorphism      315
Developable surface      40
Differential form      336
Differential form, basic, semi-basic      339
Differential form, closed      338
Differential form, homogeneous      340
Differential form, left-invariant      17
Differential form, vector-valued      338
Differential ideal      340
Differential invariant, Euclidean      3
Dual basis      311
Dual variety      87 118
Dual variety, defect of      120
Dual variety, reflexivity      119
Dual vector space      311
Dupin, cyclides of      361
Dupin, theorem of      253
e-structure      304
Embedded tangent space      76
End(V)      312
Engel structure      217
Equivalent G-structures      275
Equivalent webs      268
Euclidean group      23
Euler characteristic      62
Exterior derivative      337—338
Exterior differential system      29
Exterior differential system with independence condition      27
Exterior differential system, hyperbolic      214—215
Exterior differential system, linear Pfaffian      164
Exterior differential system, Pfaffian      341
Exterior differential system, symmetries      204—205
F(M)      49
Face of calibration      199
First fundamental form (Riemannian)      46
First-order adapted frames (Euclidean)      45
Flag variety      85 316
Flag, A-generic      154
Flag, complete      85
Flag, derived      216
Flag, partial      85
Flat 3-web      268
Flat G-structure      275
Flat path geometry      296
Flat Riemannian manifold      52
Flat Riemannian manifold, isometric immersions of      194
Flat surface      41
Flow of a vector field      6
Flowbox coordinates      6
Flowchart for Cartan's algorithm      178
Focal hypersurface      89
Focal surface      237 266
Frame bundle, general      49
Frame bundle, orthonormal      50
Frame, Darboux      42
Frenet equations      25
Frobenius ideal      11
Frobenius structure      308
Frobenius system, tableau of      146
Frobenius theorem      10—12 30
Frobenius Theorem, proof      30
Fubini cubic form      94
Fubini forms      94 107
Fulton — Hansen theorem      130
Fundamental form via spectral sequences      98
Fundamental form, effective calculation of      97
Fundamental form, k-th      97
Fundamental form, prolongation property of      97
G(k, V), Grassmannian      72
G(n,m)      198
G-structure      267—275
G-structure, 1-flat      280
G-structure, 2-flat      281
G-structure, curvature      280 282
G-structure, definition      274
G-structure, flat      275
G-structure, prolongation      281
G/H-structure of order two      296
Gauss curvature in coordinates      4
Gauss curvature via frames      36—38
Gauss curvature, geometric interpretation of      47
Gauss equation      47
Gauss image      77
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