A comprehensive introduction to differential geometry / Michael Spivak.
Editor: Boston : Publish or Perish, 1970-75Descripción: 5 v. : il. ; 28 cmISBN: 0914098004 (v. 2)Otro título: Differential geometryOtra clasificación: 53-01CONTENTS OF VOLUME I CHAPTER 1. MANIFOLDS (1-1 to 1-32) Elementary properties of manifolds 1-1 Examples of manifolds 1-8 PROBLEMS 1-24 CHAPTER 2. DIFFERENTIABLE STRUCTURES (2-1 to 2-47) C∞ structures 2-1 C∞ functions 2-6 Partial derivatives 2-11 Critical points 2-17 Immersion theorems 2-20 Partitions of unity 2-31 PROBLEMS [33] 2-36 CHAPTER 3. THE TANGENT BUNDLE (3-1 to 3-58 ) The tangent space of Rn 3-1 The tangent space of an imbedded manifold 3-6 Vector bundles 3-11 The tangent bundle of a manifold 3-16 Equivalence classes of curves, and derivations 3-20 Uniqueness of the tangent bundle 3-23 Vector fields 3-33 Orientation 3-36 PROBLEMS [1,2,16,23,24,26,30,31] 3-43 CHAPTER 4. TENSORS (4-1 to 4-39) The dual bundle 4-1 The differential of a map 4-4 CONTENTS OF VOLUME I CHAPTER 4. TENSORS (4-1 to 4-39) - Cont. Classical versus modern terminology 4-6 Multilinear functions 4-12 Covariant and contravariant tensors 4-14 Mixed tensors, and contraction 4-20 PROBLEMS [5] 4-29 CHAPTER 5- VECTOR FIELDS AND DIFFERENTIAL EQUATIONS (5-1 to 5-61) Integral curves 5-1 Existence and uniqueness theorems 5-7 The local flow 5-15 One-parameter groups of diffeomorphisms 5-21 Lie derivatives 5-24 Brackets 5-29 Addendum on Differential Equations 5-43 PROBLEMS [5,6,15,17] 5-47 CHAPTER 6. INTEGRAL MANIFOLDS (6-1 to 6-31) Prologue; classical integrability theorems 6-1 Local Theory; Frobenius Integrability Theorem 6-15 Global Theory 6-21 PROBLEMS [5,7,8] 6-26 CHAPTER 7. DIFFERENTIAL FORMS (7-1 to 7-54) Alternating functions 7-1 The wedge product 7-4 Forms 7-10 Differential of a form 7-14 Frobenius integrability theorem (differential form version) 7-22 Closed and exact forms 7-25 The Poincaré Lemma 7-35 PROBLEMS [12,14,15,19,23,26] 7-38 CHAPTER 8. INTEGRATION (8-1 to 8-82) Classical line and surface integrals 8-1 Integrals over singular k-cubes 8-9 The boundary of a chain 8-13 Stokes * Theorem 8-18 CHAPTER 8. INTEGRATION (8-1 to 8-82) - Cont. Integrals over manifolds 8-22 Volume elements 8-25 Stokes' Theorem 8-29 De Rham cohomology 8-32 PROBLEMS [23,24,25,26,31] 8-61 CHAPTER 9. RIEMANNIAN METRICS (9-1 to 9-89) Inner products 9-1 Riemannian metrics 9-12 Length of curves 9-18 The calculus of variations 9-24 The First Variation Formula and geodesics 9-38 The exponential map 9-46 Geodesic completeness 9-55 Addendum on Tubular Neighborhoods 9-59 PROBLEMS [23,27,28,29,32, 41] 9-63 CHAPTER 10. LIE GROUPS (10-1 to 10-68) Lie groups 10-1 Left invariant vector fields 10-6 Lie algebras 10-8 Subgroups and subalgebras 10-13 Homomorphisms 10-14 One-parameter subgroups 10-20 The exponential map 10-23 Closed subgroups 10-32 Left invariant forms 10-35 Bi-invariant metrics 10-46 The equations of structure 10-50 PROBLEMS [7,15,19,24] 10-53 CHAPTER 11. EXCURSION I: IN THE REALM OF ALGEBRAIC TOPOLOGY (11-1 to 11-52) Complexes and exact sequences 11-1 The Mayer-Vietoris Sequence 11-8 Triangulations 11-10 The Euler Characteristic 11-12 Mayer-Vietoris sequence for compact supports 11-16 The exact sequence of a pair 11-19 Poincare Duality 11-30 The Thom class 11-31
CONTENTS OF VOLUME II CHAPTER 1. CURVES IN THE PLANE AND IN SPACE (1-1 to 1-61) Curvature of plane curves 1-1 Convex curves 1-16 Curvature and torsion of space curves 1-30 The Serret-Frenet formulas 1-43 The natural form on a Lie group 1-46 Classification of plane curves under the group of special and proper affine motions 1-50 Classification of curves in Rn under the group of proper Euclidean motions 1-58 CHAPTER. 2. WHAT THEY KNEW ABOUT SURFACES BEFORE GAUSS (2-1 to 2-9) Euler’s Theorem 2-1 Meusnier’s Theorem 2-6 CHAPTER 3A. HOW TO READ GAUSS (3A-I to 3A-12) CHAPTER 3B. GAUSS’ THEORY OF SURFACES (3B-1 to 3B-b6) The Gauss map 3B-1 Gaussian curvature 3B-4 The Weingarten map 3B-12 The first and second fundamental forms 3B-14 The Theorems Egreguim 3B-27 Geodesics on a surface 3B-30 The metric in geodesic polar coordinates 3B-32 The integral of the curvature over a geodesic triangle 3B-38 Addendum. The Formula of Bertrand and Puiseux; Diquet’s formula 3B-43 CHAPTER 4A. AN INAUGURAL LECTURE (4A-1 to UA-20) Introduction 4a-1 "On the Hypotheses which lie at the Foundations of Geometry" 4A-4 CHAPTER 4B. WHAT DID RIEMANN SAY? (4B-1 to 4B-38) The form of the metric in Riemannian normal coordinates 4B-1 Addendum. Finsler Metrics 4B-27 CHAPTER 4C. A PRIZE ESSAY (4C-1 to 4C-5) CHAPTER 4D. THE BIRTH OF THE RIEMANN CURVATURE TENSOR (4D-1 to 4D-26) Necessary conditions for a metric to be flat 4D-1 The Riemann curvature tensor 4D-8 Sectional curvature 4D-15 The Test Case; first version 4D-19 Addendum. Riemann’s Invariant Definition of the Curvature Tensor 4D-24 CHAPTER 5. THE ABSOLUTE DIFFERENTIAL CALCULUS (THE RICCI CALCULUS); OR, THE DEBAUCH OF INDICES (5-1 to 5-24) Covariant derivatives 5-1 Ricci’s Lemma 5-6 Ricci’s Identities 5-8 The curvature tensor 5-9 The Test Case; second version 5-12 Classical connections 5-17 The torsion tensor 5-18 Geodesics 5-19 Bianchi’s identities 5-21 CHAPTER 6. THE V OPERATOR (6-1 to 6-42) Koszul connections 6-1 Covariant derivatives 6-3 Parallel translation 6-10 The torsion tensor 6-14 The Levi-Civita connection 6-17 The curvature tensor 6-18 The Test Case; third version 6-21 HAPTER 6. The v OPERATOR (6-1 to 6-42) - Cont. Bianchi’s identities 6-25 Geodesics 6-28 The First Variation Formula 6-29 Addendum 1. Connections with the same Geodesics 6-32 Addendum 2. Riemann’s Invariant Definition of the Curvature Tensor 6-40 CHAPTER 7. THE REPERE MOBILE (THE MOVING FRAME) (7-1 to 7-58) Moving frames 7-1 The structural equations of Eculidean space 7-4 The structural equations of a Riemannian manifold 7-12 The Test Case; fourth version 7-14 Adapted frames 7-17 The structural equations in polar coordinates 7-20 The Test Case; fifth version 7-22 The Test Case; sixth version - 7-23 ’’The curvature determines the metric” 7-26 The 2-dimensional case 7-29 Cartan connections 7-32 Covariant derivatives and the torsion and curvature tensors 7-37 Bianchi’s identities ' 7-41 Addendum 1. Manifolds of Constant Curvature 7-44 Schur’s Theorem 7-46 The form of the metric in normal coordinates 7-49 Isothermic coordinates 7-51 Addendum 2. E. Cartan’s Treatment of Normal Coordinates 7-56 CHAPTER 8. CONNECTIONS IN PRINCIPAL BUNDLES (8-1 to 8-62) Principal bundles 8-1 Lie groups acting on manifolds 8-6 A new definition of Cartan connections 8-10 Ehresmann connections 8-16 Lifts 8-19 Parallel translation and covariant derivatives 8-22 The covariant differential and the curvature form 8-29 The dual form and the torsion form 8-30 The structural equations 8-33 The torsion and curvature tensors 8-36 The Test Case; seventh version 8-41 Bianchi’s identities 8-42 Summary 8-44 Addendum 1. The Tangent Bundle of F(M) 8-52 Addendum 2. Complete Connections 8-54 Addendum 3, Connections in Vector Bundles 8-56 Addendum 4. Flat Connections, and an Apology 8-61
| Item type | Home library | Shelving location | Call number | Materials specified | Status | Date due | Barcode | Course reserves |
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Libros
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Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 53 Sp761 (Browse shelf) | Vol. I | Available | A-3985 | ||
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Libros
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Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 53 Sp761 (Browse shelf) | Vol. II | Available | A-3986 |
La biblioteca posee los vols. 1 y 2. Para los vols. 3, 4, 5 consulte la 2.ª ed. AR-BbIMB.
CONTENTS OF VOLUME I --
CHAPTER 1. MANIFOLDS (1-1 to 1-32) --
Elementary properties of manifolds 1-1 --
Examples of manifolds 1-8 --
PROBLEMS 1-24 --
CHAPTER 2. DIFFERENTIABLE STRUCTURES (2-1 to 2-47) --
C∞ structures 2-1 --
C∞ functions 2-6 --
Partial derivatives 2-11 --
Critical points 2-17 --
Immersion theorems 2-20 --
Partitions of unity 2-31 --
PROBLEMS [33] 2-36 --
CHAPTER 3. THE TANGENT BUNDLE (3-1 to 3-58 ) --
The tangent space of Rn 3-1 --
The tangent space of an imbedded manifold 3-6 --
Vector bundles 3-11 --
The tangent bundle of a manifold 3-16 --
Equivalence classes of curves, and derivations 3-20 --
Uniqueness of the tangent bundle 3-23 --
Vector fields 3-33 --
Orientation 3-36 --
PROBLEMS [1,2,16,23,24,26,30,31] 3-43 --
CHAPTER 4. TENSORS (4-1 to 4-39) --
The dual bundle 4-1 --
The differential of a map 4-4 --
CONTENTS OF VOLUME I --
CHAPTER 4. TENSORS (4-1 to 4-39) - Cont. --
Classical versus modern terminology 4-6 --
Multilinear functions 4-12 --
Covariant and contravariant tensors 4-14 --
Mixed tensors, and contraction 4-20 --
PROBLEMS [5] 4-29 --
CHAPTER 5- VECTOR FIELDS AND DIFFERENTIAL EQUATIONS (5-1 to 5-61) --
Integral curves 5-1 --
Existence and uniqueness theorems 5-7 --
The local flow 5-15 --
One-parameter groups of diffeomorphisms 5-21 --
Lie derivatives 5-24 --
Brackets 5-29 --
Addendum on Differential Equations 5-43 --
PROBLEMS [5,6,15,17] 5-47 --
CHAPTER 6. INTEGRAL MANIFOLDS (6-1 to 6-31) --
Prologue; classical integrability theorems 6-1 --
Local Theory; Frobenius Integrability Theorem 6-15 --
Global Theory 6-21 --
PROBLEMS [5,7,8] 6-26 --
CHAPTER 7. DIFFERENTIAL FORMS (7-1 to 7-54) --
Alternating functions 7-1 --
The wedge product 7-4 --
Forms 7-10 --
Differential of a form 7-14 --
Frobenius integrability theorem (differential form version) 7-22 --
Closed and exact forms 7-25 --
The Poincaré Lemma 7-35 --
PROBLEMS [12,14,15,19,23,26] 7-38 --
CHAPTER 8. INTEGRATION (8-1 to 8-82) --
Classical line and surface integrals 8-1 --
Integrals over singular k-cubes 8-9 --
The boundary of a chain 8-13 --
Stokes * Theorem 8-18 --
CHAPTER 8. INTEGRATION (8-1 to 8-82) - Cont. --
Integrals over manifolds 8-22 --
Volume elements 8-25 --
Stokes' Theorem 8-29 --
De Rham cohomology 8-32 --
PROBLEMS [23,24,25,26,31] 8-61 --
CHAPTER 9. RIEMANNIAN METRICS (9-1 to 9-89) --
Inner products 9-1 --
Riemannian metrics 9-12 --
Length of curves 9-18 --
The calculus of variations 9-24 --
The First Variation Formula and geodesics 9-38 --
The exponential map 9-46 --
Geodesic completeness 9-55 --
Addendum on Tubular Neighborhoods 9-59 --
PROBLEMS [23,27,28,29,32, 41] 9-63 --
CHAPTER 10. LIE GROUPS (10-1 to 10-68) --
Lie groups 10-1 --
Left invariant vector fields 10-6 --
Lie algebras 10-8 --
Subgroups and subalgebras 10-13 --
Homomorphisms 10-14 --
One-parameter subgroups 10-20 --
The exponential map 10-23 --
Closed subgroups 10-32 --
Left invariant forms 10-35 --
Bi-invariant metrics 10-46 --
The equations of structure 10-50 --
PROBLEMS [7,15,19,24] 10-53 --
CHAPTER 11. EXCURSION I: IN THE REALM OF ALGEBRAIC TOPOLOGY (11-1 to 11-52) --
Complexes and exact sequences 11-1 --
The Mayer-Vietoris Sequence 11-8 --
Triangulations 11-10 --
The Euler Characteristic 11-12 --
Mayer-Vietoris sequence for compact supports 11-16 --
The exact sequence of a pair 11-19 --
Poincare Duality 11-30 --
The Thom class 11-31 --
CONTENTS OF VOLUME II --
CHAPTER 1. CURVES IN THE PLANE AND IN SPACE (1-1 to 1-61) --
Curvature of plane curves 1-1 --
Convex curves 1-16 --
Curvature and torsion of space curves 1-30 --
The Serret-Frenet formulas 1-43 --
The natural form on a Lie group 1-46 --
Classification of plane curves under the group of special --
and proper affine motions 1-50 --
Classification of curves in Rn under the group of proper --
Euclidean motions 1-58 --
CHAPTER. 2. WHAT THEY KNEW ABOUT SURFACES BEFORE GAUSS (2-1 to 2-9) --
Euler’s Theorem 2-1 --
Meusnier’s Theorem 2-6 --
CHAPTER 3A. HOW TO READ GAUSS (3A-I to 3A-12) --
CHAPTER 3B. GAUSS’ THEORY OF SURFACES (3B-1 to 3B-b6) --
The Gauss map 3B-1 --
Gaussian curvature 3B-4 --
The Weingarten map 3B-12 --
The first and second fundamental forms 3B-14 --
The Theorems Egreguim 3B-27 --
Geodesics on a surface 3B-30 --
The metric in geodesic polar coordinates 3B-32 --
The integral of the curvature over a geodesic triangle 3B-38 --
Addendum. The Formula of Bertrand and Puiseux; Diquet’s formula 3B-43 --
CHAPTER 4A. AN INAUGURAL LECTURE (4A-1 to UA-20) --
Introduction 4a-1 --
"On the Hypotheses which lie at the Foundations of Geometry" 4A-4 --
CHAPTER 4B. WHAT DID RIEMANN SAY? (4B-1 to 4B-38) --
The form of the metric in Riemannian normal coordinates 4B-1 --
Addendum. Finsler Metrics 4B-27 --
CHAPTER 4C. A PRIZE ESSAY (4C-1 to 4C-5) --
CHAPTER 4D. THE BIRTH OF THE RIEMANN CURVATURE TENSOR (4D-1 to 4D-26) --
Necessary conditions for a metric to be flat 4D-1 --
The Riemann curvature tensor 4D-8 --
Sectional curvature 4D-15 --
The Test Case; first version 4D-19 --
Addendum. Riemann’s Invariant Definition of the Curvature Tensor 4D-24 --
CHAPTER 5. THE ABSOLUTE DIFFERENTIAL CALCULUS (THE RICCI CALCULUS); OR, THE DEBAUCH OF INDICES (5-1 to 5-24) --
Covariant derivatives 5-1 --
Ricci’s Lemma 5-6 --
Ricci’s Identities 5-8 --
The curvature tensor 5-9 --
The Test Case; second version 5-12 --
Classical connections 5-17 --
The torsion tensor 5-18 --
Geodesics 5-19 --
Bianchi’s identities 5-21 --
CHAPTER 6. THE V OPERATOR (6-1 to 6-42) --
Koszul connections 6-1 --
Covariant derivatives 6-3 --
Parallel translation 6-10 --
The torsion tensor 6-14 --
The Levi-Civita connection 6-17 --
The curvature tensor 6-18 --
The Test Case; third version 6-21 --
HAPTER 6. The v OPERATOR (6-1 to 6-42) - Cont. --
Bianchi’s identities 6-25 --
Geodesics 6-28 --
The First Variation Formula 6-29 --
Addendum 1. Connections with the same Geodesics 6-32 --
Addendum 2. Riemann’s Invariant Definition of the Curvature Tensor 6-40 --
CHAPTER 7. THE REPERE MOBILE (THE MOVING FRAME) (7-1 to 7-58) --
Moving frames 7-1 --
The structural equations of Eculidean space 7-4 --
The structural equations of a Riemannian manifold 7-12 --
The Test Case; fourth version 7-14 --
Adapted frames 7-17 --
The structural equations in polar coordinates 7-20 --
The Test Case; fifth version 7-22 --
The Test Case; sixth version - 7-23 --
’’The curvature determines the metric” 7-26 --
The 2-dimensional case 7-29 --
Cartan connections 7-32 --
Covariant derivatives and the torsion and curvature tensors 7-37 --
Bianchi’s identities ' 7-41 --
Addendum 1. Manifolds of Constant Curvature 7-44 Schur’s Theorem 7-46 --
The form of the metric in normal coordinates 7-49 --
Isothermic coordinates 7-51 --
Addendum 2. E. Cartan’s Treatment of Normal Coordinates 7-56 --
CHAPTER 8. CONNECTIONS IN PRINCIPAL BUNDLES (8-1 to 8-62) --
Principal bundles 8-1 --
Lie groups acting on manifolds 8-6 --
A new definition of Cartan connections 8-10 --
Ehresmann connections 8-16 --
Lifts 8-19 --
Parallel translation and covariant derivatives 8-22 --
The covariant differential and the curvature form 8-29 --
The dual form and the torsion form 8-30 --
The structural equations 8-33 --
The torsion and curvature tensors 8-36 --
The Test Case; seventh version 8-41 --
Bianchi’s identities 8-42 --
Summary 8-44 --
Addendum 1. The Tangent Bundle of F(M) 8-52 --
Addendum 2. Complete Connections 8-54 --
Addendum 3, Connections in Vector Bundles 8-56 --
Addendum 4. Flat Connections, and an Apology 8-61 --
MR, 42 #2369 (v. 1)
MR, 42 #6726 (v. 2)
MR, 51 #8962 (v. 3)
MR, 52 #15254a, 52 #15254b (v. 4-5)
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