## A comprehensive introduction to differential geometry / Michael Spivak.

Editor: Berkeley : Publish or Perish, 1979Edición: 2nd edDescripción: 5 v. : il. ; 24 cmISBN: 0914098837 (v. 15); 0914098829 (v. 3-5); 0914098799 (pbk.)Otro título: Differential geometryOtra clasificación: 53-01CONTENTS Introduction v Chapter 1. The Fundamental Equations for Hypersurfaces Covariant differentiation in a submanifold of a Riemannian manifold [1] The second fundamental form, the Gauss formulas, and Gauss’ equation; Synge’s inequality [5] The Weingarten equations and the Codazzi-Mainardi equations for hypersurfaces [10] The classical tensor analysis description [17] The moving frame description [23] Addendum. Auto-parallel and Totally Geodesic Submanifolds [32] Problems [40] Chapter 2. Elements of the Theory of Surfaces in IR3 The first and second fundamental forms [45] Classification of points on a surface; the osculating paraboloid and the Dupin indicative [51] Principal directions and curvatures, asymptotic directions, flat points and umbilics; all-umbilic surfaces [68] The classical Gauss formulas, Weingarten equations, Gauss equation, and Codazzi-Mainardi equations [73] Fundamental theorem of surface theory [79] The third fundamental form [87] Convex surfaces; Hadamard’s theorem [90] The fundamental equations via moving frames [96] Review of Lie groups and their role in curve theory and affine curve theory [100] Application of Lie groups to surface theory; the fundamental equations and the structural equations of S0(3) [109] Affine surface theory; the osculating paraboloids and the affine invariant conformal structure [113] The special affine first fundamental form [123] Quadratic and cubic forms; apolarity [135] The affine normal direction; the special affine normal [144] The special affine Gauss formulas, and special affine second fundamental form [153] The Pick invariant; surfaces with Pick invariant 0 [164] The special affine Weingarten formulas [178] The special affine Codazzi-Mainardi equations; the fundamental theorem of special affine surface theory [187] Problems [195] Chapter 3. A Compendium of Surfaces Basic calculations [197] The classical flat surfaces [205] Ruled surfaces [213] Quadric surfaces [220] Surfaces of revolution [228] rotation surfaces of constant curvature [235] Minimal surfaces [242] Addendum. Envelopes of 1-parameter families of planes [255] Problems [264] Chapter 4, Curves on Surfaces Normal and geodesic curvature [272] The Darboux frame; geodesic torsion [277] Laguerre’s theorem [282] General properties of lines of curvature, asymptotic curves, and geodesics 281i The Beltrami -Enneper theorem [291] Lines of curvature and Dupin’s theorem [295] Conformal maps of IR3 ; Liouville ’s theorem [302] Geodesics and Clairaut ’ s theorem [313] Addendum 1. Special Parameter Curves [320] Addendum 2. Singularities of Line Fields [324] Problems [333] Chapter 5. Complete Surfaces of Constant Curvature Hilbert’s lemma; complete surfaces of constant curvature K > 0 [347] Analysis of flat surfaces; the classical classification of developable surfaces [349] Complete flat surfaces [363] Complete surfaces of constant curvature K < 0 [367] Chapter 6. The Gauss-Bonnet Theorem and Related Topics The connection form for an orthonormal moving frame on a surface; the change in angle under parallel translation [386] The integral of K dA over a polygonal region [392] The Gauss-Bonnet theorem; consequences [399] Total absolute curvature of surfaces [409] Surfaces of minimal total absolute curvature [413] Total curvature of curves; Fenchel’s theorem, and Fary-Milnor theorem [421] Addendum 1. Compact Surfaces with Constant Negative Curvature 430 Addendum 2. The Degree of the Normal Map [439] Problems [444] Corrections [447] Mini-Bibliography for Volume III [451] Notation Index [453] Index [457]

Chapter 7* Higher Dimensions and Codimensions A» The Geometry of Constant Curvature Manifolds The standard models of Sn(K0) and Hn(K0) in Rn+1 [1] Stereographic projection and the conformal model of H [6] Conformal maps of Rn and the isometries of Hn [11] Totally geodesic submanifolds and geodesic spheres of Hn [15] Horospheres and equidistant hypersurfaces [19] Geodesic mappings; the projective model of Hn; Beltrami’s theorem [23] B* Curves in a Riemannian manifold Frenet frames and curvatures [29] Curves whose jtn curvature vanish [38] C, The Fundamental Equations for Submanifolds The normal connection and the Weingarten equations [46] Second fundamental forms and normal fundamental forms; the Codazzi-Mainardi equations [50] The Ricci equations [56] The fundamental theorem for submanifolds of Euclidean space [61] The fundamental theorem for submanifolds of constant curvature manifolds [74] D. First Consequences The curvatures of a hypersurface; Theorema Egregium; formula for the Gaussian curvature [95] The mean curvature normal; umbilics; all-umbilic submanifolds of Euclidean space [104] All-umbilic submanifolds of constant curvature manifolds [111] Positive curvature and convexity [117] E. Further Results Flat ruled surfaces in Rm [125] Flat ruled surfaces in constant curvature manifolds [127] Curves on hypersurfaces [131] F. Complete Surfaces of Constant Curvature Modifications of results for surfaces in R3 [134] Surfaces of constant curvature in [138] surfaces with constant curvature 0 [139] the Hopf map [158] Surfaces of constant curvature in H3 [163] Jorgens theorem; surfaces of constant curvature 0 [165] surfaces of constant curvature -1 [172] rotation surfaces of constant curvature between -1 and 0 [173] G. Hypersurfaces of Constant Curvature in Higher Dimensions Hypersurfaces of constant curvature in dimensions > 3 [174] The Ricci tensor; Einstein spaces, hypersurfaces which are Einstein spaces [176] Hypersurfaces of the same constant curvature as the ambient manifold [180] Addendum 1. The Laplacian [187] Addendum 2. The * Operator and the Laplacian on Forms; Hodge’s Theorem [202] Addendum 3. When are two Riemannian Manifolds Isometric? [219] Addendum 4. Better Imbedding Invariants [235] Problems. [276] Chapter 8. The Second Variation Two-parameter variations; the second variation formula [296] Jacobi fields; conjugate points [307] Minimizing and non-minimizing geodesics [315] The Hadamard-Cartan theorem [328] The Sturm comparison theorem; Bonnet’s theorem [332] Generalizations to higher dimensions; the Morse-Schoenberg comparison theorem; Myer’s theorem; the Rauch comparison theorem [338] Synge’s lemma; Synge’s theorem [352] Cut points; Klingenberg’s theorem [362] Problems. [377] Chapter 9. Variations of Length, Area, and Volume [3] Variation of area for normal variations of surfaces in JR ; minimal surfaces [379] Isothermal coordinates on minimal surfaces; Bernstein's theorem [385] Weierstrass-Enneper representation [391] Associated minimal surfaces; Schwarz’s theorem [400] Change of orientation; Henneberg's minimal surface [403] Classical calculus of variations in n dimensions [410] Variation of volume formula [416] Isoperimetric problems [426] Addendum 1. Isothermal Coordinates [455] Addendum 2. Immersed Spheres with Constant Mean Curvature [501] Addendum 3. Imbedded Surfaces with Constant Mean Curvature [507] Addendum 4. The Second Variation of Volume [513] Problems. [540] Mini-Bibliography for Volume IV [547] Notation Index [549] Index [553]

Chapter 10. And Now a Brief Message from our Sponsor 1. First Order PDEs Linear first order PDEs; characteristic curves; Cauchy problem for free initial curves [1] Quasi-linear first order PDEs; characteristic curves; Cauchy problem for free initial conditions; characteristic initial conditions [13] General first order PDEs; Monge cone; characteristic curves of a solution; characteristic strips; Cauchy problem for free initial data; characteristic initial data [19] First order PDEs in n variables [36] 2. Free Initial Manifolds for Higher Order Equations [40] 3. Systems of First Order PDEs [50] 4. The Cauchy-Kowalewski Theorem [57] 5. Classification of Second Order PDEs Classification of semi-linear equations [68] Reduction to normal forms [73] Classification of general second order equations [83] 6. The Prototypical PDEs of Physics The wave equation; the heat equation; Laplace’s equation [87] Elementary properties [99] 7. Hyperbolic Systems in Two Variables [105] 8. Hyperbolic Second Order Equations in Two Variables First reduction of the problem [118] New system of characteristic equations [122] Characteristic initial data [136] Monge-Ampere equations [138] 9. Elliptic Solutions of Second Order Equations in Two Variables [140] Addendum 1. Differential Systems; The Cartan-Kahler Theorem [158] Addendum 2. An Elementary Maximum Principal [181] Problem. [191] Chapter 11. Existence and Non-existence of Isometric Imbeddings Non-imbeddability theorems; exteriorly orthogonal bilinear forms; index of nullity and index of relative nullity [192] The Darboux equation [206] Burstin-Janet-Cartan theorem [214] Addendum. The Embedding Problem via Differential Systems [230] Problems. [241] Chapter 12. Rigidity Rigidity in higher dimensions; type number [244] Bendings, warpings, and infinitesimal bendings [249] R3-valued differential forms, the support function, and Minkowski's formulas [265] Infinitesimal rigidity of convex surfaces 273 Cohn-Vossen's theorem 280 Minkowski’s theorem 293 Christoffel’s theorem [299] Other problems, solved and unsolved [302] Local problems — the role of the asymptotic curves [314] Other classical results [323] E.E. Levi’s theorems and Schilt’s theorem [331] Surfaces in and [343] Rigidity for higher codimension [361] Addendum. Infinitesimal Bendings of Rotation Surfaces 370 Problems. [381] Chapter 13. The Generalized Gauss-Bonnet Theorem and What It Means for Mankind Historical remarks 1. Operations on Bundles [385] Bundle maps and principal bundle maps; Whitney sums and induced bundles; the covering homotopy theorem 2. Grassmannians and Universal Bundles [388] 3. The Pfaffian [400] 4. Defining the Euler Class in Terms of a Connection [416] The Euler class [427] The Class C(£) [431] The Gauss-Bonnet-Chern Theorem [437] 5. The Concept of Characteristic Classes [443] 6. The Cohomology of Homogeneous Spaces The C∞ structure of homogeneous spaces [447] Invariant forms [453] 7. A Smattering of Classical Invariant Theory The Capelli identities [466] The first fundamental theorem of invariant theory for O(n) and SO(n) [480] 8. An Easier Invariance Problem [486] 9. The Cohomology of the Oriented Grassmannians Computation of the cohomology; Pontryagin classes [495] Describing the characteristic classes in terms of a connection [507] 10. The Weil Homomorphism [519] 11. Complex Bundles Hermitian inner products, the unitary group, and complex Grassmannians [524] The cohomology of the complex Grassmannians; Chern classes [530] Relations between the Chern classes and the Pontryagin and Euler classes [537] 12. Valedictory [543] Addendum 1. Invariant Theory for the Unitary Group [546] Addendum 2. Recovering the Differential Forms; The Gauss-Bonnet-Chern Theorem for Manifolds—with—boundary [558] Bibliography [577] A. Other Topics in Differential Geometry [578] B. Books [604] Supplement [632] C. Journal Articles [634] Notation Index [647] Index [651] Contents of Volume III Chapter 1. The Fundamental Equations for Hypersurfaces Chapter 2. Elements of the Theory of Surfaces in R3 Chapter 3. A Compendium of Surfaces Chapter 4. Curves on Surfaces Chapter [5.] Complete Surfaces of Constant Curvature Chapter [6.] The Gauss-Bonnet Theorem and Related Topics Contents of Volume IV Chapter 7. Higher Dimensions and Codimensions Chapter 8. The Second Variation Chapter 9. Variations of Length, Area, and Volume

Item type | Home library | Shelving location | Call number | Materials specified | Status | Date due | Barcode | Course reserves |
---|---|---|---|---|---|---|---|---|

Libros | Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 53 Sp761-2 (Browse shelf) | Vol. III | Available | A-4842 | ||

Libros | Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 53 Sp761-2 (Browse shelf) | Vol. IV | Available | A-4843 | ||

Libros | Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 53 Sp761-2 (Browse shelf) | Vol. V | Available | A-4844 |

La biblioteca posee los vols. 3, 4 y 5. Para los vols. 1 y 2, consulte la 1ª ed. AR-BbIMB

Incluye referencias bibliográficas.

CONTENTS --

Introduction v --

Chapter 1. The Fundamental Equations for Hypersurfaces --

Covariant differentiation in a submanifold of a Riemannian manifold [1] --

The second fundamental form, the Gauss formulas, and Gauss’ equation; Synge’s inequality [5] --

The Weingarten equations and the Codazzi-Mainardi equations for hypersurfaces [10] --

The classical tensor analysis description [17] --

The moving frame description [23] --

Addendum. Auto-parallel and Totally Geodesic Submanifolds [32] --

Problems [40] --

Chapter 2. Elements of the Theory of Surfaces in IR3 --

The first and second fundamental forms [45] --

Classification of points on a surface; the osculating paraboloid and the Dupin indicative [51] --

Principal directions and curvatures, asymptotic directions, flat points and umbilics; all-umbilic surfaces [68] --

The classical Gauss formulas, Weingarten equations, Gauss equation, and Codazzi-Mainardi equations [73] --

Fundamental theorem of surface theory [79] --

The third fundamental form [87] --

Convex surfaces; Hadamard’s theorem [90] --

The fundamental equations via moving frames [96] --

Review of Lie groups and their role in curve theory and affine curve theory [100] --

Application of Lie groups to surface theory; the fundamental equations and the structural equations of S0(3) [109] --

Affine surface theory; the osculating paraboloids and the affine invariant conformal structure [113] --

The special affine first fundamental form [123] --

Quadratic and cubic forms; apolarity [135] --

The affine normal direction; the special affine normal [144] --

The special affine Gauss formulas, and special affine second fundamental form [153] --

The Pick invariant; surfaces with Pick invariant 0 [164] --

The special affine Weingarten formulas [178] --

The special affine Codazzi-Mainardi equations; the fundamental theorem of special affine surface theory [187] --

Problems [195] --

Chapter 3. A Compendium of Surfaces --

Basic calculations [197] --

The classical flat surfaces [205] --

Ruled surfaces [213] --

Quadric surfaces [220] --

Surfaces of revolution [228] --

rotation surfaces of constant curvature [235] --

Minimal surfaces [242] --

Addendum. Envelopes of 1-parameter families of planes [255] --

Problems [264] --

Chapter 4, Curves on Surfaces --

Normal and geodesic curvature [272] --

The Darboux frame; geodesic torsion [277] --

Laguerre’s theorem [282] --

General properties of lines of curvature, asymptotic curves, and geodesics 281i --

The Beltrami -Enneper theorem [291] --

Lines of curvature and Dupin’s theorem [295] --

Conformal maps of IR3 ; Liouville ’s theorem [302] --

Geodesics and Clairaut ’ s theorem [313] --

Addendum 1. Special Parameter Curves [320] --

Addendum 2. Singularities of Line Fields [324] --

Problems [333] --

Chapter 5. Complete Surfaces of Constant Curvature --

Hilbert’s lemma; complete surfaces of constant curvature K > 0 [347] --

Analysis of flat surfaces; the classical classification of developable surfaces [349] --

Complete flat surfaces [363] --

Complete surfaces of constant curvature K < 0 [367] --

Chapter 6. The Gauss-Bonnet Theorem and Related Topics --

The connection form for an orthonormal moving frame on a surface; the change in angle under parallel translation [386] --

The integral of K dA over a polygonal region [392] --

The Gauss-Bonnet theorem; consequences [399] --

Total absolute curvature of surfaces [409] --

Surfaces of minimal total absolute curvature [413] --

Total curvature of curves; Fenchel’s theorem, and Fary-Milnor theorem [421] --

Addendum 1. Compact Surfaces with Constant Negative Curvature 430 Addendum 2. The Degree of the Normal Map [439] --

Problems [444] --

Corrections [447] --

Mini-Bibliography for Volume III [451] --

Notation Index [453] --

Index [457] --

Chapter 7* Higher Dimensions and Codimensions --

A» The Geometry of Constant Curvature Manifolds --

The standard models of Sn(K0) and Hn(K0) in Rn+1 [1] --

Stereographic projection and the conformal model of H [6] --

Conformal maps of Rn and the isometries of Hn [11] --

Totally geodesic submanifolds and geodesic spheres of Hn [15] --

Horospheres and equidistant hypersurfaces [19] --

Geodesic mappings; the projective model of Hn; Beltrami’s theorem [23] --

B* Curves in a Riemannian manifold --

Frenet frames and curvatures [29] --

Curves whose jtn curvature vanish [38] --

C, The Fundamental Equations for Submanifolds --

The normal connection and the Weingarten equations [46] --

Second fundamental forms and normal fundamental forms; the Codazzi-Mainardi equations [50] --

The Ricci equations [56] --

The fundamental theorem for submanifolds of Euclidean space [61] --

The fundamental theorem for submanifolds of constant curvature manifolds [74] --

D. First Consequences --

The curvatures of a hypersurface; Theorema Egregium; formula for the Gaussian curvature [95] --

The mean curvature normal; umbilics; all-umbilic submanifolds of Euclidean space [104] --

All-umbilic submanifolds of constant curvature manifolds [111] --

Positive curvature and convexity [117] --

E. Further Results --

Flat ruled surfaces in Rm [125] --

Flat ruled surfaces in constant curvature manifolds [127] --

Curves on hypersurfaces [131] --

F. Complete Surfaces of Constant Curvature --

Modifications of results for surfaces in R3 [134] --

Surfaces of constant curvature in [138] --

surfaces with constant curvature 0 [139] --

the Hopf map [158] --

Surfaces of constant curvature in H3 [163] --

Jorgens theorem; surfaces of constant curvature 0 [165] --

surfaces of constant curvature -1 [172] --

rotation surfaces of constant curvature between -1 and 0 [173] --

G. Hypersurfaces of Constant Curvature in Higher Dimensions --

Hypersurfaces of constant curvature in dimensions > 3 [174] --

The Ricci tensor; Einstein spaces, hypersurfaces which are Einstein spaces [176] --

Hypersurfaces of the same constant curvature as the ambient manifold [180] --

Addendum 1. The Laplacian [187] --

Addendum 2. The * Operator and the Laplacian on Forms; Hodge’s Theorem [202] --

Addendum 3. When are two Riemannian Manifolds Isometric? [219] --

Addendum 4. Better Imbedding Invariants [235] --

Problems. [276] --

Chapter 8. The Second Variation --

Two-parameter variations; the second variation formula [296] --

Jacobi fields; conjugate points [307] --

Minimizing and non-minimizing geodesics [315] --

The Hadamard-Cartan theorem [328] --

The Sturm comparison theorem; Bonnet’s theorem [332] --

Generalizations to higher dimensions; the Morse-Schoenberg comparison theorem; Myer’s theorem; the Rauch comparison theorem [338] --

Synge’s lemma; Synge’s theorem [352] --

Cut points; Klingenberg’s theorem [362] --

Problems. [377] --

Chapter 9. Variations of Length, Area, and Volume [3] --

Variation of area for normal variations of surfaces in JR ; minimal surfaces [379] --

Isothermal coordinates on minimal surfaces; Bernstein's theorem [385] --

Weierstrass-Enneper representation [391] --

Associated minimal surfaces; Schwarz’s theorem [400] --

Change of orientation; Henneberg's minimal surface [403] --

Classical calculus of variations in n dimensions [410] --

Variation of volume formula [416] --

Isoperimetric problems [426] --

Addendum 1. Isothermal Coordinates [455] --

Addendum 2. Immersed Spheres with Constant Mean Curvature [501] --

Addendum 3. Imbedded Surfaces with Constant Mean Curvature [507] --

Addendum 4. The Second Variation of Volume [513] --

Problems. [540] --

Mini-Bibliography for Volume IV [547] --

Notation Index [549] --

Index [553] --

Chapter 10. And Now a Brief Message from our Sponsor --

1. First Order PDEs --

Linear first order PDEs; characteristic curves; Cauchy problem for free initial curves [1] --

Quasi-linear first order PDEs; characteristic curves; Cauchy problem for free initial conditions; characteristic initial conditions [13] --

General first order PDEs; Monge cone; characteristic curves of a solution; characteristic strips; Cauchy problem for free initial data; characteristic initial data [19] --

First order PDEs in n variables [36] --

2. Free Initial Manifolds for Higher Order Equations [40] --

3. Systems of First Order PDEs [50] --

4. The Cauchy-Kowalewski Theorem [57] --

5. Classification of Second Order PDEs --

Classification of semi-linear equations [68] --

Reduction to normal forms [73] --

Classification of general second order equations [83] --

6. The Prototypical PDEs of Physics --

The wave equation; the heat equation; Laplace’s equation [87] --

Elementary properties [99] --

7. Hyperbolic Systems in Two Variables [105] --

8. Hyperbolic Second Order Equations in Two Variables --

First reduction of the problem [118] --

New system of characteristic equations [122] --

Characteristic initial data [136] --

Monge-Ampere equations [138] --

9. Elliptic Solutions of Second Order Equations in Two Variables [140] --

Addendum 1. Differential Systems; The Cartan-Kahler Theorem [158] --

Addendum 2. An Elementary Maximum Principal [181] --

Problem. [191] --

Chapter 11. Existence and Non-existence of Isometric Imbeddings --

Non-imbeddability theorems; exteriorly orthogonal bilinear forms; index of nullity and index of relative nullity [192] --

The Darboux equation [206] --

Burstin-Janet-Cartan theorem [214] --

Addendum. The Embedding Problem via Differential Systems [230] --

Problems. [241] --

Chapter 12. Rigidity --

Rigidity in higher dimensions; type number [244] --

Bendings, warpings, and infinitesimal bendings [249] --

R3-valued differential forms, the support function, and Minkowski's formulas [265] --

Infinitesimal rigidity of convex surfaces 273 Cohn-Vossen's theorem 280 Minkowski’s theorem 293 Christoffel’s theorem [299] --

Other problems, solved and unsolved [302] --

Local problems — the role of the asymptotic curves [314] --

Other classical results [323] --

E.E. Levi’s theorems and Schilt’s theorem [331] --

Surfaces in and [343] --

Rigidity for higher codimension [361] --

Addendum. Infinitesimal Bendings of Rotation Surfaces 370 Problems. [381] --

Chapter 13. The Generalized Gauss-Bonnet Theorem and What It Means for Mankind Historical remarks --

1. Operations on Bundles [385] --

Bundle maps and principal bundle maps; Whitney sums and induced bundles; the covering homotopy theorem --

2. Grassmannians and Universal Bundles [388] --

3. The Pfaffian [400] --

4. Defining the Euler Class in Terms of a Connection [416] --

The Euler class [427] --

The Class C(£) [431] --

The Gauss-Bonnet-Chern Theorem [437] --

5. The Concept of Characteristic Classes [443] --

6. The Cohomology of Homogeneous Spaces --

The C∞ structure of homogeneous spaces [447] --

Invariant forms [453] --

7. A Smattering of Classical Invariant Theory --

The Capelli identities [466] --

The first fundamental theorem of invariant theory for O(n) and SO(n) [480] --

8. An Easier Invariance Problem [486] --

9. The Cohomology of the Oriented Grassmannians --

Computation of the cohomology; Pontryagin classes [495] --

Describing the characteristic classes in terms of a connection [507] --

10. The Weil Homomorphism [519] --

11. Complex Bundles --

Hermitian inner products, the unitary group, and complex Grassmannians [524] --

The cohomology of the complex Grassmannians; Chern classes [530] --

Relations between the Chern classes and the Pontryagin and Euler classes [537] --

12. Valedictory [543] --

Addendum 1. Invariant Theory for the Unitary Group [546] --

Addendum 2. Recovering the Differential Forms; The Gauss-Bonnet-Chern Theorem for Manifolds—with—boundary [558] --

Bibliography [577] --

A. Other Topics in Differential Geometry [578] --

B. Books [604] --

Supplement [632] --

C. Journal Articles [634] --

Notation Index [647] --

Index [651] --

Contents of Volume III --

Chapter 1. The Fundamental Equations for Hypersurfaces --

Chapter 2. Elements of the Theory of Surfaces in R3 --

Chapter 3. A Compendium of Surfaces --

Chapter 4. Curves on Surfaces --

Chapter [5.] --

Complete Surfaces of Constant Curvature --

Chapter [6.] --

The Gauss-Bonnet Theorem and Related Topics --

Contents of Volume IV --

Chapter 7. Higher Dimensions and Codimensions --

Chapter 8. The Second Variation --

Chapter 9. Variations of Length, Area, and Volume --

MR, 82g:53003a

There are no comments on this title.