A comprehensive introduction to differential geometry / Michael Spivak.

Por: Spivak, MichaelEditor: 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: 53-01
Contenidos:
 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
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Libros Libros Instituto de Matemática, CONICET-UNS
Libros ordenados por tema 53 Sp761-2 (Browse shelf) Vol. III Available A-4842

GEOMETRÍA II

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

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