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Lectures on Differential Geometry: Second Edition
 
Shlomo Sternberg
Lectures on Differential Geometry
AMS Chelsea Publishing: An Imprint of the American Mathematical Society
Softcover ISBN:  978-1-4704-7898-8
Product Code:  CHEL/316.S
List Price: $69.00
MAA Member Price: $62.10
AMS Member Price: $62.10
eBook ISBN:  978-1-4704-7910-7
Product Code:  CHEL/316.E
List Price: $65.00
MAA Member Price: $58.50
AMS Member Price: $58.50
Softcover ISBN:  978-1-4704-7898-8
eBook: ISBN:  978-1-4704-7910-7
Product Code:  CHEL/316.S.B
List Price: $134.00 $101.50
MAA Member Price: $120.60 $91.35
AMS Member Price: $120.60 $91.35
Add to cart
Lectures on Differential Geometry
Click above image for expanded view
Lectures on Differential Geometry: Second Edition
Shlomo Sternberg
AMS Chelsea Publishing: An Imprint of the American Mathematical Society
Softcover ISBN:  978-1-4704-7898-8
Product Code:  CHEL/316.S
List Price: $69.00
MAA Member Price: $62.10
AMS Member Price: $62.10
eBook ISBN:  978-1-4704-7910-7
Product Code:  CHEL/316.E
List Price: $65.00
MAA Member Price: $58.50
AMS Member Price: $58.50
Softcover ISBN:  978-1-4704-7898-8
eBook ISBN:  978-1-4704-7910-7
Product Code:  CHEL/316.S.B
List Price: $134.00 $101.50
MAA Member Price: $120.60 $91.35
AMS Member Price: $120.60 $91.35
Add to cart
  • Book Details
     
     
    AMS Chelsea Publishing
    Volume: 316; 1983; 442 pp
    MSC: Primary 53

    This book is based on lectures given at Harvard University during the academic year 1960–1961. The presentation assumes knowledge of the elements of modern algebra (groups, vector spaces, etc.) and point-set topology and some elementary analysis. Rather than giving all the basic information or touching upon every topic in the field, this work treats various selected topics in differential geometry. The author concisely addresses standard material and spreads exercises throughout the text. his reprint has two additions to the original volume: a paper written jointly with V. Guillemin at the beginning of a period of intense interest in the equivalence problem and a short description from the author on results in the field that occurred between the first and the second printings.
  • Table of Contents
     
     
    • Front Cover
    • Preface to the Second Edition
    • Preface to the First Edition
    • Contents
    • CHAPTER I Algebraic Preliminaries
    • 1. Tensor products of vector spaces.
    • 2. The tensor algebra of a vector space.
    • 3. The contravariant and symmetric algebras.
    • 4. Exterior algebra.
    • 5. Exterior equations.
    • CHAPTER II Differentiable Manifolds
    • 1. Definitions.
    • 2. Differentiable maps.
    • 3. Sard's theorem.
    • 4. Partitions of unity, approximation theorems.
    • 5. The tangent space.
    • 6. The principal bundle.
    • 7. The tensor bundles.
    • 8. Vector fields and Lie derivatives.
    • CHAPTER Ill Integral Calculus On Manifolds
    • 1. The operator d.
    • 2. Chains and integration.
    • 3. Integration of densities.
    • 4. 0 and n-dimensional cohomology, degree.
    • 5. Frobenius' theorem.
    • 6. Darboux's theorem.
    • 7. Hamiltonian structures.
    • CHAPTER IV The Calculus of Variations
    • 1. Legendre transformations.
    • 2. Necessary conditions.
    • 3. Conservation laws.
    • 4. Sufficient conditions.
    • 5. Conjugate and focal points, Jacobi's condition.
    • 6. The Riemannian case.
    • 7. Completeness.
    • 8. lsometries.
    • CHAPTER V Lie Groups
    • I. Definitions.
    • 2. The invariant forms and the Lie algebra.
    • 3. Normal coordinates, exponential map.
    • 4. Closed subgroups.
    • 5. Invariant metrics.
    • 6. Forms with values in a vector space.
    • CHAPTER VI Differential Geometry of Euclidean Space
    • I. The equations of structure of Euclidean space.
    • 2. The equations of structure of a submanifold.
    • 3. The equations of structure of a Riemann manifold.
    • 4. Curves in Euclidean space.
    • 5. The second fundamental form.
    • 6. Surfaces.
    • CHAPTER VII The Geometry of G-Structures
    • 1. Principal and associated bundles, connections.
    • 2. G-structures.
    • 3. Prolongations.
    • 4. Structures of finite type.
    • 5. Connections on G-structures.
    • 6. The spray of a linear connection.
    • APPENDIX I Two Existence Theorems
    • APPENDIX II Outline of Theory of Integration on En
    • APPENDIX Ill AN ALGEBRAIC MODEL OF TRANSITIVE DIFFERENTIAL GEOMETRY
    • APPENDIX IV The Integrability Problem for Geometrical Structures
    • References for Appendix IV
    • References
    • Index
    • Back Cover
  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
AMS Chelsea Publishing
Volume: 316; 1983; 442 pp
MSC: Primary 53

This book is based on lectures given at Harvard University during the academic year 1960–1961. The presentation assumes knowledge of the elements of modern algebra (groups, vector spaces, etc.) and point-set topology and some elementary analysis. Rather than giving all the basic information or touching upon every topic in the field, this work treats various selected topics in differential geometry. The author concisely addresses standard material and spreads exercises throughout the text. his reprint has two additions to the original volume: a paper written jointly with V. Guillemin at the beginning of a period of intense interest in the equivalence problem and a short description from the author on results in the field that occurred between the first and the second printings.
  • Front Cover
  • Preface to the Second Edition
  • Preface to the First Edition
  • Contents
  • CHAPTER I Algebraic Preliminaries
  • 1. Tensor products of vector spaces.
  • 2. The tensor algebra of a vector space.
  • 3. The contravariant and symmetric algebras.
  • 4. Exterior algebra.
  • 5. Exterior equations.
  • CHAPTER II Differentiable Manifolds
  • 1. Definitions.
  • 2. Differentiable maps.
  • 3. Sard's theorem.
  • 4. Partitions of unity, approximation theorems.
  • 5. The tangent space.
  • 6. The principal bundle.
  • 7. The tensor bundles.
  • 8. Vector fields and Lie derivatives.
  • CHAPTER Ill Integral Calculus On Manifolds
  • 1. The operator d.
  • 2. Chains and integration.
  • 3. Integration of densities.
  • 4. 0 and n-dimensional cohomology, degree.
  • 5. Frobenius' theorem.
  • 6. Darboux's theorem.
  • 7. Hamiltonian structures.
  • CHAPTER IV The Calculus of Variations
  • 1. Legendre transformations.
  • 2. Necessary conditions.
  • 3. Conservation laws.
  • 4. Sufficient conditions.
  • 5. Conjugate and focal points, Jacobi's condition.
  • 6. The Riemannian case.
  • 7. Completeness.
  • 8. lsometries.
  • CHAPTER V Lie Groups
  • I. Definitions.
  • 2. The invariant forms and the Lie algebra.
  • 3. Normal coordinates, exponential map.
  • 4. Closed subgroups.
  • 5. Invariant metrics.
  • 6. Forms with values in a vector space.
  • CHAPTER VI Differential Geometry of Euclidean Space
  • I. The equations of structure of Euclidean space.
  • 2. The equations of structure of a submanifold.
  • 3. The equations of structure of a Riemann manifold.
  • 4. Curves in Euclidean space.
  • 5. The second fundamental form.
  • 6. Surfaces.
  • CHAPTER VII The Geometry of G-Structures
  • 1. Principal and associated bundles, connections.
  • 2. G-structures.
  • 3. Prolongations.
  • 4. Structures of finite type.
  • 5. Connections on G-structures.
  • 6. The spray of a linear connection.
  • APPENDIX I Two Existence Theorems
  • APPENDIX II Outline of Theory of Integration on En
  • APPENDIX Ill AN ALGEBRAIC MODEL OF TRANSITIVE DIFFERENTIAL GEOMETRY
  • APPENDIX IV The Integrability Problem for Geometrical Structures
  • References for Appendix IV
  • References
  • Index
  • Back Cover
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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