Elementary Differential Geometry - 1st Edition - ISBN: 9781483231709, 9781483268118

Elementary Differential Geometry

1st Edition

Authors: Barrett O'Neill
eBook ISBN: 9781483268118
Imprint: Academic Press
Published Date: 1st January 1966
Page Count: 422
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Elementary Differential Geometry focuses on the elementary account of the geometry of curves and surfaces.

The book first offers information on calculus on Euclidean space and frame fields. Topics include structural equations, connection forms, frame fields, covariant derivatives, Frenet formulas, curves, mappings, tangent vectors, and differential forms. The publication then examines Euclidean geometry and calculus on a surface. Discussions focus on topological properties of surfaces, differential forms on a surface, integration of forms, differentiable functions and tangent vectors, congruence of curves, derivative map of an isometry, and Euclidean geometry.

The manuscript takes a look at shape operators, geometry of surfaces in E, and Riemannian geometry. Concerns include geometric surfaces, covariant derivative, curvature and conjugate points, Gauss-Bonnet theorem, fundamental equations, global theorems, isometries and local isometries, orthogonal coordinates, and integration and orientation.

The text is a valuable reference for students interested in elementary differential geometry.

Table of Contents



Chapter I. Calculus on Euclidean Space

1. Euclidean Space

2. Tangent Vectors

3. Directional Derivatives

4. Curves in E3

5. 1-Forms

6. Differential Forms

7. Mappings

8. Summary

Chapter II. Frame Fields

1. Dot Product

2. Curves

3. The Frenet Formulas

4. Arbitrary-Speed Curves

5. Covariant Derivatives

6. Frame Fields

7. Connection Forms

8. The Structural Equations

9. Summary

Chapter III. Euclidean Geometry

1. Isometries of E3

2. The Derivative Map of an Isometry

3. Orientation

4. Euclidean Geometry

5. Congruence of Curves

6. Summary

Chapter IV. Calculus on a Surface

1. Surfaces in E3

2. Patch Computations

3. Differentiable Functions and Tangent Vectors

4. Differential Forms on a Surface

5. Mappings of Surfaces

6. Integration of Forms

7. Topological Properties of Surfaces

8. Manifolds

9. Summary

Chapter V. Shape Operators

1. The Shape Operator of M C E3

2. Normal Curvature

3. Gaussian Curvature

4. Computational Techniques

5. Special Curves in a Surface

6. Surfaces of Revolution

7. Summary

Chapter VI. Geometry of Surfaces in E3

1. The Fundamental Equations

2. Form Computations

3. Some Global Theorems

4. Isometries and Local Isometries

5. Intrinsic Geometry of Surfaces in E3

6. Orthogonal Coordinates

7. Integration and Orientation

8. Congru


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© Academic Press 1966
Academic Press
eBook ISBN:

About the Author

Barrett O'Neill

Barrett O'Neill is currently a Professor in the Department of Mathematics at the University of California, Los Angeles. He has written two other books in advanced mathematics.

Affiliations and Expertise

University of California, Los Angeles, California, U.S.A.

Ratings and Reviews