Elementary Differential Geometry
Published on 12. May 2014
422 pages
978-1-4832-6811-8 (ISBN)
Description
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.
More details
Person
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.
Content
PrefaceIntroductionChapter 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. SummaryChapter 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. SummaryChapter III. Euclidean Geometry 1. Isometries of E3 2. The Derivative Map of an Isometry 3. Orientation 4. Euclidean Geometry 5. Congruence of Curves 6. SummaryChapter 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. SummaryChapter 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. SummaryChapter 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. Congruence of Surfaces 9. SummaryChapter VII. Riemannian Geometry 1. Geometric Surfaces 2. Gaussian Curvature 3. Covariant Derivative 4. Geodesics 5. Length-Minimizing Properties of Geodesics 6. Curvature and Conjugate Points 7. Mappings that Preserve Inner Products 8. The Gauss-Bonnet Theorem 9. SummaryBibliographyAnswer to Odd-Numbered ExercisesIndex
