Differential Geometry and Its Visualization is suitable for graduate level courses in differential geometry, serving both students and teachers. It can also be used as a supplementary reference for research in mathematics and the natural and engineering sciences.
Differential geometry is the study of geometric objects and their properties using the methods of mathematical analysis. The classical theory of curves and surfaces in three-dimensional Euclidean space is presented in the first three chapters. The abstract and modern topics of tensor algebra, Riemannian spaces and tensor analysis are studied in the last two chapters. A great number of illustrating examples, visualizations and genuine figures created by the authors’ own software are included to support the understanding of the presented concepts and results, and to develop an adequate perception of the shapes of geometric objects, their properties and the relations between them.
Features
- Extensive, full colour visualisations
- Numerous exercises
- Self-contained and comprehensive treatment of the topic
Author(s): Eberhard Malkowsky, Ćemal Dolićanin, Vesna Veličković
Publisher: CRC Press/Chapman & Hall
Year: 2023
Language: English
Pages: 492
City: Boca Raton
Cover
Half Title
Title Page
Copyright Page
Contents
List of Figures
Preface
Symbol Description
Authors’ Biographies
CHAPTER 1: Curves in Three-Dimensional Euclidean Space
1.1. POINTS AND VECTORS
1.2. VECTOR-VALUED FUNCTIONS OF A REAL VARIABLE
1.3. THE GENERAL CONCEPT OF CURVES
1.4. SOME EXAMPLES OF PLANAR CURVES
1.5. THE ARC LENGTH OF A CURVE
1.6. THE VECTORS OF THE TRIHEDRON OF A CURVE
1.7. FRENET’S FORMULAE
1.8. THE GEOMETRIC SIGNIFICANCE OF CURVATURE AND TORSION
1.9. OSCULATING CIRCLES AND SPHERES
1.10. INVOLUTES AND EVOLUTES
1.11. THE FUNDAMENTAL THEOREM OF CURVES
1.12. LINES OF CONSTANT SLOPE
1.13. SPHERICAL IMAGES OF A CURVE
CHAPTER 2: Surfaces in Three-Dimensional Euclidean Space
2.1. SURFACES AND CURVES ON SURFACES
2.2. THE TANGENT PLANES AND NORMAL VECTORS OF A SURFACE
2.3. THE ARC LENGTH, ANGLES AND GAUSS’S FIRST FUNDAMENTAL COEFFICIENTS
2.4. THE CURVATURE OF CURVES ON SURFACES, GEODESIC AND NORMAL CURVATURE
2.5. THE NORMAL, PRINCIPAL, GAUSSIAN AND MEAN CURVATURE
2.6. THE SHAPE OF A SURFACE IN THE NEIGHBOURHOOD OF A POINT
2.7. DUPIN’S INDICATRIX
2.8. LINES OF CURVATURE AND ASYMPTOTIC LINES
2.9. TRIPLE ORTHOGONAL SYSTEMS
2.10. THE WEINGARTEN EQUATIONS
CHAPTER 3: The Intrinsic Geometry of Surfaces
3.1. THE CHRISTOFFEL SYMBOLS
3.2. GEODESIC LINES
3.3. GEODESIC LINES ON SURFACES WITH ORTHOGONAL PARAMETERS
3.4. GEODESIC LINES ON SURFACES OF REVOLUTION
3.5. THE MINIMUM PROPERTY OF GEODESIC LINES
3.6. ORTHOGONAL AND GEODESIC PARAMETERS
3.7. LEVI-CIVITÀ PARALLELISM
3.8. THEOREMA EGREGIUM
3.9. MAPS BETWEEN SURFACES
3.10. THE GAUSS-BONNET THEOREM
3.11. MINIMAL SURFACES
CHAPTER 4: Tensor Algebra and Riemannian Geometry
4.1. DIFFERENTIABLE MANIFOLDS
4.2. TRANSFORMATION OF BASES
4.3. LINEAR FUNCTIONALS AND DUAL SPACES
4.4. TENSORS OF SECOND ORDER
4.5. SYMMETRIC BILINEAR FORMS AND INNER PRODUCTS
4.6. TENSORS OF ARBITRARY ORDER
4.7. SYMMETRIC AND ANTI-SYMMETRIC TENSORS
4.8. RIEMANN SPACES
4.9. THE CHRISTOFFEL SYMBOLS
CHAPTER 5: Tensor Analysis
5.1. COVARIANT DIFFERENTIATION
5.2. THE COVARIANT DERIVATIVE OF AN (R, S)–TENSOR
5.3. THE INTERCHANGE OF ORDER FOR COVARIANT DIFFERENTIATION AND RICCI’S IDENTITY
5.4. BIANCHI’S IDENTITIES FOR THE COVARIANT DERIVATIVE OF THE TENSORS OF CURVATURE
5.5. BELTRAMI’S DIFFERENTIATORS
5.6. A GEOMETRIC MEANING OF THE COVARIANT DIFFERENTIATION, THE LEVI–CIVIT`A PARALLELISM
5.7. THE FUNDAMENTAL THEOREM FOR SURFACES
5.8. A GEOMETRIC MEANING OF THE RIEMANN TENSOR OF CURVATURE
5.9. SPACES WITH VANISHING TENSOR OF CURVATURE
5.10. AN EXTENSION OF FRENET’S FORMULAE
5.11. RIEMANN NORMAL COORDINATES AND THE CURVATURE OF SPACES
Bibliography
Index
