The book contains a mathematical background section in the first chapter to outline some important pre-required mathematical issues. Apart from general background knowledge in a number of mathematical branches such as calculus, geometry and algebra, an important requirement for the reader and user of this book is familiarity with the terminology, notation and concepts of tensor calculus at reasonable level since many of the notations and concepts of differential geometry in its modern style are based on tensor calculus. It can therefore be used as part of a course on tensor calculus as well as a textbook or a reference for an intermediate-level course on differential geometry of curves and surfaces. The book is prepared, to some extent, as part of tutorials about topics and applications related to tensor calculus.

The formulation and presentation are largely based on a tensor calculus approach, which is the dominant trend in the modern mathematical literature of this subject, rather than the geometric approach which is usually found in some old style books.

Preface The present book is about differential geometry of space curves and surfaces. Table of contents : Preface Nomenclature 1: Preliminaries 1.1: Differential Geometry 1.2: General Remarks, Conventions and Notations 1.3: Classifying the Properties of Curves and Surfaces 1.3.1: Local versus Global Properties 1.3.2: Intrinsic versus Extrinsic Properties 1.4: General Mathematical Background 1.4.1: Geometry and Topology 1.4.2: Functions 1.4.3: Coordinates, Transformations and Mappings 1.4.4: Intrinsic Distance 1.4.5: Basis Vectors 1.4.6: Flat and Curved Spaces 1.4.7: Homogeneous Coordinate Systems 1.4.8: Geodesic Coordinates 1.4.9: Christoffel Symbols for Curves and Surfaces 1.4.10: Riemann-Christoffel Curvature Tensor 1.4.11: Ricci Curvature Tensor and Scalar 1.5: Exercises 2: Curves in Space 2.1: General Background about Curves 2.2: Mathematical Description of Curves 2.3: Curvature and Torsion of Space Curves 2.3.1: Curvature 2.3.2: Torsion 2.4: Geodesic Torsion 2.5: Relationship between Curve Basis Vectors and their Derivatives 2.6: Osculating Circle and Sphere 2.7: Parallelism and Parallel Propagation 2.8: Exercises 3: Surfaces in Space 3.1: General Background about Surfaces 3.2: Mathematical Description of Surfaces 3.3: Surface Metric Tensor 3.3.1: Arc Length 3.3.2: Surface Area 3.3.3: Angle Between Two Surface Curves 3.4: Surface Curvature Tensor 3.5: First Fundamental Form 3.6: Second Fundamental Form 3.6.1: Dupin Indicatrix 3.7: Third Fundamental Form 3.8: Fundamental Forms 3.9: Relationship between Surface Basis Vectors and their Derivatives 3.9.1: Codazzi-Mainardi Equations 3.10: Sphere Mapping 3.11: Global Surface Theorems 3.12: Exercises 4: Curvature 4.1: Curvature Vector 4.2: Normal Curvature 4.2.1: Meusnier Theorem 4.3: Geodesic Curvature 4.4: Principal Curvatures and Directions 4.5: Gaussian Curvature 4.6: Mean Curvature 4.7: Theorema Egregium 4.8: Gauss-Bonnet Theorem 4.9: Local Shape of Surface 4.10: Umbilical Point 4.11: Exercises 5: Special Curves 5.1: Straight Line 5.2: Plane Curve 5.3: Involute and Evolute 5.4: Bertrand Curve 5.5: Spherical Indicatrix 5.6: Spherical Curve 5.7: Geodesic Curve 5.8: Line of Curvature 5.9: Asymptotic Line 5.10: Conjugate Direction 5.11: Exercises 6: Special Surfaces 6.1: Plane Surface 6.2: Quadratic Surface 6.3: Ruled Surface 6.4: Developable Surface 6.5: Isometric Surface 6.6: Tangent Surface 6.7: Minimal Surface 6.8: Exercises 7: Tensor Differentiation over Surfaces 7.1: Exercises References Author Notes 8: Footnotes Citation preview

Differential geometry surfaces