Differential Geometry

This textbook provides a concise introduction to the differential geometry of curves and surfaces in three-dimensional space, tailored for undergraduate students with a solid foundation in mathematical analysis and linear algebra. The book emphasizes the geometric content of the subject, aiming to quickly cover fundamental topics such as the isoperimetric inequality and the Gauss–Bonnet theorem. This approach allows the author to extend beyond the typical content of introductory books and include additional important geometric results, such as curves and surfaces of constant width, the classification of complete surfaces of non-negative constant curvature, and Hadamard’s theorem on surfaces of non-positive curvature. This range of topics offers greater variety for an introductory course. Preface Contents Chapter 1 Differentiable Curves 1.1 Velocity and Arc Length 1.2 Acceleration, Curvature and the Frenet Trihedron 1.3 Planar Curves 1.4 Contact of Curves 1.5 Convex Curves 1.6 Curves of Constant Width 1.7 Theorem of the Four Vertices 1.8 The Isoperimetric Inequality Chapter 2 Regular Surfaces 2.1 Definition and Examples 2.2 Change of Parameters, Level Surfaces 2.3 Differentiable Functions on Surfaces, Tangent Space 2.4 Orientability 2.5 Areas, Lengths, and Angles: The First Fundamental Form Chapter 3 The Geometry of the Gauss Map 3.1 The Gauss Map and its Derivative 3.2 The Second Fundamental Form 3.3 Vector Fields Chapter 4 The Intrinsic Geometry of Surfaces 4.1 Conformal Mappings and Isometries 4.2 Gauss’s Theorema Egregium 4.3 Covariant Derivative, Parallel Transport, Geodesic Curvature 4.4 The Divergence Theorem. First Variation of Area 4.5 The Gauss-Bonnet Theorem 4.6 Minimizing Properties of Geodesics Appendix: Rotation Index Chapter 5 The Global Geometry of Surfaces 5.1 Complete Surfaces 5.2 Coverings 5.3 Complete Surfaces of Non-Positive Curvature 5.4 Ovals (First Part): The Rigidity of the Sphere 5.5 Ovals: Areas and Volumes; Surfaces of Constant Width 5.6 Abstract Surfaces. The Hyperbolic Plane 5.7 Complete Surfaces of Constant Curvature References Index