A Comprehensive Introduction to Differential Geometry, Vol. 2, Third Edition

About The Book

The volume begins the study of modern differential geometry, following a semi-historical path and encountering classical differential geometry in Chapters 3 and 4. Chapter 1 focuses on curves in the plane and in space, discussing the curvature of plane curves, convex curves, curvature and torsion of space curves, Serret-Frenet formulas, and the classification of curves under special affine motions. This chapter provides a solid foundation for understanding the geometric properties of curves in different dimensions. In Chapter 2, the book explores what was known about surfaces before Gauss, including Euler’s Theorem and Meusnier’s Theorem. This historical perspective helps readers appreciate the contributions of early mathematicians to the field of differential geometry. Chapter 3 delves into the curvature of surfaces in space, discussing Gauss’s theory of surfaces, the Gauss map, Gaussian curvature, the Weingarten map, and the first and second fundamental forms. The chapter also covers important concepts such as the Theorema Egregium, geodesics on a surface, and the integral of curvature over a geodesic triangle. Chapter 4 extends the discussion to the curvature of higher-dimensional manifolds, including Riemann’s inaugural lecture, the hypotheses at the foundations of geometry, and the birth of the Riemann curvature tensor. This chapter provides a comprehensive understanding of the curvature properties of higher-dimensional spaces. Chapter 5 introduces the absolute differential calculus (Ricci calculus), covering covariant derivatives, Ricci’s Lemma, Ricci’s identities, the curvature tensor, classical connections, the torsion tensor, geodesics, and Bianchi’s identities. This chapter is essential for understanding the mathematical framework used to describe the curvature of manifolds. Chapter 6 discusses the r operator, including Koszul connections, covariant derivatives, parallel translation, the torsion tensor, the Levi-Civita connection, the curvature tensor, Bianchi’s identities, geodesics, and the first variation formula. This chapter provides a detailed exploration of the mathematical tools used in differential geometry. Chapter 7 focuses on the repère mobile (the moving frame), covering topics such as moving frames, structural equations of Euclidean space, structural equations of a Riemannian manifold, adapted frames, and Cartan connections. The chapter also discusses the curvature determines the metric, manifolds of constant curvature, Schur’s Theorem, and conformally equivalent manifolds. Finally, Chapter 8 explores connections in principal bundles, including principal bundles, Lie groups acting on manifolds, Cartan connections, Ehresmann connections, parallel translation, covariant derivatives, the covariant differential and curvature form, the dual form and torsion form, structural equations, and Bianchi’s identities. This chapter provides a comprehensive understanding of the geometric structures that arise in the study of differential geometry. Overall, A Comprehensive Introduction to Differential Geometry, Volume Two, Third Edition is an invaluable resource for students, educators, and lifelong learners. Spivak’s writing style is clear and precise, making complex topics approachable for readers at all levels. The book’s focus on problem-solving, mathematical rigor, and detailed explanations makes it a must-have for anyone interested in the field of differential geometry.