A classic text on differential geometry, this book offers a comprehensive introduction to the subject for advanced undergraduate and graduate students. It covers topics such as tangent spaces, vector fields, and the curvature tensor, and provides numerous examples and exercises to aid understanding.
Focusing on the geometry of curves and surfaces, this book serves as an introductory resource for graduate students, reflecting years of teaching experience. It begins with twisted curves and introduces moving axes, linking to established methods by Darboux and Cesaro. The following chapters explore surface geometry using Gauss's approach, emphasizing differential equations without delving into algebraic treatments. It covers various surface types, applicability, and the recent Weingarten method, supplemented by numerous examples to enhance understanding and geometrical thinking.
The book delves into the evolution of tensor calculus since its inception, highlighting its significance in Einstein's General Theory of Relativity and Riemannian Geometry. It begins with a thorough development of tensor calculus in Euclidean 3-space before extending it to Riemannian spaces of any dimension. The latter chapters focus on applying this calculus to the differential geometry of surfaces in 3-space, incorporating advanced concepts like Levi-Civita parallelism, enhancing the foundational material from the author's earlier work.
A thorough, complete, and unified introduction, this volume affords exceptional insights into coordinate geometry. Invariants of conic sections and quadric surfaces receive full treatments. Algebraic equations on the first degree in two and three unknowns are carefully reviewed. Throughout the book, results are formulated precisely, with clearly stated theorems. More than 500 helpful exercises. 1939 edition.