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.

Table of Contents

Chapter 1 Differentiable Curves

Chapter 2 Regular Surfaces

Chapter 3 The Geometry of the Gauss Map

Chapter 4 The Intrinsic Geometry of Surfaces

Chapter 5 The Global Geometry of Surfaces

About the Author

Paulo Ventura Araújo graduated with a degree in Mathematics from the University of Porto and obtained his doctorate from the University of Warwick. Currently, he is a professor at the Faculty of Sciences in Porto.