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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.
This book is based on the lecture notes of the course Differential Geometry taught at the Faculty of Sciences of the University of Porto (Portugal) in the academic years 1992–93 and 1993–94. The exercises included in the text are rarely routine, although few are really difficult, and were chosen on the assumption that a good exercise, with a medium level of difficulty, should reward the students’ effort by teaching them something