Publisher finelybook 出版社: World Scientific Publishing
Publication Date 出版日期: October 29, 2025
Language 语言: English
Print length 页数: 376 pages
ISBN-10: 9819816165
ISBN-13: 9789819816163
Book Description
This textbook focuses on the study of curves and surfaces, applying modern differential equation theory to geometric problems. By introducing isothermal parameters, it simplifies the fundamental equations of surface theory, leading to clear derivations of results like those of H Hopf and S Bernstein for surfaces of constant and vanishing mean curvature. Deviating from traditional approaches, the book first treats n-dimensional Riemannian spaces by a corresponding metric, then constructs Riemannian manifolds through transition conditions. The ultimate goal is to prove the Hadamard–Cartan theorem on the diffeomorphic character of the exponential mapping in Riemannian manifolds with nonpositive sectional curvature. By considering curves and surfaces in their optimal parametrization, the resulting ODEs and complex PDEs can be analytically solved, eliminating the need for intricate tensor calculus. The approach follows that of G Monge in his treatise L'Application de l'Analyse à la Géométrie, applying analytical techniques to geometric problems. Building on this foundation, the book uses modern theory of ODEs and PDEs to study the local and global results for curves and surfaces and their relevant curvatures.
Editorial Reviews
About the Author
Friedrich Sauvigny studied Mathematics with Physics as a minor from 1971 to 1978 at the University of Göttingen under the supervision of Professor Erhard Heinz. He earned his Diploma in 1978, his Dr. rer. nat. in 1981, and his Habilitation (Dr. rer. nat. habil.) in 1989 at the same institution.
During this time, he worked as a Scientific Assistant at RWTH Aachen and TU Clausthal from 1978 to 1989. He then spent a research stay at the University of Bonn, invited by Professor Stefan Hildebrandt, from 1989 to 1990, where an intensive scientific collaboration began and continued until 2015.
From 1990 to 1992, Friedrich Sauvigny worked as an Assistant Professor at the University of Wisconsin–Milwaukee. In 1992, he was appointed Professor on the Chair of Mathematics, especially Analysis, at the Brandenburg University of Technology. He retired in 2019. From 2022 to 2024, he taught specialized courses on Differential Geometry and Riemannian Geometry, which contributed to the development of this book project.
Friedrich Sauvigny has authored a two-volume textbook on partial differential equations, a basic textbook on analysis, and a graduate-level textbook on the spectral theory of self-adjoint operators. Together with Ulrich Dierkes and Stefan Hildebrandt, he co-authored a comprehensive book on minimal surfaces. In the research area of surfaces with prescribed mean curvature, he has published numerous original papers in established mathematical journals.
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