Overview
- Differential Geometry from an extrinsic and an intrinsic view point
- One semester lecture course for students of mathematics or STEM disciplines
- Contains a lot of examples and exercises
Part of the book series: Springer Studium Mathematik (Master) (SSMM)
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Table of contents (7 chapters)
Keywords
About this book
The first chapters give a historical overview of the field and contain an introduction to basic concepts such as manifolds and smooth maps, vector fields and flows, and Lie groups, leading up to the theorem of Frobenius. Subsequent chapters deal with the Levi-Civita connection, geodesics, the Riemann curvature tensor, a proof of the Cartan-Ambrose-Hicks theorem, as well as applications to flat spaces, symmetric spaces, and constant curvature manifolds. Also included are sections about manifolds with nonpositive sectional curvature, the Ricci tensor, the scalar curvature, and the Weyl tensor.
An additional chapter goes beyond the scope of a one semester lecture course and deals with subjects such as conjugate points and the Morse index, the injectivity radius, the group of isometries and the Myers-Steenrod theorem, and Donaldson's differential geometric approach to Lie algebra theory.
Authors and Affiliations
About the authors
Joel Robbin began his research in mathematical logic (and wrote a text book on this subject) and later moved on to dynamical systems and symplectic topology. In 1970 he proved a conjecture by Stephen Smale which asserts that Axiom A implies structural stability. His publications include a book on "Matrix Algebra" and joint book with Ralph Abraham on "Transversal Mappings and Flows".
He is a Fellow of the American Mathematical Society.
Dietmar Salamon was born in Bremen in 1953 and completed his PhD at the University of Bremen in 1982 under the direction of Diederich Hinrichsen. After postdoctoral positions in Madison and Zurich, hetook up a position at the University Warwick in 1986, and moved to ETH Zurich in 1998, where he has been emeritus since 2018. His field of research is symplectic topology and related subjects.
He was an invited speaker at the ECM 1992 in Paris, at the ICM 1994 in Zurich, and at the ECM 2000 in Barcelona. He delivered the Andrejewski Lectures in Goettingen (1998) and at the Humboldt Unversity Berlin (2005), and the Xth Lisbon Summer Lectures in Geometry (2009).
He is the author of several textbooks and research momgraphs including two joint books with Dusa McDuff entitled "Introduction to Sympectic Topology" and "J-holomorphic Curves and Symplectic Topology" for which they were jointly awarded the 2017 Leroy P Steele Prize for Mathematical Exposition. He is a Fellow of the American Mathematical Society and a member of the Academia Europaea.
Bibliographic Information
Book Title: Introduction to Differential Geometry
Authors: Joel W. Robbin, Dietmar A. Salamon
Series Title: Springer Studium Mathematik (Master)
DOI: https://doi.org/10.1007/978-3-662-64340-2
Publisher: Springer Spektrum Berlin, Heidelberg
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer-Verlag GmbH, DE, part of Springer Nature 2022
Softcover ISBN: 978-3-662-64339-6Published: 13 January 2022
eBook ISBN: 978-3-662-64340-2Published: 12 January 2022
Series ISSN: 2509-9310
Series E-ISSN: 2509-9329
Edition Number: 1
Number of Pages: XIII, 418
Number of Illustrations: 45 illustrations in colour
Topics: Differential Geometry