This text focuses on developing an intimate acquaintance with the geometric meaning of curvature and thereby introduces and demonstrates all the main technical tools needed for a more advanced course on Riemannian manifolds. It covers proving the four most fundamental theorems relating curvature and topology: the Gauss-Bonnet Theorem, the Cartan-Hadamard Theorem, Bonnet's Theorem, and a special case of the Cartan-Ambrose-Hicks Theorem.
Rezensionen / Stimmen
"One interesting aspect of the book is the decision of which audience to target it towards. ... Overall, this would make a very appropriate text for a graduate course, or a programme of individual study in Riemannian geometry, whether to give a thorough treatment of the fundamentals, or to introduce the more advanced topics in global geometry." (Robert J. Low, Mathematical Reviews, November, 2019)
"This material is carefully developed and several useful examples and exercises are included in each chapter. The reviewer's belief is that this excellent edition will become soon a standard text for several graduate courses as well as an frequent citation in articles." (Mircea Crasmareanu, zbMATH 1409.53001, 2019)
"One interesting aspect of the book is the decision of which audience to target it towards. ... Overall, this would make a very appropriate text for a graduate course, or a programme of individual study in Riemannian geometry, whether to give a thorough treatment of the fundamentals, or to introduce the more advanced topics in global geometry." (Robert J. Low, Mathematical Reviews, November, 2019)
"This material is carefully developed and several useful examples and exercises are included in each chapter. The reviewer's belief is that this excellent edition will become soon a standard text for several graduate courses as well as an frequent citation in articles." (Mircea Crasmareanu, zbMATH 1409.53001, 2019)
Produkt-Info
Reihe
Auflage
Sprache
Verlagsort
Verlagsgruppe
Springer International Publishing
Zielgruppe
Editions-Typ
Illustrationen
210
210 s/w Abbildungen
XIII, 437 p. 210 illus.
Maße
Höhe: 235 mm
Breite: 155 mm
Dicke: 25 mm
Gewicht
ISBN-13
978-3-030-80106-9 (9783030801069)
DOI
10.1007/978-3-319-91755-9
Schweitzer Klassifikation
?John "Jack" M. Lee is a professor of mathematics at the University of Washington. Professor Lee is the author of three highly acclaimed Springer graduate textbooks : Introduction to Smooth Manifolds, (GTM 218) Introduction to Topological Manifolds (GTM 202), and Riemannian Manifolds (GTM 176). Lee's research interests include differential geometry, the Yamabe problem, existence of Einstein metrics, the constraint equations in general relativity, geometry and analysis on CR manifolds.
<b>?John "Jack" M. Lee</b> is a professor of mathematics at the University of Washington. Professor Lee is the author of three highly acclaimed Springer graduate textbooks : <i>Introduction to Smooth Manifolds</i>, (GTM 218) <i>Introduction to Topological Manifolds </i>(GTM 202), and <i>Riemannian Manifolds </i>(GTM 176). Lee's research interests include differential geometry, the Yamabe problem, existence of Einstein metrics, the constraint equations in general relativity, geometry and analysis on CR manifolds.
Preface.- 1. What Is Curvature?.- 2. Riemannian Metrics.- 3. Model Riemannian Manifolds.- 4. Connections.- 5. The Levi-Cevita Connection.- 6. Geodesics and Distance.- 7. Curvature.- 8. Riemannian Submanifolds.- 9. The Gauss-Bonnet Theorem.- 10. Jacobi Fields.- 11. Comparison Theory.- 12. Curvature and Topology.- Appendix A: Review of Smooth Manifolds.- Appendix B: Review of Tensors.- Appendix C: Review of Lie Groups.- References.- Notation Index.- Subject Index.
