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Universitext

Surfaces in Classical Geometries

A Treatment by Moving Frames

Authors: Jensen, Gary R., Musso, Emilio, Nicolodi, Lorenzo

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  • Contains nearly 300 compelling problems and exercises
  • Contains several fascinating threads that emerge as larger groups of transformations
  • Presents isothermic immersions which have enjoyed a recent rebirth in the field of integrable systems
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eBook $64.99
price for USA in USD (gross)
  • ISBN 978-3-319-27076-0
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $84.99
price for USA in USD
  • ISBN 978-3-319-27074-6
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
About this Textbook

Designed for intermediate graduate studies, this text will broaden students' core knowledge of differential geometry providing foundational material to relevant topics in classical differential geometry. The method of moving frames, a natural means for discovering and proving important results, provides the basis of treatment for topics discussed. Its application in many areas helps to connect the various geometries and to uncover many deep relationships, such as the Lawson correspondence.  The nearly 300 problems and exercises range from simple applications to open problems. Exercises are embedded in the text as essential parts of the exposition. Problems are collected at the end of each chapter; solutions to select problems are given at the end of the book. Mathematica®, Matlab™,  and Xfig are used to illustrate selected concepts and results. The careful selection of results serves to show the reader how to prove the most important theorems in the subject, which may become the foundation of future progress.


The book pursues significant results beyond the standard topics of an introductory differential geometry course. A sample of these results includes the Willmore functional, the classification of cyclides of Dupin, the Bonnet problem, constant mean curvature immersions, isothermic immersions, and the duality between minimal surfaces in Euclidean space and constant mean curvature surfaces in hyperbolic space. The book concludes with Lie sphere geometry and its spectacular result that all cyclides of Dupin are Lie sphere equivalent. The exposition is restricted to curves and surfaces in order to emphasize the geometric interpretation of invariants and other constructions. Working in low dimensions helps students develop a strong geometric intuition. Aspiring geometers will acquire a working knowledge of curves and surfaces in classical geometries. Students will learn the invariants of conformal geometry and how these relate to the invariants of Euclidean, spherical, and hyperbolic geometry. They will learn the fundamentals of Lie sphere geometry, which require the notion of Legendre immersions of a contact structure. Prerequisites include a completed one semester standard course on manifold theory.

About the authors

Gary R. Jensen is currently professor of mathematics at Washington University in St. Louis, Missouri. Emilio Musso is professor of mathematics at the Politecnico di Torino, Torino, Italy and Lorenzo Nicolodi is professor of mathematics at the Univerita di Parma, Parma, Italy.

Reviews

“300 problems and exercices, from simple until open problems, are contained in this book and they are the essential part of it. In conclusion, it is an interesting book written carefully and with originality. Undoubtedly its reading will be very helpful to any geometer.” (Charalampos Charitos, zbMATH 1347.53001, 2016)


Table of contents (15 chapters)

Table of contents (15 chapters)

Buy this book

eBook $64.99
price for USA in USD (gross)
  • ISBN 978-3-319-27076-0
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $84.99
price for USA in USD
  • ISBN 978-3-319-27074-6
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Surfaces in Classical Geometries
Book Subtitle
A Treatment by Moving Frames
Authors
Series Title
Universitext
Copyright
2016
Publisher
Springer International Publishing
Copyright Holder
Springer International Publishing Switzerland
eBook ISBN
978-3-319-27076-0
DOI
10.1007/978-3-319-27076-0
Softcover ISBN
978-3-319-27074-6
Series ISSN
0172-5939
Edition Number
1
Number of Pages
XIII, 571
Number of Illustrations
16 b/w illustrations, 61 illustrations in colour
Topics