Lie Sphere Geometry (eBook)

With Applications to Submanifolds

Thomas E. Cecil (Autor)

eBook Download: PDF

2007 | 2. Auflage
224 Seiten
Springer New York (Verlag)
978-0-387-74656-2 (ISBN)

Infos

This book provides a clear and comprehensive modern treatment of Lie sphere geometry and its applications to the study of Euclidean submanifolds. It begins with the construction of the space of spheres, including the fundamental notions of oriented contact, parabolic pencils of spheres, and Lie sphere transformations. The link with Euclidean submanifold theory is established via the Legendre map, which provides a powerful framework for the study of submanifolds, especially those characterized by restrictions on their curvature spheres. This new edition contains revised sections on taut submanifolds, compact proper Dupin submanifolds, reducible Dupin submanifolds, and the cyclides of Dupin. Completely new material on isoparametric hypersurfaces in spheres and Dupin hypersurfaces with three and four principal curvatures is also included. The author surveys the known results in these fields and indicates directions for further research and wider application of the methods of Lie sphere geometry. Further key features of Lie Sphere Geometry 2/e: Provides the reader with all the necessary background to reach the frontiers of research in this area; Fills a gap in the literature; no other thorough examination of Lie sphere geometry and its applications to submanifold theory; Complete treatment of the cyclides of Dupin, including 11 computer-generated illustrations; Rigorous exposition driven by motivation and ample examples.

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This book provides a modern treatment of Lie's geometry of spheres, its applications and the study of Euclidean space. It begins with Lie's construction of the space of spheres, including the fundamental notions of oriented contact, parabolic pencils of spheres and Lie sphere transformation. The link with Euclidean submanifold theory is established via the Legendre map. This provides a powerful framework for the study of submanifolds, especially those characterized by restrictions on their curvature spheres. This new edition contains revised sections on taut submanifolds, compact proper Dupin submanifolds, reducible Dupin submanifolds, Lie frames and frame reductions. Completely new material on isoparametric hyperspaces in spheres, Dupin hyperspaces with three and four principle curvatures is also included.

Produkt-Details

• Erscheint lt. Verlag: 29.10.2007
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• ISBN-10: 0-387-74656-0 / 0387746560
• ISBN-13: 978-0-387-74656-2 / 9780387746562