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Books > Science & Mathematics > Mathematics > Geometry > Algebraic geometry
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Surfaces in Classical Geometries - A Treatment by Moving Frames (Paperback, 1st ed. 2016)
Loot Price: R4,018
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Surfaces in Classical Geometries - A Treatment by Moving Frames (Paperback, 1st ed. 2016)
Series: Universitext
Expected to ship within 10 - 15 working days
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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 (R), Matlab (TM), 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.
General
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