Surfaces in Classical Geometries: A Treatment by Moving Frames (Universitext)

December 8, 2020
Surfaces in Classical Geometries: A Treatment by Moving Frames (Universitext)

Contains nearly 300 compelling problems and exercisesContains several fascinating threads that emerge as largergroups of transformationsPresents isothermic immersions which have enjoyed a recentrebirth in the field of integrable systems——————————Designed for intermediate graduate studies, this text willbroaden students' core knowledge of differential geometry providingfoundational material to relevant topics in classical differentialgeometry. The method of moving frames, a natural means fordiscovering and proving important results, provides the basis oftreatment for topics discussed. Its application in many areas helpsto connect the various geometries and to uncover many deeprelationships, such as the Lawson correspondence. The nearly 300problems and exercises range from simple applications to openproblems. Exercises are embedded in the text as essential parts ofthe 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 selectedconcepts and results. The careful selection of results serves toshow the reader how to prove the most important theorems in thesubject, which may become the foundation of future progress.The book pursues significant results beyond the standard topicsof an introductory differential geometry course. A sample of theseresults includes the Willmore functional, the classification ofcyclides of Dupin, the Bonnet problem, constant mean curvatureimmersions, isothermic immersions, and the duality between minimalsurfaces in Euclidean space and constant mean curvature surfaces inhyperbolic space. The book concludes with Lie sphere geometry andits spectacular result that all cyclides of Dupin are Lie sphereequivalent. The exposition is restricted to curves and surfaces inorder to emphasize the geometric interpretation of invariants andother constructions. Working in low dimensions helps studentsdevelop a strong geometric intuition. Aspiring geometers willacquire a working knowledge of curves and surfaces in classicalgeometries. Students will learn the invariants of conformalgeometry and how these relate to the invariants of Euclidean,spherical, and hyperbolic geometry. They will learn thefundamentals of Lie sphere geometry, which require the notion ofLegendre immersions of a contact structure. Prerequisites include acompleted one semester standard course on manifold theory.