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(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.
