Differential Geometry of Curves and Surfaces

Students and professors of an undergraduate course in differential geometry will appreciate the clear exposition and comprehensive exercises in this book that focuses on the geometric properties of curves and surfaces, one- and two-dimensional objects in Euclidean space. The problems generally relate to questions of local properties (the properties observed at a point on the curve or surface) or global properties (the properties of the object as a whole). Some of the more interesting theorems explore relationships between local and global properties. A special feature is the availability of accompanying online interactive java applets coordinated with each section. The applets allow students to investigate and manipulate curves and surfaces to develop intuition and to help analyze geometric phenomena.

Table of Contents:
Preface Acknowledgements Plane Curves: Local Properties Parameterizations Position, Velocity, and Acceleration Curvature Osculating Circles, Evolutes, and Involutes Natural Equations Plane Curves: Global Properties Basic Properties Rotation Index Isoperimetric Inequality Curvature, Convexity, and the Four-Vertex Theorem Curves in Space: Local Properties Definitions, Examples, and Differentiation Curvature, Torsion, and the Frenet Frame Osculating Plane and Osculating Sphere Natural Equations Curves in Space: Global Properties Basic Properties Indicatrices and Total Curvature Knots and Links Regular Surfaces Parametrized Surfaces Tangent Planes and Regular Surfaces Change of Coordinates The Tangent Space and the Normal Vector Orientable Surfaces The First and Second Fundamental Forms The First Fundamental Form The Gauss Map The Second Fundamental Form Normal and Principal Curvatures Gaussian and Mean Curvature Ruled Surfaces and Minimal Surfaces The Fundamental Equations of Surfaces Tensor Notation Gauss’s Equations and the Christoffel Symbols Codazzi Equations and the Theorema Egregium The Fundamental Theorem of Surface Theory Curves on Surfaces Curvatures and Torsion Geodesics Geodesic Coordinates Gauss-Bonnet Theorem and Applications Intrinsic Geometry Bibliography

About the Author :
Thomas F. Banchoff is a geometer and has been a professor at Brown University since 1967. Banchoff was president of the MAA from 1999-2000. He is published widely and known to a broad audience as editor and commentator on Abbotts Flatland. He has been the recipient of such awards as the MAA National Award for Distinguished College or University Teaching of Mathematics and most recently the 2007 Teaching with Technology Award. Stephen Lovett is an associate professor of mathematics at Wheaton College in Illinois. Lovett has also taught at Eastern Nazarene College and has taught introductory courses on differential geometry for many years. Lovett has traveled extensively and has given many talks over the past several years on differential and algebraic geometry, as well as cryptography.

Review :
Students and professors of an undergraduate course in differential geometry will appreciate the clear exposition and comprehensive exercises in this book ! Some of the more interesting theorems explore relationships between local and global properties. A special feature is the availability of accompanying online interactive java applets coordinated with each section. The applets allow students to investigate and manipulate curves and surfaces to develop intuition and to help analyze geometric phenomena. --L'Enseignement Mathematique (2) 57 (2011) Coming from intuitive considerations to precise definitions the authors have written a very readable book. Every section contains many examples, problems and figures visualizing geometric properties. The understanding of geometric phenomena is supported by a number of available Java applets. This special feature distinguishes the textbook from others and makes it recommendable for self studies too. ! highly recommendable ! --F. Manhart, International Mathematical News, August 2011