Differential Geometry: A First Course

Differential Geometry: A First Course is an introduction to the classical theory of space curves and surfaces offered in graduate and postgraduate courses in mathematics. Based on Serret-Frenet formulae, the theory of space curves is developed and concluded with a detailed discussion on fundamental existence theorem. The theory of surfaces includes the first fundamental form with local intrinsic properties, geodesics on surfaces, the second fundamental form with local non-intrinsic properties and the fundamental equations of the surface theory with several applications.

Table of Contents:
Theory of Space Curves: Introduction Representation of space curves Unique parametric representation of a space curve Arc – length Tangent and Osculating Plane Principal normal and binormal Curvature and Torsion Behaviour of a curve near one of its points Curvature and torsion of the curve of intersection of two surfaces Contact between curves and surfaces Osculating circle and osculating sphere Locus of centres of spherical curvature Tangent surfaces, Involutes and evolutes Betrand curves Spherical Indicatrix Intrinsic equations of space curves Fundamental Existence Theorem for space curves Helices Examples 1 Exercises 1 The First Fundamental Form and Local Intrinsic Properties of a Surface: Introduction Definition of a surface Nature of points on a surface Representation of a surface Curves on surfaces Tangent plane and surface normal The general surface of revolution Helicoids Metric on a surface Direction coefficients on a surface Families of curves Orthogonal Trajectories Double Family of curves Isometric correspondence Intrinsic properties Examples II Exercises II Geodesics on a Surface: Introduction Geodesics and their differential equations Canonical geodesic equations Geodesics on surfaces of revolution Normal property of geodesics Differential equations of geodesics using normal property Existence theorems Geodesic parallels Geodesic curvature Gauss – Bonnet theorem Gaussian Curvature Surfaces of constant curvature Conformal mapping Geodesic mapping Examples III Exercises III The Second Fundamental form and local Non - Intrinsic Properties of Surfaces: Introduction The second fundamental form The Classification of points on a surface Principal curvatures Lines of curvature The Dupin indicatrix Developable surfaces Developables associated with space curves Developables associated with curves on surfaces Minimal surfaces Ruled surfaces Three fundamental forms Examples IV Exercises IV The Fundamental Equations of Surface Theory: Introduction Tensor notations Gauss equations Weingarten Equations Mainardi – Codazzi equations Parallel Surfaces Fundamental existence theorem for surfaces Examples V Exercises V

About the Author :
D. Somasundaram.: Professor of Mathematics (Retd.), University of Madras