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