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DIFFERENTIAL GEOMETRY: A FIRST COURSE (pb)2008
Author: SOMASUNDARAM
ISBN: 9788173195464
Year: 2008
Publisher: NAROSA PUBLISHING HOUSE PVT. LTD.
Category: MATHEMATICS
Edition: 1
Format: Paperback
Language: English
Pages: 458
 ABOUT THE TITLE
Differential Geometry: A First Course is an introduction to the classical theory of space curves and surfaces offered at the Graduate and Post- Graduate 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. Author: D. Somasundaram
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 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
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