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DIFFERENTIAL GEOMETRY

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DIFFERENTIAL GEOMETRY

Poonam Kumari

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Blurb

The book is intended to serve as a textbook for the undergraduate students in Mathematics. It is also useful for the postgraduate students. The book deals with the local theory of curves and surfaces in three dimensional Euclidean space. The subject is presented in a simple manner using the methods of differential calculus, integral calculus and vector algebra. Chapter 1 begins with the theory of curves in three dimensional Euclidean space. The curves are represented in a parametric form as a function of one variable.

The geometric properties and various quantities associated with the curves such as the unit tangent vector, curvature and torsion are expressed via derivatives. The subsequent chapters discuss representation of a surface in various forms; concepts of characteristics, envelopes, and developable surfaces; fundamental magnitudes and their properties; normal curvature; principal curvature and principal directions; geodesics and their properties. A large number of solved examples followed by a carefully selected set of exercises are given at the end of each section.

Chapters Pages

1. Curves in Space 1-124

1.1 Introduction 1

1.2 Parametric Representation of Space Curves 1

1.3 Arc Length of a Curve between Two Points 1

1.4 Tangent Line 3

1.5 Order of Contact 5

1.6 Osculating Plane or Plane of Curvature 14

1.7 Osculating Plane at a Point of the Curve of

Intersection of Two Surfaces 17

1.8 Normal Plane 18

1.9 Principal Normal and Binormal 19

1.10 Rectifying Plane 20

1.11 Curvature, Torsion and Screw Curvature 21

1.12 Serret-Frenet Formulae 25

1.13 Curvature and Torsion of the Curve r = r (t) 27

1.14 Helices 29

1.15 Fundamental Theorems for Space Curves 73

1.16 Osculating Circle (or Circle of Curvature) 79

1.17 Osculating Sphere 83

1.18 Spherical Indicatrix 106

1.19 Involutes and Evolutes 110

1.20 Bertrand Curves 116

2. Concept of a Surface and Envelopes and

Developable 125-185

2.1 Introduction 125

2.2 Ordinary Points and Singularities 126

2.3 Curvilinear Equations of the Curve on the

Surface 127

vi

2.4 Parametric Curves 127

2.5 Tangent Plane and Normal 128

2.6 Characteristics and Envelope of One

Parameter Family of Surfaces 136

2.7 Edge of Regression 138

2.8 Developable Surfaces or Developables 145

2.9 Developables associated with a Space Curve 149

2.10 Envelope of Two Parameter Family of

Surfaces 175

3. Fundamental Forms and Directions on a Surface 186-226

3.1 First Fundamental Form 186

3.2 Second Fundamental Form 191

3.3 Derivatives of N (Weingarten Equations) 193

3.4 Fundamental Magnitudes for Some Important

Surfaces 195

3.5 Directions on a Surface 206

3.6 Direction Coefficients 208

3.7 Family of Curves 212

3.8 Orthogonal Trajectories 213

3.9 Double Family of Curves 215

4. Curves on a Surface 227-387

4.1 Curvature of Normal Section or Normal

Curvature 227

4.2 Principal Directions and Principal Curvatures 229

4.3 Lines of Curvature 237

4.4 Elliptic, Hyperbolic and Parabolic Points on a

Surface 250

4.5 Dupin’s Indicatrix 251

4.6 Third Fundamental Form 253

4.7 Conjugate Directions 302

4.8 Conjugate Systems 307

4.9 Asymptotic Lines 310

4.10 Ruled Surfaces 333

vii

4.11 Isometric Lines 357

4.12 Null Lines 360

4.13 The Fundamental Equations of Surface

Theory 363

4.14 Parallel Surfaces 382

5. Geodesics 388-476

5.1 Definition 388

5.2 Differential Equations of Geodesics 388

5.3 Canonical Geodesic Equations 396

5.4 Normal Property of Geodesics 397

5.5 Differential Equations of Geodesics using

Normal Property 399

5.6 Geodesics on a Surface of Revolution 406

5.7 Clairaut’s Theorem 408

5.8 Torsion and Curvature of a Geodesic 427

5.9 Curves in relation to Geodesics 436

5.10 Geodesic Curvature 440

5.11 Gauss-Bonnet Theorem 450

5.12 Geodesic Parallels 454

5.13 Geodesic Polar Coordinates 455

5.14 Minding Theorem 459

5.15 Mapping of Surfaces 463

5.16 Tissot’s Theorem 469

5.17 Dini’s Theorem 470

Index 477-479

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