Differential geometry of curves and surfaces

345 Pages·2010·8.338 MB·English

by  Banchoff, Thomas F., Lovett, Stephen T

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Differential Geometry
of Curves and Surfaces
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Differential Geometry
of Curves and Surfaces
Thomas Banchoff
Stephen Lovett
AKPeters,Ltd.
Natick,Massachusetts
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CRC Press
Taylor & Francis Group
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Boca Raton, FL 33487-2742
© 2010 by Taylor & Francis Group, LLC
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Version Date: 20110714
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Contents
Preface vii
Acknowledgements xv
1 PlaneCurves: LocalProperties 1
1.1 Parametrizations . . . . . . . . . . . . . . . . . . . 1
1.2 Position, Velocity, and Acceleration . . . . . . . . . 10
1.3 Curvature . . . . . . . . . . . . . . . . . . . . . . . 20
1.4 Osculating Circles, Evolutes, and Involutes . . . . 27
1.5 Natural Equations . . . . . . . . . . . . . . . . . . 34
2 PlaneCurves: GlobalProperties 39
2.1 Basic Properties . . . . . . . . . . . . . . . . . . . 39
2.2 Rotation Index . . . . . . . . . . . . . . . . . . . . 43
2.3 Isoperimetric Inequality . . . . . . . . . . . . . . . 51
2.4 Curvature,Convexity,andtheFour-VertexTheorem 53
3 CurvesinSpace: LocalProperties 61
3.1 Definitions, Examples, and Differentiation . . . . . 61
3.2 Curvature, Torsion, and the Frenet Frame . . . . . 68
3.3 Osculating Plane and Osculating Sphere . . . . . . 78
3.4 Natural Equations . . . . . . . . . . . . . . . . . . 84
4 CurvesinSpace: GlobalProperties 87
4.1 Basic Properties . . . . . . . . . . . . . . . . . . . 87
4.2 Indicatrices and Total Curvature . . . . . . . . . . 89
4.3 Knots and Links . . . . . . . . . . . . . . . . . . . 97
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vi Contents
5 RegularSurfaces 107
5.1 Parametrized Surfaces . . . . . . . . . . . . . . . . 107
5.2 Tangent Planes and Regular Surfaces . . . . . . . . 114
5.3 Change of Coordinates . . . . . . . . . . . . . . . . 130
5.4 The Tangent Space and the Normal Vector . . . . 134
5.5 Orientable Surfaces . . . . . . . . . . . . . . . . . . 138
6 TheFirstandSecondFundamentalForms 145
6.1 The First Fundamental Form . . . . . . . . . . . . 145
6.2 The Gauss Map . . . . . . . . . . . . . . . . . . . . 160
6.3 The Second Fundamental Form . . . . . . . . . . . 166
6.4 Normal and Principal Curvatures . . . . . . . . . . 177
6.5 Gaussian and Mean Curvature . . . . . . . . . . . 189
6.6 Ruled Surfaces and Minimal Surfaces . . . . . . . . 197
7 TheFundamentalEquationsofSurfaces 209
7.1 Tensor Notation . . . . . . . . . . . . . . . . . . . 210
7.2 Gauss’s Equations and the Christoffel Symbols . . 237
7.3 Codazzi Equations and the Theorema Egregium . 245
7.4 The Fundamental Theorem of Surface Theory . . . 255
8 CurvesonSurfaces 259
8.1 Curvatures and Torsion . . . . . . . . . . . . . . . 260
8.2 Geodesics . . . . . . . . . . . . . . . . . . . . . . . 270
8.3 Geodesic Coordinates . . . . . . . . . . . . . . . . 285
8.4 Gauss-Bonnet Theorem and Applications . . . . . 297
8.5 Intrinsic Geometry . . . . . . . . . . . . . . . . . . 320
Bibliography 325
Index 327
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Preface
What is Differential Geometry?
Differential geometry studies the properties of curves, surfaces, and
higher-dimensionalcurvedspacesusingtoolsfromcalculusandlinear
algebra. Justas the introduction of calculus expands the descriptive
andpredictiveabilitiesofnearlyeveryfieldofscientificstudy,theuse
of calculus in geometry brings about avenues of inquiry that extend
far beyond classical geometry.
Before the advent of calculus, much of geometry consisted of
proving consequences of Euclid’s postulates. Even conics, which
came into vogue in the physical sciences after Kepler observed that
planets travel around the sun in ellipses, arise as the intersection of
a double cone and a plane, two shapes that fit comfortably within
the paradigm of Euclidean geometry. One cannot underestimate
the impact of geometry on science, philosophy, and civilization as a
whole. Not only did the geometry books in Euclid’s Elements serve
as the model for mathematical proof for over two thousand years in
the Western tradition of a liberal arts education, but geometry also
produced an unending flow of applications in surveying, architec-
ture, ballistics, astronomy, astrology, and natural philosophy more
generally.
TheobjectsofstudyinEuclideangeometry(points,lines,planes,
circles, spheres, cones, and conics) are limited in what they can de-
scribe. A boundless variety of curves and surfaces and manifolds
arise naturally in areas of inquiry that employ geometry. To ad-
dress these new classes of objects, various branches of mathematics
brought their tools to bear on the expanding horizons of geometry,
each withadifferentbentandsetoffruitfulresults. Techniquesfrom
calculus and analysis led to differential geometry, pure set theoretic
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viii Preface
methods led to topology, and modern algebra contributed the field
of algebraic geometry.
The types of questions one typically asks in differential geometry
extend far beyond what one can ask in classical geometry and yet
the former do not entirely subsume the latter. Differential geometry
questions often fall into two categories: local properties, by which
one means properties of a curve or surface defined in the neighbor-
hood of a point, or global properties, which refer to properties of
the curve or surface taken as a whole. As a comparison to func-
tions of one variable, the derivative of a function f at a point a is
a local property, since one only needs information about f near a,
whereas the integral of f between a and b is a global property. Some
of the most interesting theorems in differential geometry relate local
properties to global ones. As a case in point, the celebrated Gauss-
Bonnet Theorem single-handedly encapsulates many global results
of curves in the plane at the same time as it proves results about
spherical and hyperbolic geometry.
Using This Textbook
This book is the first in a pair of books that together are intended
to bring the reader through classical differential geometry into the
modern formulation of the differential geometry of manifolds. The
second book in the pair, by Lovett, is entitled Differential Geometry
of Manifolds with Applications to Physics [22]. Neitherbookdirectly
relies on the other but knowledge of the content of this book is quite
beneficial for [22].
On its own, the present book is intended as a textbook for a
single semester undergraduate course in the differential geometry of
curves and surfaces, with only vector calculus and linear algebra
as prerequisites. The interactive computer graphics applets that
are provided for this book can be used for computer labs, in-class
illustrations, exploratory exercises, or simply as intuitive aides for
the reader. Each section concludes with a collection of exercises
that range from perfunctoryto challenging and are suitable for daily
or weekly problem sets.
However, the self-contained text, the careful introduction of con-
cepts, the many exercises, and the interactive computer graphics
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