Table Of ContentDi(cid:11)erential Geometry
of Curves and Surfaces
Di(cid:11)erential Geometry
of Curves and Surfaces
Second Edition
Thomas Bancho(cid:11)
Stephen Lovett
CRC Press
Taylor & Francis Group
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Contents
Preface vii
Acknowledgements xv
1 Plane Curves: Local Properties 1
1.1 Parametrizations . . . . . . . . . . . . . . . . . . . 1
1.2 Position, Velocity, and Acceleration . . . . . . . . . 13
1.3 Curvature . . . . . . . . . . . . . . . . . . . . . . . 25
1.4 Osculating Circles, Evolutes, and Involutes . . . . 33
1.5 Natural Equations . . . . . . . . . . . . . . . . . . 40
2 Plane Curves: Global Properties 47
2.1 Basic Properties . . . . . . . . . . . . . . . . . . . 47
2.2 Rotation Index . . . . . . . . . . . . . . . . . . . . 52
2.3 Isoperimetric Inequality . . . . . . . . . . . . . . . 61
2.4 Curvature, Convexity, and the Four-Vertex Theorem 64
3 Curves in Space: Local Properties 71
3.1 De(cid:12)nitions, Examples, and Di(cid:11)erentiation . . . . . 71
3.2 Curvature, Torsion, and the Frenet Frame . . . . . 80
3.3 Osculating Plane and Osculating Sphere . . . . . . 90
3.4 Natural Equations . . . . . . . . . . . . . . . . . . 97
4 Curves in Space: Global Properties 103
4.1 Basic Properties . . . . . . . . . . . . . . . . . . . 103
4.2 Indicatrices and Total Curvature . . . . . . . . . . 106
4.3 Knots and Links . . . . . . . . . . . . . . . . . . . 115
v
vi Contents
5 Regular Surfaces 125
5.1 Parametrized Surfaces . . . . . . . . . . . . . . . . 125
5.2 Tangent Planes and Regular Surfaces . . . . . . . . 133
5.3 Change of Coordinates . . . . . . . . . . . . . . . . 150
5.4 The Tangent Space and the Normal Vector . . . . 155
5.5 Orientable Surfaces . . . . . . . . . . . . . . . . . . 159
6 The First and Second Fundamental Forms 165
6.1 The First Fundamental Form . . . . . . . . . . . . 165
6.2 Map Projections (Optional) . . . . . . . . . . . . . 180
6.3 The Gauss Map . . . . . . . . . . . . . . . . . . . . 192
6.4 The Second Fundamental Form . . . . . . . . . . . 198
6.5 Normal and Principal Curvatures . . . . . . . . . . 209
6.6 Gaussian and Mean Curvatures . . . . . . . . . . . 221
6.7 Developable Surfaces and Minimal Surfaces . . . . 231
7 The Fundamental Equations of Surfaces 247
7.1 Gauss’s Equations and the Christo(cid:11)el Symbols . . 248
7.2 Codazzi Equations and the Theorema Egregium . 258
7.3 The Fundamental Theorem of Surface Theory . . . 268
8 The Gauss-Bonnet Theorem and Geometry of Geodesics 273
8.1 Curvatures and Torsion . . . . . . . . . . . . . . . 274
8.2 Gauss-Bonnet Theorem, Local Form . . . . . . . . 284
8.3 Gauss-Bonnet Theorem, Global Form . . . . . . . 295
8.4 Geodesics . . . . . . . . . . . . . . . . . . . . . . . 305
8.5 Geodesic Coordinates . . . . . . . . . . . . . . . . 323
8.6 Applications to Plane, Spherical, and Elliptic Ge-
ometry . . . . . . . . . . . . . . . . . . . . . . . . . 335
8.7 Hyperbolic Geometry . . . . . . . . . . . . . . . . 342
9 Curves and Surfaces in n-dimensional Euclidean Space 355
9.1 Curves in n-dimensional Euclidean Space . . . . . 355
9.2 Surfaces in n-dimensional Euclidean Space . . . . . 364
A Tensor Notation 377
A.1 Tensor Notation . . . . . . . . . . . . . . . . . . . 377
Bibliography 405
Preface
What Is Di(cid:11)erential Geometry?
Di(cid:11)erential geometry studies properties of curves, surfaces, and
higher-dimensionalcurvedspacesusingtoolsfromcalculusandlinear
algebra. Just as the introduction of calculus expands the descriptive
and predictive abilities of nearly every (cid:12)eld of scienti(cid:12)c study, so
the use 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 which (cid:12)t comfortably within
the paradigm of Euclidean geometry. One cannot underestimate
the impact of geometry on science, philosophy, and civilization as a
whole. The geometric proofs in Euclid’s Elements served as models
of mathematical proof for over two thousand years in the Western
tradition of a liberal arts education. Geometry also produced an
unending (cid:13)ow of applications in surveying, architecture, ballistics,
astronomy, 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,
eachwithadi(cid:11)erentbentandsetoffruitfulresults. Techniquesfrom
calculus and analysis led to di(cid:11)erential geometry, pure set theoretic
methods led to topology, and modern algebra contributed the (cid:12)eld
of algebraic geometry.
vii
viii Preface
The types of questions one typically asks in di(cid:11)erential geometry
extend far beyond what one can ask in classical geometry and yet
the former do not entirely subsume the latter. Di(cid:11)erential geometry
questions often fall into two categories: local properties, by which
one means properties of a curve or surface de(cid:12)ned 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 f near a whereas
the integral of f between a and b is a global property. Some of the
most interesting theorems in di(cid:11)erential geometry relate local prop-
erties to global ones. The culminating theorem in this book, the
Gauss-Bonnet Theorem, relates global properties of curves and sur-
faces to the topology of a surface and leads to fundamental results
in non-Euclidean spherical and hyperbolic geometry.
Using This Textbook
This new edition is intended as a textbook for a single semester
undergraduate course in the di(cid:11)erential geometry of curves and sur-
faces, with only multivariable calculus and linear algebra as prereq-
uisites. 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 aids for the reader.
Each section concludes with a collection of exercises which range
fromperfunctorytochallenging,suitablefordailyorweeklyproblem
sets.
However, the self-contained text, the careful introduction of con-
cepts, the many exercises, and the interactive computer graphics
also make this text well-suited for self-study. Such a reader should
feel free to primarily follow the textbook and use the software as
supporting material; primarily follow the presentation in the soft-
ware package and consult the textbook for de(cid:12)nitions, theorems,
and proofs; or try to follow both with equal weight. Either way, the
authors hope that the dual nature of software applets and classic
textbook structure will o(cid:11)er the reader both a rigorous and intuitive
introduction to the (cid:12)eld of di(cid:11)erential geometry.
Preface ix
This book is the (cid:12)rst in a pair of books which together are in-
tended to bring the reader through classical di(cid:11)erential geometry
into the modern formulation of the di(cid:11)erential geometry of mani-
folds. The second book in the pair, by Lovett, is entitled Di(cid:11)erential
Geometry of Manifolds [24]. Neitherbookdirectlyreliesontheother
but knowledge of the content of this book is quite bene(cid:12)cial for [24].
Computer Applets
Anintegralpartofthisbookistheaccesstoon-linecomputergraph-
ics applets that illustrate many concepts and theorems introduced
in the text. Though one can explore the computer demos indepen-
dently of the text, the two are intended as complementary modes
of studying the same material: a visual/intuitive approach and an
analytical/theoretical approach. Though the text does its best to
explain the reason for various de(cid:12)nitions and why one might be in-
terested in such and such a topic, the graphical applets can often
providemotivationforcertainde(cid:12)nitions,allowthereadertoexplore
examples further, and give a visual explanation for complicated the-
orems. The ability to change the choice of the parametric curve or
the parametrized surface in an applet or to change other properties
allows the reader to explore the concepts far beyond what a static
book permits.
Any element in the text (Example, Problem, De(cid:12)nition, Theo-
rem, etc.) that has an associated applet is indicated by the symbol
showninthismargin. Eachdemocomeswithsomeexplanationtext.
Theauthorsintendedtheappletstobeintuitiveenoughsothatafter
using just one or two (and reading the supporting text) any reader
can quickly understand their functionality. At present the applets
runonaJavaplatformbutwemayaddotherformatspossiblybased
on standard computer algebra systems. As an interest to an instruc-
tor of this course, the Java applets are extensible in that they are
designed with considerable (cid:13)exibility so that the reader can often
change whether certain elements are displayed or not. Often, there
are additional elements that one can display either by accessing the
Controls menu on the Demo window or the Plot/Add Plot menu on
anydisplaywindow. Appletsgiveninacomputeralgebrasystemare
naturally extensible through the capabilities of the given program.
