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Publisher:

Chapman & Hall/CRC

Publication Date:

2006

Number of Pages:

984

Format:

Hardcover

Edition:

3

Series:

Studies in Advanced Mathematics

Price:

89.95

ISBN:

1584884487

Category:

Textbook

[Reviewed by , on ]

Fernando Q. Gouvêa

01/26/2007

When it comes to the use of mathematical software as a pedagogical tool, I am something of an old fogey. I find *Mathematica* and *Maple* useful, and I encourage students to use them, but I am leery of allowing such tools to occupy a very large part of the classroom or the textbook, for two reasons. First, I worry that learning to use these programs will become one more hurdle, one more difficult thing in a subject that is already quite difficult. Second, I worry about the fascination that these programs seem to exert. Some mathematicians, and some students, seem to get so fascinated with the tools that the mathematics ends up relegated to a supporting role.

Alfred Gray's *Modern Differential Geometry of Curves and Surfaces* was one of the first textbooks to fully integrate *Mathematica* into an undergraduate course on differential geometry. Doing so is clearly a good idea. Geometry should be visual, and using software allows us to make it so. Furthermore, differential geometry deals with lots of important things that are easy to define and very hard to compute. Even computing the curvature and torsion of a curve in Euclidean space can be a pain. As a result, we are either reduced to having students work on carefully selected toy examples or we tell our students to learn to use symbolic manipulation software.

In his introduction to his second edition, Gray made the first point forcefully. He pointed out that differential geometry was out of fashion in the mid-twentieth century, and said, "I attribute the decline of differential geometry, especially in the United States, to the rise of tensor analysis. Instead of drawing pictures it became fashionable to raise and lower indices." Computers made it possible once again to draw pictures. Computer graphics has, in fact, created a whole new range of applications of differential geometry. And computers make it easy to do complicated computations. To use the most obvious example, the horrible formula for the Gaussian curvature in terms of the first fundamental form that comes out of Gauss's *Theorema Egregium* is much too complicated to use when computing by hand, but it doesn't bother *Mathematica* one bit.

All of this makes me ambivalent about this new (third) edition of Gray's book, revised after his death by Elsa Abbena and Simon Salamon. It is *huge*, bringing to mind those scary massive software manuals that used to come with *Mathematica*. Opening the book reveals that much of this size is due to the decision to *print* the *Mathematica* notebooks corresponding to each chapter as a appendices to their chapters. This instantly doubles the size of the book. It is a puzzling decision. First of all, the notebooks are all available online. Second, there is very little benefit to reading a *Mathematica* notebook without being able to execute the code. Yes, it's nice to see the code written out, but one could see that in the electronic form too.

Seeing the size of the book and the presence of all the code, my initial reaction was to conclude that my fear had come true: the software had eaten up the mathematics. There is certainly some truth to this. In fact, someone wanting to teach a course on *Mathematica* and graphics could use parts of this book as a text. (Gray even says so!) But there is more to the book than that.

If one ignores those many pages of *Mathematica* notebooks, what is left is very good. The treatment of differential geometry is quite classical, but it is also intelligently and charmingly presented, with many more pictures than usual. For example, I had never before seen a simultaneous plot of a plane curve and its curvature function, or a plot of the Gaussian curvature of a surface. There are pictures of tangent developables, generalized cones, parallel surfaces, and other constructions that usually only show up in problem sets. There are *lots* of pictures of minimal surfaces.

There are also some nice theoretical touches, such as relating the existence of a signed curvature function for plane curves to the existence of a complex structure on the plane. Links to complex analysis and to quaternions are made when appropriate, which is also helpful. And the ugly equations with awful indices are also there for those who like them.

The book covers far too many topics for a one-semester course, but that is fine: one can select, and much will be left over for students to consider and study. There are little biographical footnotes and attempts at history. These are mostly inadequate and sometimes even annoying. (What's the point of pointing out that "ellipse" comes from "falling short" if one does not then explain what has fallen short of what?)

One final complaint is that the Gauss-Bonnet theorem is relegated to the very last chapter, and the proof uses (lightly) the theory of abstract surfaces developed earlier. This is unfortunate, since in an undergraduate course it is very hard to cover abstract surfaces and still get to Gauss-Bonnet… but it would be educational malpractice *not* to get to Gauss-Bonnet.

In the end, though I would have preferred to have the notebooks on a CD-ROM and a thinner book, I think this is a very good introductory textbook, particularly if your students are already familiar with *Mathematica* and can therefore concentrate on learning the geometry. I would probably not choose it as a main text, but I would certainly recommend it to appropriately inclined students and to their professors.

Fernando Q. Gouvêa just finished teaching a one-semester undergraduate introduction to differential geometry at Colby College.

CURVES IN THE PLANE

Euclidean Spaces

Curves in Space

The Length of a Curve

Curvature of Plane Curves

Angle Functions

First Examples of Plane Curves

The Semicubical Parabola and Regularity

1.8 Exercises

Notebook 1

FAMOUS PLANE CURVES

Cycloids

Lemniscates of Bernoulli

Cardioids

The Catenary

The Cissoid of Diocles

The Tractrix

Clothoids

Pursuit Curves

Exercises

Notebook

ALTERNATIVE WAYS OF PLOTTING CURVES

Implicitly Defined Plane Curves

The Folium of Descartes

Cassinian Ovals

Plane Curves in Polar Coordinates

A Selection of Spirals

Exercises

Notebook 3

NEW CURVES FROM OLD

Evolutes

Iterated Evolutes

Involutes

Osculating Circles to Plane Curves

Parallel Curves

Pedal Curves

Exercises

Notebook 4

DETERMINING A PLANE CURVE FROM ITS CURVATURE

Euclidean Motions

Isometries of the Plane

Intrinsic Equations for Plane Curves

Examples of Curves with Assigned Curvature

Exercises

Notebook 5

GLOBAL PROPERTIES OF PLANE CURVES

Total Signed Curvature

Trochoid Curves

The Rotation Index of a Closed Curve

Convex Plane Curves

The Four Vertex Theorem

Curves of Constant Width

Reuleaux Polygons and Involutes

The Support Function of an Oval

Exercises

Notebook 6

CURVES IN SPACE

The Vector Cross Product

Curvature and Torsion of Unit-Speed Curves

The Helix and Twisted Cubic

Arbitrary-Speed Curves in R3

More Constructions of Space Curves

Tubes and Tori

Torus Knots

Exercises

Notebook 7

CONSTRUCTION OF SPACE CURVES

The Fundamental Theorem of Space Curves

Assigned Curvature and Torsion

Contact

Space Curves that Lie on a Sphere

Curves of Constant Slope

Loxodromes on Spheres

8.7 Exercises

Notebook 8

CALCULUS ON EUCLIDEAN SPACE

Tangent Vectors to Rn

Tangent Vectors as Directional Derivatives

Tangent Maps or Differentials

Vector Fields on R n

Derivatives of Vector Fields

Curves Revisited

Exercises

Notebook 9

SURFACES IN EUCLIDEAN SPACE

Patches in Rn

Patches in R3 and the Local Gauss Map

The Definition of a Regular Surface

Examples of Surfaces

Tangent Vectors and Surface Mappings

Level Surfaces in R3

Exercises

Notebook 10

NONORIENTABLE SURFACES

Orientability of Surfaces

Surfaces by Identification

The Möbius Strip

The Klein Bottle

Realizations of the Real Projective Plane

Twisted Surfaces

Exercises

Notebook 11

METRICS ON SURFACES

The Intuitive Idea of Distance

Isometries between Surfaces

Distance and Conformal Maps

The Intuitive Idea of Area

Examples of Metrics

Exercises

Notebook 12

SHAPE AND CURVATURE

The Shape Operator

Normal Curvature

Calculation of the Shape Operator

Gaussian and Mean Curvature

More Curvature Calculations

A Global Curvature Theorem

Nonparametrically Defined Surfaces

Exercises

Notebook 13

RULED SURFACES

Definitions and Examples

Curvature of a Ruled Surface

Tangent Developables

Noncylindrical Ruled Surfaces

Exercises

Notebook 14

SURFACES OF REVOLUTION AND CONSTANT CURVATURE

Surfaces of Revolution

Principal Curves

Curvature of a Surface of Revolution

Generalized Helicoids

Surfaces of Constant Positive Curvature

Surfaces of Constant Negative Curvature

More Examples of Constant Curvature

Exercises

Notebook 15

A SELECTION OF MINIMAL SURFACES

Normal Variation

Deformation from the Helicoid to the Catenoid

Minimal Surfaces of

More Examples of Minimal Surfaces

Monge Patches and Scherk's Minimal Surface

The Gauss Map of a Minimal Surface

Isothermal Coordinates

Exercises

Notebook 16

INTRINSIC SURFACE GEOMETRY

Intrinsic Formulas for the Gaussian Curvature

Gauss's Theorema Egregium

Christoffel Symbols

Geodesic Curvature of Curves on Surfaces

Geodesic Torsion and Frenet Formulas

Exercises

Notebook 17

ASYMPTOTIC CURVES AND GEODESICS ON SURFACES

Asymptotic Curves

Examples of Asymptotic Curves and Patches

The Geodesic Equations

First Examples of Geodesics

Clairaut Patches

Use of Clairaut Patches

Exercises

Notebook 18

PRINCIPAL CURVES AND UMBILIC POINTS

The Differential Equation for Principal Curves

Umbilic Points

The Peterson-Mainardi-Codazzi Equations

Hilbert's Lemma and Liebmann's Theorem

Triply Orthogonal Systems of Surfaces

Elliptic Coordinates

Parabolic Coordinates and a General Construction

Parallel Surfaces

The Shape Operator of a Parallel Surface

Exercises

Notebook 19

CANAL SURFACES AND CYCLIDES OF DUPIN

Surfaces Whose Focal Sets are 2-Dimensional

Canal Surfaces

Cyclides of Dupin via Focal Sets

The Definition of Inversion

Inversion of Surfaces

Exercises

Notebook 20

THE THEORY OF SURFACES OF CONSTANT NEGATIVE CURVATURE

Intrinsic Tchebyshef Patches

Patches on Surfaces of Constant Negative Curvature

The Sine-Gordon Equation

Tchebyshef Patches on Surfaces of Revolution

The Bianchi Transform

Moving Frames on Surfaces in R3

Kuen's Surface as Bianchi Transform of the Pseudosphere

The B¨ acklund Transform

Exercises

Notebook 21

MINIMAL SURFACES VIA COMPLEX VARIABLES

Isometric Deformations of Minimal Surfaces

Complex Derivatives

Minimal Curves

Finding Conjugate Minimal Surfaces

The Weierstrass Representation

Minimal Surfaces via Björling's Formula

Costa's Minimal Surface

Exercises

Notebook 22

ROTATION AND ANIMATION USING QUATERNIONS

Orthogonal Matrices

Quaternion Algebra

Unit Quaternions and Rotations

Imaginary Quaternions and Rotations

Rotation Curves

Euler Angles

Further Topics

Exercises

Notebook 23

DIFFERENTIABLE MANIFOLDS

The Definition of a Differentiable Manifold

Differentiable Functions on Manifolds

Tangent Vectors on Manifolds

Induced Maps

Vector Fields on Manifolds

Tensor Fields

Exercises

Notebook 24

RIEMANNIAN MANIFOLDS

Covariant Derivatives

Pseudo-Riemannian Metrics

The Classical Treatment of Metrics

The Christoffel Symbols in Riemannian Geometry

The Riemann Curvature Tensor

Exercises

Notebook 25

ABSTRACT SURFACES AND THEIR GEODESICS

Christoffel Symbols on Abstract Surfaces

Examples of Abstract Metrics

The Abstract Definition of Geodesic Curvature

Geodesics on Abstract Surfaces

The Exponential Map and the Gauss Lemma

Length Minimizing Properties of Geodesics

Exercises

Notebook 26

THE GAUSS-BONNET THEOREM

Turning Angles and Liouville's Theorem

The Local Gauss-Bonnet Theorem

An Area Bound

A Generalization to More Complicated Regions

The Topology of Surfaces

The Global Gauss-Bonnet Theorem .

Applications of the Gauss-Bonnet Theorem

Exercises

Notebook

Bibliography

Name Index

Subject Index

Notebook Index

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