# Textbook for undergraduate course in geometry

I've been assigned to teach our undergraduate course in geometry next semester. This course originally was intended for future high-school teachers and focused on axiomatic, Euclid-style geometry (planar, spherical, and hyperbolic). Rice University has changed a lot since this course began being taught (many, many years ago); we now have very few students who want to be high school teachers, and in general the level of our students is such that most of our math majors perceive the course to be beneath them.

My assignment is to redesign the course. I have almost complete freedom except that I cannot require any prerequisites beyond multivariable calculus and ODE's.

Question : What textbook should I use?

Here are my thoughts about what I am looking for.

1. As I said, I cannot require any prerequisites beyond multivariable calculus and ODE's. However, our undergraduate students are very strong (based on test scores and high school grades, they are pretty similar to the students at eg Cornell or Northwestern). So I want a book that has plenty of meat in it.

2. It should contain a mixture of proofs and computation, but plenty of proofs.

3. There are no topics that I am required to cover, though of course it has to be geometric (in particular, this course is not a prerequisite for anything else).

4. I find axiomatic treatments of geometry boring.

5. I don't want to develop any machinery unless it has an immediate payoff. However, I am not at all adverse to developing some tools from scratch as long as they lead to something cool.

6. I want there to be lots of good problems.

Does anyone have any suggestions?

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It sounds like your students are ideal for a baby differential geometry course. As a perk, you could develop from DG the spherical and hyperbolic geometry models. This can satisfy 1-6 handily. –  Ryan Budney Nov 1 '12 at 3:12
A polytopes course could also satisfy 1-6 handily, but there are fewer books for this than for differential geometry. –  Alexander Woo Nov 1 '12 at 3:30
I had a course much like what you describe, Andy, but at U.Vic I had a more diverse array of backgrounds than you'd (likely) encounter at Rice. Baby DG went over fine. I used Millman and Parker as my text. –  Ryan Budney Nov 1 '12 at 3:41
Differential Geometry is a good idea, but we already have an undergraduate course in differential geometry, so I probably should do something else. Is there a good undergraduate level book on polytopes? That could be a lot of fun (and I might learn something too). –  Andy Putman Nov 1 '12 at 3:42
I graded Peter Shalen's such course at one time. I think they used Artin's Geometric Algebra. You might find an axiomatic treatment boring, but Shalen had a number of applications, one being digging a railroad tunnel, that fostered part of my desire to attend the course rather than grade it. If you can, you might email him for suggestions. Gerhard "Still Went The Algebraic Route" Paseman, 2012.10.31 –  Gerhard Paseman Nov 1 '12 at 3:54

I wonder whether Igor Pak's "Lectures on Discrete and Polyhedral Geometry" might be appropriate as a textbook for an undergraduate geometry course. This is still in preliminary form, available on his website. In the introduction he describes a selection of topics from the book that could be used for a basic undergraduate course. There seem to be lots of exercises, and at a quick glance a lot of the topics look quite interesting. This whole subject is way outside my expertise, however, so I have no idea if the book would make a good basis for a course like the one you'll be teaching.

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I accepted this because I decided to try teaching the course with Igor's book. Thanks to everyone else for their suggestions! –  Andy Putman Nov 16 '12 at 19:22

Concerning the fourth item in your list, I believe that axiomatic approach is not only boring, but (and it is more important) it is almost useless for further mathematical courses.

In my view the best geometry you can teach your first year undergraduates is the one based on modern treatment of linear algebra. The syllabus might look like this (it is based on the course I've taken in the recent years):

1. The language of vector spaces and linear transformations (bases, determinants, dual spaces)

2. Euclidean structure (Gram matrices, orthogonal bases, orthogonal projections, any orthogonal operator is a composition of reflections in hyperplanes, orthogonal operator acts as a rotation in two-dimensional subspaces)

3. Affine geometry and topology (norms, metrics, topology; convex sets, supporting halfspaces; polytopes as intersections of halfspaces)

4. Projective spaces (homogeneous coordinates, atlases on projective space, Veronese embedding, projective transformations, duality of points and hyperplanes)

5. Projective and affine quadrics and conics(rank, kernel; tangent space to a quadric; polar transformations; pencils of quadrics)

Textbooks with this kind of geometry are:

1. Michele Audin - Geometry
2. Elmer Rees - Notes on geometry
3. Gruenberg, Weir - Linear geometry
4. Jean Gallier - Geometric Methods and Applications
5. Mark Steinberger - A course in low-dimensional geometry
6. Tarrida - Affine maps, Euclidean motions and Quadrics
7. Dieudonne - Linear algebra and geometry
8. Berger - Geometry
9. Vinberg - A course in algebra, chapter "affine and projective spaces"

Such a course would give your students better understanding of the geometric nature of linear algebra (personally I think that the material one learns in a linear algebra course should be called "linear geometry"), it would show how modern mathematics simplifies classic material such as euclidean geometry and it would provide strong geometric basis for courses like algebraic geometry and topology (where familiarity with projective spaces helps a lot).

I suggest reading the preface to Dieudonne's book where he elaborates on these issues.

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In the same spirit: Efimov, N. V.; Rozendorn, È. R. Linear algebra and multidimensional geometry. Translated from the Russian by George Yankovsky [G. Jankovskiĭ]. Mir Publishers, Moscow, 1975. 495 pp. There are few exercises, though. –  Margaret Friedland Feb 26 '13 at 17:44
• Introduction to Geometry by Coxeter.
• Elementary Geometry From An Advanced Viewpoint by Moise.
• Geometry: Euclid and Beyond by Hartshorne.
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I would recommend Continuous Symmetry : From Euclid to Klein by Barker and Howe. I took the course as an undergraduate and enjoyed it very much. The first chapter gives an axiomatic treatment of geometry, and is meant to be a short part of the course. The rest of the book is a transformational approach to geometry, introducing isometries and similarities. Felix Klein's Erlanger Programm is the guiding principle for the course.

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How about John McCleary's book Geometry from a differentiable viewpoint''. I've always thought that would be a nice basis for a course like this.

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Here are few ideas:

1) I like the idea of a course about polytopes. Few books but some are excellent: "Lecture on polytopes" by Ziegler or "Convex polytopes" by Grunbaum are the obvious choices.

2) A course about curves and surfaces + an introduction to manifolds should satisfy 1-6 without troubles. "Differential geometry of curves and surfaces" by Do Carmo is a very good book; there are plenty of excellent books about manifolds.

3) A basic course on algebraic varieties require the use of algebra and differential calculus and gives example of spaces with pathological spaces (i.e. non Hausdorff and/or with singularities)

4) I guessed you want a more modern geometry course but without leaving the view of the formation of high school teachers. Michèle Audin wrote a very good book about affine, projective, curves and surfaces. It is aimed to future (French) high school teachers. I guess the title is "Geometry" (it is "Géométrie" in the French version).

I don't know the curriculum of a typical American student so I hope my suggestions are still pertinent (especially the point 3).

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Even for the Rice undergraduate, I think Ziegler or Grunbaum would be too much as a textbook. I do not mean to say their content could not be taught to undergraduates, only that the presentation would be too dense for them to follow. –  Alexander Woo Nov 1 '12 at 7:18

Last semester I taught (at Colby College) a geometry course based on two books: Bonahon's "Low dimensional geometry" and Schwartz's "Mostly Surfaces". Both are relatively inexpensive as far as textbooks go, so I could require both from the students. The students really enjoyed reading both books simultaneously as the authors have very different styles but some overlap of content. The students would certainly need to know some linear algebra in addition to multivariable calculus. The course was challenging, but reasonably successful at helping students develop some "geometric imagination" and proof-writing skills.

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Have you seen the book "Geometry by Its History" by Ostermann/Wanner?It looks anything but boring.I liked it very much!It has plenty of interestig mathematics.Maybe you record the lectures and put them on net...

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For a more applied course: What about Jean Gallier: Geometric Methods andApplications.

From the preface:

"Novelties: As far as we know, there is no fully developed modern exposition integrating the basic concepts of affine geometry, projective geometry, Euclidean geometry, Hermitian geometry, basics of Hilbert spaces with a touch of Fourier series, basics of Lie groups and Lie algebras, as well as a presentation of curves and surfaces both from the standard differential point of view and from the algorithmic point of view in terms of control points (in the polynomial and rational case).

Preface 1Introduction 1.1 Geometries: Their Origin, Their Uses 1.2 Prerequisites and Notation2Basics of Affine Geometry 2.1 Affine Spaces 2.2 Examples of Affine Spaces 2.3 Chasles's Identity 2.4 Affine Combinations, Barycenter 2.5 Affine Subspaces 2.6 Affine Independence and Affine Frames 2.7 Affine Maps2.8 Affine Groups 2.9 Affine Geometry: A Glimpse 2.10 Affine Hyperplanes 2.11 Intersection of Affine Spaces 2.12 Problems

3Properties of Convex Sets: A Glimpse 3.1 Convex Sets3.2 Caratheodory's Theorem 3.3 Radon's and Helly's Theorems Contents 3.4 Problems

4Embedding an Affine Space in a Vector Space 4.1 The "Hat Construction," or Homogenizing 4.2 Affine Frames of E and Bases of Ё 4.3 Another Construction of E 4.4 Extending Affine Maps to Linear Map 4.5 Problems

5 Basics of Projective Geometry 5.1 Why Projective Spaces? 5.2 Projective Spaces 5.3 Projective Subspaces 5.4 Projective Frames 5.5 Projective Maps 5.6 Projective Completion of an Affine Space, AffinePatches 5.7 Making Good Use of Hyperplanes at Infinity 5.8 The Cross-Ratio 5.9 Duality in Projective Geometry 5.10 Cross-Ratios of Hyperplanes 5.11 Complexification of a Real Projective Space 5.12 Similarity Structures on a Projective Space 5.13 Some Applications of Projective Geometry 5.14 Problems

6Basics of Euclidean Geometry 6.1 Inner Products, Euclidean Spaces 6.2 Orthogonality, Duality, Adjoint of a Linear Map 6.3 Linear Isometries (Orthogonal Transformations) 6.4 The Orthogonal Group, Orthogonal Matrices 6.5 Qi?-Decomposition for Invertible Matrices 6.6 Some Applications of Euclidean Geometry 6.7 Problems

7The Cartan-Dieudonne Theorem 7.1 Orthogonal Reflections 7.2 The Cartan-Dieudonne Theorem for Linear Isometries 7.3 (^-Decomposition Using Householder Matrices 7.4 Affine Isometries (Rigid Motions) 7.5 Fixed Points of Affine Maps 7.6 Affine Isometries and Fixed Points 7.7 The Cartan-Dieudonne Theorem for Affine Isometries 7.8 Orientations of a Euclidean Space, Angles 7.9 Volume Forms, Cross Products 7.10 Problems 8The Quaternions and the Spaces S3, SUB), SOC),and RP3 8.1 The Algebra M of Quaternions 8.2 Quaternions and Rotations in SOC) 8.3 Quaternions and Rotations in SOD) 8.4 Applications of Euclidean Geometry to MotionInterpolation 8.5 Problems

9Dirichlet—Voronoi Diagrams and DelaunayTriangulations 9.1 Dirichlet-Voronoi Diagrams 9.2 Simplicial Complexes and Triangulations 9.3 Delaunay Triangulations 9.4 Delaunay Triangulations and Convex Hulls 9.5 Applications of Voronoi Diagrams and DelaunayTriangulations 9.6 Problems10 Basics of Hermitian Geometry 10.1 Sesquilinear and Hermitian Forms, Pre-Hilbert Spacesand Hermitian Spaces 10.2 Orthogonality, Duality, Adjoint of a Linear Map 10.3 Linear Isometries (Also Called UnitaryTransformations) 10.4 The Unitary Group, Unitary Matrices 10.5 Problems11 Spectral Theorems in Euclidean and Hermitian Spaces 11.1 Introduction: What's with Lie Groups and LieAlgebras? 11.2 Normal Linear Maps 11.3 Self-Adjoint, Skew Self-Adjoint, and OrthogonalLinear Maps 11.4 Normal, Symmetric, Skew Symmetric, Orthogonal,Hermitian, Skew Hermitian, and Unitary Matrices .... 11.5 Problems

12 Singular Value Decomposition (SVD) and Polar Form 12.1 Polar Form 12.2 Singular Value Decomposition (SVD) 12.3 Problems

13 Applications of Euclidean Geometry to VariousOptimization Problems 13.1 Applications of the SVD and Qi^-Decomposition toLeast Squares Problems : 13.2 Minimization of Quadratic Functions UsingLagrange Multipliers 13.3 Problems

14 Basics of Classical Lie Groups: The Exponential Map,Lie Groups, and Lie Algebras 14.1 The Exponential Map 14.2 The Lie Groups GL(n,i), SL(n,M), O(n), SO(n),the Lie Algebras gZ(rc, R), sl(n,R), o(n), so(n), and theExponential Map 14.3 Symmetric Matrices, Symmetric Positive DefiniteMatrices, and the Exponential Map 14.4 The Lie Groups GL(n, C), SL(n, C), U(n), SU(n),the Lie Algebras gZ(rc, C), sZ(n,C), u(n), su(n),and the Exponential Map14.5 Hermitian Matrices, Hermitian Positive DefiniteMatrices, and the Exponential Map 14.6 The Lie Group SE(n) and the Lie Algebra se(n)14.7 Finale: Lie Groups and Lie Algebras14.8 Applications of Lie Groups and Lie Algebras 14.9 Problems

15 Basics of the Differential Geometry of Curves 15.1 Introduction: Parametrized Curves15.2 Tangent Lines and Osculating Planes15.3 Arc Length15.4 Curvature and Osculating Circles (Plane Curves) ....15.5 Normal Planes and Curvature CD Curves)15.6 The Frenet Frame CD Curves)15.7 Torsion CD Curves)15.8 The Frenet Equations CD Curves)15.9 Osculating Spheres CD Curves)15.10 The Frenet Frame for nD Curves (n > 4)15.11 Applications15.12 Problems

16 Basics of the Differential Geometry of Surfaces16.1 Introduction16.2 Parametrized Surfaces16.3 The First Fundamental Form (Riemannian Metric) . . .16.4 Normal Curvature and the Second Fundamental Form 16.5 Geodesic Curvature and the Christoffel Symbols16.6 Principal Curvatures, Gaussian Curvature, MeanCurvature16.7 The Gauss Map and Its Derivative dN16.8 The Dupin Indicatrix16.9 The Theorema Egregium of Gauss, the Equationsof Codazzi-Mainardi, and Bonnet's Theorem16.10 Lines of Curvature, Geodesic Torsion, AsymptoticLines16.11 Geodesic Lines, Local Gauss-Bonnet Theorem16.12 Applications16.13 Problems

17 Appendix17.1 Hyperplanes and Linear Forms17.2 Metric Spaces and Normed Vector Spaces

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I like Euclidean and Non-Euclidean Geometries: Development and History by Marvin J. Greenberg. I will warn you: it is certainly an axiomatic treatment. However, I really enjoyed the way that the book develops it. For example, the distinction between the axioms of a geometry and theorems you can prove about them, versus the models of geometry and their various properties, is clearly drawn. I dare say that, despite how advanced your undergraduates feel, they will learn a lot about the axiomatic method from this book. I recommend that you give it a look; even if it is not the primary textbook for the course, you can use it as a convenient source of motivation, problems, examples, and history. (There is a lot of history in this book, and many exercises.)

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I'm a a fan off Episodes in 19th and 20th Century Geometry [Ross Honsberger] It may have an old-fashioned outlook, but you would have to have simply amazing undergraduate students for none of them to find this material a challenge. I would supplement this with perhaps some chapters from Miles Reid's Undergraduate Algebraic Geometry (especially his treatment of the 27 lines on a cubic surface) and perhaps Alain Connes' little paper on Morley's Theorem.

I realize that most people prefer more foundation/methodological approaches (comparisons of various axiomatic systems). My bias: I want undergraduates to see more phenomena. But for that other sort of course, I like the Strasbourg Master Class on Geometry.

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