# Geometry with a view towards differential geometry textbook

I am scheduled to teach an upper-division undergraduate class on "Geometry" and I get to choose more or less what that means. Common choices seem to be non-Euclidean, hyperbolic, projective, or Erlangen geometry.

I would have liked to do differential geometry, since it seems to me to be a more central part of a mathematics education, but right now I think that that would probably be too difficult. In particular, linear algebra is not a prerequisite for the class. Also several of the students are future high-school teachers and would probably appreciate something that feels more relevant to them.

So I am wondering whether there is any textbook on non-Euclidean/hyperbolic/projective geometry (any or all) which at least leads up towards a "modern" differential-geometric point of view, e.g. the importance of a local metric and perhaps the intrinsic curvature. All the textbooks I have looked at so far seem to emphasize either the Euclid/Hilbert axiomatic style or the Klein Erlangen/transformation style of geometry, which are all well and good for the traditional (mostly homogeneous) examples; but I would like to at least expose the students to some of the ideas that lead later on to things like manifolds (and which are necessary for applications like relativity theory).

• I think that the typical "differential geometry of curves/surfaces" course was useful for me before ever learning any real "modern differential geometry". Many universities have such a course and there are quite a few books. Is this the kind of course you are worried your students wouldn't be prepared for?
– PVAL
Commented Mar 21, 2015 at 5:18
• @PVAL Yes. In particular, all the textbooks for such a class that I've found assume a linear algebra prerequisite. My students have had multivariable calculus, so they should know about things like vectors in $R^3$ and cross products, but not linear maps, matrices, bases, or eigenvalues. Commented Mar 22, 2015 at 15:25
• I really like Berger, Cole, Levy's two-volume Geometry (Universitext). But in terms of pre-requisites it may be too high for your course. (The use of Lebesgue measure and integration theory, as well as topology, is minimal (see Chapter 0) and probably can be glossed over. But it does rely on some abstract algebra.) Commented Mar 23, 2015 at 8:51
• @WillieWong yes, he appears to define affine spaces in terms of vector spaces in chapter 2, which would be way too much for students who haven't had any linear algebra. Commented Mar 23, 2015 at 17:09
• @MikeShulman Are course notes welcome? I believe I can get a copy of set of "advanced geometry" course notes from Herb Clemens. IIRC it does heavily utilize multivariable calculus, but does not use too many linear algebra concepts overtly. I think these notes are written in a "workbook" style, where the students are meant to fill in a lot of the details. Commented Mar 24, 2015 at 8:39

I think you may be looking for Geometry from a Differentiable Viewpoint 2nd Edition by John McCleary. I'm not sure it's the right level for your course, but, here is the advertisement:

The development of geometry from Euclid to Euler to Lobachevsky, Bolyai, Gauss, and Riemann is a story that is often broken into parts - axiomatic geometry, non-Euclidean geometry, and differential geometry. This poses a problem for undergraduates: Which part is geometry? What is the big picture to which these parts belong? In this introduction to differential geometry, the parts are united with all of their interrelations, motivated by the history of the parallel postulate. Beginning with the ancient sources, the author first explores synthetic methods in Euclidean and non-Euclidean geometry and then introduces differential geometry in its classical formulation, leading to the modern formulation on manifolds such as space-time. The presentation is enlivened by historical diversions such as Hugyens's clock and the mathematics of cartography. The intertwined approaches will help undergraduates understand the role of elementary ideas in the more general, differential setting. This thoroughly revised second edition includes numerous new exercises and a new solution key. New topics include Clairaut's relation for geodesics, Euclid's geometry of space, further properties of cycloids and map projections, and the use of transformations such as the reflections of the Beltrami disk.

If this is not what you want, it at least should give some ideas. It seems to have both axiomatic geometry and the basic differential geometry in a somewhat classical fashion.

• I'll have to check it out the next time I teach this class! (Last year I ended up doing projective geometry.) Commented Jan 5, 2017 at 9:42

You might consider basing your course on the Differential Geometry lectures given by Dr. N. J. Wildberger [found on youtube, link below.]

He has some idiosyncrasies that you can clearly ignore, but his approach does not assume a large amount of linear algebra at all. He does employ matrices later on in the game, but I think with a lecture or two on the use of them, you could be fine. His approach is super algebraic (not as in abstract algebra), which would be great for strengthening your students' skills before/concurrently with linear algebra!

His math history lectures on geometric topics might be good to flesh out the course if the latter part of his differential geometry series goes too far in depth.

His course also employs a lot of projective geometry -- which I think is definitely lacking in the curriculum as a whole and would be very worthwhile!

Here is the link to his playlist: Diff Geo Wildberger

I cannot answer your question. But since no one else answered, let me reply with another question: Would it be feasible to base an undergraduate geometry class on Thurston's great book?

William P. Thurston. Three-Dimensional Geometry and Topology: Volume 1. Edited by Silvio Levy. (Princeton Univ Press link.)

• Based on its description as originating in a graduate-level course, I expect the answer would be No. Commented Mar 23, 2015 at 17:10
• I am perpetually stunned by the sheer (high) quality of your book recommendations. Commented Mar 31, 2015 at 2:01
• Actually, I think it might be feasible to base an undergraduate course on just Chapter 1, "What is a manifold?", supplementing with additional material. Commented Mar 31, 2015 at 12:26
• A "softening" of Thurston's book is Bonahon's book "Low-dimensional geometry: from euclidean surfaces to hyperbolic knots" but even that requires extremely keen undergraduates. I imagine it would be a very rare opportunity where one could pull such a course off with a good undergraduate population. Commented Apr 19, 2017 at 5:03

I've been put in the same situation as you. I wound up "hijacking" the course and teaching it along the lines of the Millman and Parker intro to DG book. It went really well -- perhaps the highest teaching evaluations I've ever received. I'm not sure if that means much, though! The class had a population of about 20 or 22, and there were 2 or 3 education students in the course.

I viewed the course as something of an opportunity to cement the students' understanding of calculus and linear algebra. For those that did not know linear algebra, it was a "purposeful" first pass.

Saul Stahl's ** A Gateway to Modern Geometry: the Poincare Upper Half Plane ** looks extremely promising. I haven't read it, but based on the table of contents, it seems to begin with Euclidean Geometry, then inversion (circles), then focus for most of the book on hyperbolic geometry, and in particular the upper half plane model.

Chapter 12 is * Differential Geometry and Gaussian Curvature * and has the following sections:

• 12.1 Differential Geometry 157
• 12.2 A Review of Length and Area on Surfaces 163
• 12.3 Gauss's Formula for the Curvature at a Point 168
• 12.4 Riemannian Geometry Revisited 170
• This is a good choice if you are supposed to cover more axiomatic stuff but want to have some actual calculus in there somewhere, and you have the knowledge/experience of your specific clientele to figure out how much to slow down (or not) in the first few chapters. (For example, Menelaus' Theorem is an exercise with little context, which isn't quite appropriate for a class with people already uncertain about what they remember from high school!) But it is a good book and could definitely be used in this context. Commented May 9, 2022 at 15:52

You might try to do intrinsic geometry of convex/saddle surfaces. The subject is very visual and it makes perfect nontechnical introduction to differential geometry.

Start with Alexandrov's "Convex Polyhedra", and continue as to convex surfaces and Shefel's theorem on saddle surfaces if you time allows.