Previous publications of Tristan Needham have stressed his visual and geometric approach to mathematics. His highly praised massive book

*Visual Complex Analysis* may still be resounding in the minds of those who have read it. The original approach and the numerous graphics must have left a lasting impression. In this book, he repeats this

*tour de force*. This time he uses his unique vision to introduce differential geometry at an undergraduate level. Along the way, he stumbles upon classic results such as Kepler’s laws and Maxwell’s equations, but eventually, he also goes deep into the applications of general relativity and black holes, and he ends with Cartan’s differential forms to discuss more advanced vector and tensor calculus on higher-dimensional manifolds. It is all starting from first geometric principles mixed with some physical insight. The graphics and pictures used on most of the 530 pages are obviously an essential element to realize this objective. Most are computer generated, but there are also pictures of fruits and vegetables decorated with toothpicks to represent curved surfaces and vectors.

The key observation (and this is where the book starts) is that when doing geometry on a curved surface, one has to leave Euclidean geometry. For creatures living on a spherical or hyperbolic surface, their geometry and their physical world, differs from the one of external spectators. The intrinsic and the extrinsic analysis of a local situation is quite different. Just like Abbott’s flatlanders have no idea about a third dimension, neither have spherelanders or inhabitants of any two-dimensional manifold, and yet, by measuring angles in a geodesic triangle, they can learn about the curvature of their sphere. This is also how we can learn about the curvature of our expanding universe. So, there is this surprising interplay between the intrinsic and the extrinsic formulas to describe some local geometry. Proving this connection is one of the main objectives (Needham calls it a climax) of this book.

The mathematical approach Needham takes is strongly influenced by reading Newton’s *Principia*, not only because of the explicit physical motivation (a lot of new mathematics originated from trying to understand a physical phenomenon) but also by Newton’s method. Leibniz’ approach to calculus was more successful mainly because of his easier notational formalism, but what if Newton had been a better salesman with a better notation for his theory of fluxions? This is where Needham picks up the challenge introducing a notation for Ultimate Equality \( a \asymp b \) meaning that \( a \) and \( b \) are asymptotically the same (as some small perturbation goes to zero). This is a key instrument throughout the book.

This text can be seen as an introduction to differential geometry, accessible for undergraduates, but I guess that professional mathematicians, even those who are already somewhat familiar with the subject, will appreciate the book even more, not because of the results, but because of the original approach. I here use ’original’ with its historical meaning referring for example to the ways Gauss or other ’originators’ treated the subject, but it also refers to ’different from’ the current approach usually taken in modern handbooks. Both meanings are linked to really understand a result. It is often relatively easy to obtain a formula by blindly applying the rules of calculus. Let the ’diabolic’ mathematical machinery spit out the formulas for you. It will tell you what the formula is, but it does not tell you why. That is why the book is not really conceived as a classroom textbook. Needham introduces and discusses formulas from a geometrical point of view, while sometimes the hard mathematical proofs come only much later (although care is taken to avoid circular reasoning). While collecting the ingredients for the main story, serendipity may give well-known, but unexpected, results derived from a (geometric) point of view, completely different from their discussion in other courses. Presenting it this way requires many more pages (and pictures!) and is more time consuming than usually allowed in a regular course.

Not hindered by the restrictions of efficiency, Needham can tell his story. A drama he calls it, and he gives it a Shakespearian allure by subdividing it into five acts. The dramatic theme is clearly ‘curvature’ in all its aspects. There is a climax in Act III where Gauss’ *Beautiful Theorem* and his *Theorema Egregium* and eventually (drum roll) the glorious *global Gauss-Bonnet theorem* (GGBT) are introduced. The latter marries intrinsic geometry with global topology. The mathematical proofs come in Act IV with a firework of different proofs of GGBT (not just one but four of them!) and, as a copious dessert, the application of the Riemann tensor to general relativity, black holes being the icing on the cake. It is not a spoiler to tell you that the main protagonists of the story (Gauss, Einstein, Riemann, Poincaré, Cartan,...) have died, although not as dramatically as in Shakespeare’s plays. Needham gives proper tribute to these initiators in historical notes. Act V is somewhat different in that it refers to the ‘Forms’ in the title of the book. General \( p \)-forms (with special emphasis on \( p = 1, 2, 3 \)) and tensor calculus in a higher dimensional space provides all the ’diabolic’ mathematical machinery (Atiyah calls it the Devil’s machine) to generalize the results that were derived so far. In the previous acts, results were mainly framed in a scenery of 2-dimensional manifolds in our familiar 3-dimensional space which allowed all the visual representations. But underlying this is still a true geometric introduction to the classical results of vector calculus (gradient, divergence, curl, Green and Stokes theorem,...). Needham writes

Our aim in Act V is to confront —succinctly, and in the plainest of Anglo-Saxon— a century-long scandal, namely, that the vast majority of undergraduates (in both mathematics and physics) will obtain their degree without ever having glimpsed Cartan’s Forms.

Although everything in the book is indeed introduced at an undergraduate level, I do not think that solving Einstein’s gravitational field equation and the physics of black holes is part of a normal undergraduate curriculum. Otherwise, the required elements are relatively low level. A bit of linear algebra and basic analysis should be sufficient. The least intuitive concept to grasp is perhaps the notion of parallel transport. This is how geometric information is moved around on the manifold. It is intensively used and thus is rather essential.

The text is basically self-contained, but there is also a long list of books for further reading (often with recommendations by Needham). Even though it is not a textbook, each Act has a long chapter with exercises. The typesetting of the book is excellent (it is nice to note that the Euler font (sic) is used for the formulas). Important formulas and results are put in display (no colors are used) and they get frames around them (one, two, or three) according to their relative importance. There are some small (harmless) typos, which is not uncommon in a voluminous first edition.

The book is a remarkable and highly original approach to the basic stem of differential geometry. And that mathematical trunk has roots and branches in so many other unexpected yet related subjects, each of which can be equally well approached from the same geometrical point of view. Needham throws away all the usual textbooks to re-think the material almost from scratch, using his geometrical glasses and following the footsteps of the giants of the past.

Adhemar Bultheel is emeritus professor at the Department of Computer Science of the KU Leuven (Belgium). He has been teaching mainly undergraduate courses in analysis, algebra, and numerical mathematics. More information can be found on his homepage

https://people.cs.kuleuven.be/~adhemar.bultheel/.