Does learning classical approach to differential geometry before modern approach help or hinder?

Question

To my mind, there seem to be two main paths to differential geometry. There is the classical approach, focusing on curves and surfaces in $\mathbb{R}^n$, especially $\mathbb{R}^3$. Prerequisites tend to be multivariable calculus and linear algebra. Some exposure to real or complex analysis and metric spaces or point-set topology is helpful, but not strictly required. A couple of titles following this approach include Pressley's Elementary Differential Geometry and Tapp's recent Differential Geometry of Curves and Surfaces.

The modern approach defers much of the geometry and studies manifolds first. An example of this path would be Lee's trilogy: Introduction to Topological Manifolds, Introduction to Smooth Manifolds and Riemannian Manifolds: An Introduction to Curvature.

Consider a scenario in which upper-division undergraduates study classical differential geometry and then return to the subject at graduate level, the latter with a manifolds-first approach. Would the classical introduction help or hinder?

Answer 1

Looking through the Pressley book, it seems to me that a student could study the whole thing, and at the end of the semester they would believe that all of differential geometry happens on curves and 2-surfaces embedded in Euclidean 3-space, and that space is endowed by its creator with a certain natural coordinate system and a certain natural system of measurement. These are all intuitively appealing beliefs, and all of them will have to be unlearned if that student is going to go on to study general relativity or differential geometry on a manifold.

The student would imagine themself to be knowledgeable in differential geometry, but would never have heard the term "metric" (which is absent from the index). Nor have they ever heard of tensors, much less the Riemann tensor.

To me, the whole treatment has a very 19th-century flavor. I read an old-fashioned differential geometry book of this type, by do Carmo, when I was a grad student before I took general relativity, thinking it would be helpful to develop geometric intuition in a context that was more visualizable. It did me no good at all.

Answer 2

For students who mean to continue in mathematics after graduating, I would advise a combination of the old and modern approaches. The motivation is that it might be confusing later on to rely too strongly on the underlying Euclidean structure, but that you don't want your student to end you course without understanding why there is no perfectly accurate plan map of any part of the Earth or thinking all this is abstract nonsense. Differential geometry is both quite heavily formalized and deeply entrenched in physical reality, and we should convey both aspects.

An example of combination would be to consider curves in the plane, surfaces in the space, curves on surfaces (with intrinsic and extrinsic curvatures), simple intrinsic surfaces (e.g. flat tori and Riemann sphere constructed by gluing two complex lines by $z\mapsto 1/z$, so as to make connection with complex analysis) and some very simple intrinsic $3$-manifolds (flat $3$-tori, round $3$-sphere constructed by stereographical projection taking inspiration from the $2$-dimensional one).

To be honest, in France this could be done only in the very top institutions - even the basics of curves and surfaces are hopelessly impossible to teach to undergraduates in most if not all universities.

Also, a reminder that I always feel compelled to make in such occasion: in the definition of a manifold, you do not want to neglect the Hausdorff nor the paracompact assumptions. If you do, search about leaf spaces of foliations and about the long line and the wild world of non-metrizable surfaces. You'd be amazed. Back to the point, this means you may prefer to give a few example without general definition of a manifold in an undergraduate course.