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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?

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    $\begingroup$ Is this for grad students? Upper-division undergrads? $\endgroup$ – Ben Crowell Dec 30 '16 at 17:03
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    $\begingroup$ I tagged it as both graduate and undergraduate, as I am thinking of a scenario in which upper-division undergrads might study classical differential geometry and then return to the subject at graduate level, the latter with a more mature manifolds first approach. Would the classical introduction help or hinder? $\endgroup$ – J W Dec 30 '16 at 17:07
  • $\begingroup$ For upper division students, the world needs more easy, informal, descriptive, non-comprehensive books along the lines of Hartle's Gravity and Messer's Topology Now! I don't know if there is anything along those lines for calculus on a manifold. $\endgroup$ – Ben Crowell Dec 30 '16 at 17:09
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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.

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  • $\begingroup$ Which books do you suggest as an intro to differential geometry and Manifolds? Lee wasn't good for me, maybe because of the style maybe because I read it too fast. $\endgroup$ – Paracosmiste Dec 30 '16 at 16:51
  • $\begingroup$ @whatever: What I learned about differential geometry on a manifold was all from general relativity textbooks: Wald and Misner, Thorne, and Wheeler. $\endgroup$ – Ben Crowell Dec 30 '16 at 17:02
  • $\begingroup$ It's a pity to hear that it did you "no good at all" although I suspect there may be a hint of hyperbole there. Still, point taken regarding having to unlearn certain beliefs. It makes me wonder, though, if classical differential geometry has a useful purpose as a capstone course for non-advancing students. $\endgroup$ – J W Jan 1 '17 at 15:38
  • $\begingroup$ @JW: No hyperbole. Try taking a book like Pressley and a GR book like Misner and putting them on a table side by side. Try to find a topic from Pressley that is used in Misner. I don't know what it would be. I suppose you could say that Gaussian curvature prefigures the Riemann tensor, but the development of it is completely different. $\endgroup$ – Ben Crowell Jan 3 '17 at 14:46
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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.

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    $\begingroup$ This is a reasonable suggestion, but is there an existing textbook that would support this? I would think it would be horribly confusing to go back and forth between the two approaches, unless the whole development of the subject had been very carefully planned in a unified way. You seem to associate classical d.g. with real-world applications and d.g. on a manifold with abstraction. As a physicist, I find this a funny way of thinking about it, since I have never seen the classical approach applied to physics, but the manifold approach is used throughout GR. We live on a curved manifold. $\endgroup$ – Ben Crowell Jan 3 '17 at 15:38
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    $\begingroup$ @BenCrowell: in France we don't have the same textbook approach as in the US, and I can't answer your question about that. I guess both old fashioned geometry and geometry on manifold can be both quite heavily formalized and related to physics, and my attribution is more related with my own experience of the subject. Examples of physics translated in extrinsic surface geometry is minimal and CMC surfaces (modeling soap films), and the study of the bending of a thin plates put under constraints. $\endgroup$ – Benoît Kloeckner Jan 3 '17 at 22:15

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