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I want to ask if there is some book that treats Differential Geometry without assuming that the reader knows General Topology. Well, many would say: "oh, but what's the problem ? First learn General Topology, and you'll understand Differential Geometry even better!" and I agree with that, but my point is: I'm a student of Physics, however I like to do everything with rigorous math, and of course I have special interested in math made with rigour like Spivak does, however, in the course of Physics we don't have General Topology and in the moment because of some things I'm studying I'm needing Differential Geometry, so I don't have time at the moment for General Topology.

Of course I'm interested in General Topology and of course in the future I'll study it, revisit Differential Geometry and make it more general, but for now I'm looking for some place which teaches without this. I'm looking for some treatment of differential geometry that defines manifolds without resorting to topological spaces and that gives good examples of constructing atlases.

Today my background is: basic set theory, single and multivariable calculus, ordinary differential equations, linear and multilinear algebra, a little bit of abstract algebra and also the basic topology of $\mathbb{R}^n$. Also I know the construction of vectors as derivations in euclidean spaces, as well as the definitions of tangent and cotangent spaces in the euclidean space.

Is there some book that teaches differential geometry (starting from the definition of manifold) in a way that it's suitable for someone who has this background?

Thanks for your help, and sorry if this question is silly, or if it's not suitable here.

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  • $\begingroup$ I deleted my answer because I realised this isn't exactly what you want, but you still might have some interest in it, so I leave it as a comment. $\endgroup$ – Git Gud Apr 4 '13 at 21:28
  • $\begingroup$ What sort of differential geometry you want? Curves and surfaces one can do with a bare minimum of knowledge of the topology of $\mathbb R^n$. If you want manifolds and the whole package, well, learn a big of general topology: it is unavoidable. $\endgroup$ – Mariano Suárez-Álvarez Apr 4 '13 at 21:36
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    $\begingroup$ Two downvotes? Just what is wrong with this question? $\endgroup$ – Git Gud Apr 4 '13 at 21:38
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    $\begingroup$ Gauss didn't know anything about topology, and he did OK. $\endgroup$ – bubba Apr 5 '13 at 0:00
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    $\begingroup$ @Mark: sorry about that. This link works now: rybu.org/forum/6 I'll try to ensure that's stable. $\endgroup$ – Ryan Budney Dec 31 '16 at 0:23
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By ``basic topology of $\mathbb{R}^n$'' I assume that you are familiar with the notions of openness, closedness, connectedness, and compactness. If you are unclear on these notions (I found compactness hard to get used to), you should remedy that before attempting to learn differential geometry.

If you understand these, then you're probably already prepared to read an introductory book on differential geometry, such as do Carmo's Differential Geometry of Curves and Surfaces or O'Neill's Elementary Differential Geometry. Apart from the concepts I mentioned above, all the necessary topology is developed alongside the geometry in these books (e.g. homeomorphism, homotopy, Euler characteristic, and so on).

If you want to learn quickly about the topology of smooth manifolds without having to learn about general topological spaces, there is probably no better place to look than Milnor's Topology from the Differentiable Viewpoint. A more in-depth treatment along the same lines is Guillemin and Pollack's Differential Topology.

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  • $\begingroup$ Thanks very much for you answer. I've heard that also Spivak's Differential Geometry book only needs a little of metric spaces. I've already studyed his Caluclus on Manifolds, do you think it's a good idea to study what's required of metric spaces and study Spivak together with those you mentioned? Thanks for your help! $\endgroup$ – user1620696 Apr 5 '13 at 17:42
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    $\begingroup$ Yeah; Spivak seems to cover all the topology he needs in his books. If you're up for Spivak's five-volume magnum opus, go for it! $\endgroup$ – Robert Haraway Apr 8 '13 at 15:01

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