Differential Geometry without General Topology

Question

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.

Answer 1

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.