I am currently learning Differential Geometry from the "categorical" point of view. We use sheaves, ringed spaces, group objects,... to define smooth manifolds, vector bundles,...

My previous course about Differential Geometry was more "common": we defined atlases, transition maps,... and it was clearer how to make differential calculus with these objects. My favourite fields are mainly analysis and probability/statistics and it is why I am more interested in learning Differential Geometry from the easiest viewpoint to define "calculus" (in a large sense of the term).

The point of my question here is "is it possible to make analysis on manifolds even if we learn it from this more abstract viewpoint or will I be forced to use the more classical one?" This is not a question about the opinion some people might have, the point is to understand what is the goal of seeing Differential Geometry from some viewpoint instead of another and what are their respective (dis)advantages (objective ones), what one allows that the other doesn't (if so). I hope this question fits the requirements of math.stackexchange. If not, feel free to vote for it to be closed. (I'm not sure what tags are relevant for such a question)

  • $\begingroup$ If your professor's goal is noncommutative geometry then you may have to get used to the algebraic viewpoint. $\endgroup$ – Mikhail Katz Jan 20 '16 at 12:41
  • $\begingroup$ @user72694 The thing is my course is over now and I had no choice following it. However, I found it was interesting to have another viewpoint than the one I was used to. But if one does want to study analysis on manifolds, is it possible with both viewpoints? If so, what are the advantages of choosing one or the other (or using both)? $\endgroup$ – MoebiusCorzer Jan 20 '16 at 12:44
  • $\begingroup$ If he wanted to prove de Rham's theorem using Warner's approach he would need sheaves. One would need to know where this is going. $\endgroup$ – Mikhail Katz Jan 20 '16 at 12:47
  • $\begingroup$ I see. It is more a matter of perspective, then? Say I want to study PDEs on manifolds. Are there any advantages to work in one framework rather than in the other? Or using both frameworks is the better thing to do? Actually, I do not see the point of learning Differential Geometry from this abstract viewpoint (but I guess there are advantages). $\endgroup$ – MoebiusCorzer Jan 20 '16 at 12:55
  • $\begingroup$ It's not really clear to me that a viewpoint like this will help much for understanding geometric analyis or anything like that, no. $\endgroup$ – user98602 Jan 20 '16 at 18:18

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