Until recently, on my home page was a link to the scan of the paper

The topological substratum of the derivative (I), Math. Reports (Stud. Cerc. Mat.) 45, 6, (1993), 453-465

which is no longer visible now. But maybe it deserves a post here, because is my oldest attempt to understand differential calculus as an abstract matter and to look to new forms of it.

To me it became clear that differential calculus admits variants, in the same spirit as euclidean geometry admitting non-euclidean variants. At that moment I had no really intersting examples of such a “non-euclidean” differential calculus, so I switched to other research subjects. Nobody pointed to me the huge paper

Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, The Annals of Mathematics Second Series, Vol. 129, No. 1 (Jan., 1989), pp. 1-60

by Pierre Pansu. It was only luck that in 2000, at Lausanne, I met Sergey Vodop’yanov (from Sobolev Institute of Mathematics). He started to explain to me what Carnot groups are and I was thrilled to learn that examples I needed previously are numerous in sub-riemannian geometry.

With the right frame of mind (at least I think so), that of intrinsic dilations, I started then to study sub-riemannian geometry.

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I wonder if it’s the same thing Peter Saveliev was saying?

http://isomorphism.es/post/24164852462/calculus-is-topology

actually I thought about it again in an elementary example. Drawing x²/2 for a freshman and showing how the centred derivative is more accurate than the right-hand one which is usually used.

Put a point, draw two secant lines (equal ± along the abscissa).

Then the graphically intuitive fact is that, in this 1→1D picture, connecting the two ± points —- which belong to different tangent spaces! —– has “the same” slope as the real tangent. So how to state this in terms of contravariant motion? I.e., motion of the bundles themselves. Because it’s graphically obvious as well that the connected secant-line is not “the” tangent line, since it’s shifted over.

Anyway, also if you work out the average of left- and right- derivatives (limit definition) then you get a middle term which always cancels. And this reminds me of the ∂²=0 dictum, and I think it’s the same thing (as well as topologically amenable).

Anyway, then I wondered if it connects to this phenomenon: http://isomorphism.es/post/47732865342/product-rule-with-shapes

BTW, currently my analogy for top / diff top / riemannian\curvature is this:

topology = C⁰

diff top = C¹

curvature = C²