Tag Archives: derivative

Topological substratum of the derivative

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 unThe 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.