2019-04-09 17:49:05 8 Comments

The metric in sub-Riemannian geometry is often called the Carnot-Carathéodory metric.

Question 1.What is the origin of this name? Who was the first to introduce it?

I believe that the "Carathéodory" part of the name could be related to his work in theoretical thermodynamics [1], but I do not really know how it is related to his work.

Question 2.How is the notion of Carnot-Carathéodory metric related to the work of Carathéodory?

I know that Carnot groups are special examples of sub-Riemannian manifolds, but is it the reason for "Carnot" part in the name of the metric?

Question 3.What does the "Carnot" part of the name of the metric stand for?

**[1] C. Carathéodory,** *Untersuchungen uber die Grundlagen der Thermodynamik*.
*Math. Ann.* 67 (1909), 355–386.

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## 1 comments

## @Carlo Beenakker 2019-04-09 17:58:22

Pierre Pansu tells us that the terminology of the Carnot-Carathéodory metric is due to Mikhail Gromov [1].

Gromov himself explains the choice of the name:

Pansu adds

[1] M. Gromov –

Structures métriques pour les variétés Riemanniennes, Textes Mathématiques, vol. 1, Paris, 1981, Edited by J. Lafontaine and P. Pansu.## @YCor 2019-04-09 21:19:18

If I understand correctly, Carnot refers to Carnot cycles, and therefore to the French physicist Sadi Carnot (1796-1832).

## @Carlo Beenakker 2019-04-09 21:34:15

Certainly, that’s him.