Mryntu

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

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

I believe that the "Caratheodory" 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-Caratheodory metric related to the work of Caratheodory?

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. Caratheodory, Untersuchungen uber die Grundlagen der Thermodynamik.
Math. Ann. 67 (1909), 355–386.

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:

The metric is called the Carnot-Carathéodory metric because it appears
(in a more general form) in the 1909 paper by Carathéodory on
formalization of the classical thermodynamics where horizontal curves
roughly correspond to adiabatic processes. The proof of this statement
may be performed in the language of Carnot cycles and for this reason
the metric was christened Carnot-Carathéodory.

Pansu adds

While the reference to Carathéodory is fundamental, the reference to
Carnot must be seen as a collective referral to the many authors who
rediscovered accessibility criteria from the middle of the twentieth
century back to a much earlier date.

[1] M. Gromov – Structures métriques pour les variétés Riemanniennes, Textes Mathématiques, vol. 1, Paris, 1981, Edited by J. Lafontaine and P. Pansu.