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



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



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



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> **Question 3.** What does the "Carnot" part of the name of the metric stand for?
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**[1] C. Carathéodory,** [*Untersuchungen uber die Grundlagen der Thermodynamik*](https://link.springer.com/article/10.1007/BF01450409). 
*Math. Ann.* 67 (1909), 355–386.



Carnot-Carathéodory metric
[Pierre Pansu](http://www.math.polytechnique.fr/xups/textes-provisoires18/pansu.pdf) tells us that the terminology of the Carnot-Carathéodory metric is due to Mikhail Gromov [1].


Gromov himself [explains](https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/carnot_caratheodory.pdf) the choice of the name:



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> 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.
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Pansu adds



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> While the reference to Carathéodory is fundamental, the reference to
>  Carnot must be seen as a place holder for the many authors who
>  rediscovered accessibility criteria from the middle of the twentieth
>  century back to a much earlier date.
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[1] M. Gromov – *Structures métriques pour les variétés Riemanniennes*, Textes Mathématiques, vol. 1, Paris, 1981, Edited by J. Lafontaine and P. Pansu.


