Carnot-Carathéodory metric
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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.
reference-request ho.history-overview sub-riemannian-geometry heisenberg-groups
asked Apr 9 at 17:49
Piotr HajlaszPiotr Hajlasz
10.7k44077
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add a comment |
$begingroup$
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.
reference-request ho.history-overview sub-riemannian-geometry heisenberg-groups
asked Apr 9 at 17:49
Piotr HajlaszPiotr Hajlasz
10.7k44077
$endgroup$
add a comment |
$begingroup$
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.
reference-request ho.history-overview sub-riemannian-geometry heisenberg-groups
asked Apr 9 at 17:49
Piotr HajlaszPiotr Hajlasz
10.7k44077
$endgroup$
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.
reference-request ho.history-overview sub-riemannian-geometry heisenberg-groups
reference-request ho.history-overview sub-riemannian-geometry heisenberg-groups
asked Apr 9 at 17:49
Piotr HajlaszPiotr Hajlasz
10.7k44077
asked Apr 9 at 17:49
Piotr HajlaszPiotr Hajlasz
10.7k44077
edited Apr 10 at 21:44
Piotr Hajlasz
asked Apr 9 at 17:49
Piotr HajlaszPiotr Hajlasz
10.7k44077
asked Apr 9 at 17:49
Piotr HajlaszPiotr Hajlasz
10.7k44077
asked Apr 9 at 17:49
Piotr HajlaszPiotr Hajlasz
10.7k44077
10.7k44077
add a comment |
add a comment |
1 Answer
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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 place holder for 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.
answered Apr 9 at 17:58
Carlo BeenakkerCarlo Beenakker
80.6k9193295
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2
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If I understand correctly, Carnot refers to Carnot cycles, and therefore to the French physicist Sadi Carnot (1796-1832).
$endgroup$
– YCor
Apr 9 at 21:19
4
$begingroup$
Certainly, that’s him.
$endgroup$
– Carlo Beenakker
Apr 9 at 21:34
add a comment |
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1 Answer
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$begingroup$
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 place holder for 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.
answered Apr 9 at 17:58
Carlo BeenakkerCarlo Beenakker
80.6k9193295
$endgroup$
2
$begingroup$
If I understand correctly, Carnot refers to Carnot cycles, and therefore to the French physicist Sadi Carnot (1796-1832).
$endgroup$
– YCor
Apr 9 at 21:19
4
$begingroup$
Certainly, that’s him.
$endgroup$
– Carlo Beenakker
Apr 9 at 21:34
add a comment |
$begingroup$
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 place holder for 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.
answered Apr 9 at 17:58
Carlo BeenakkerCarlo Beenakker
80.6k9193295
$endgroup$
2
$begingroup$
If I understand correctly, Carnot refers to Carnot cycles, and therefore to the French physicist Sadi Carnot (1796-1832).
$endgroup$
– YCor
Apr 9 at 21:19
4
$begingroup$
Certainly, that’s him.
$endgroup$
– Carlo Beenakker
Apr 9 at 21:34
add a comment |
$begingroup$
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 place holder for 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.
answered Apr 9 at 17:58
Carlo BeenakkerCarlo Beenakker
80.6k9193295
$endgroup$
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 place holder for 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.
answered Apr 9 at 17:58
Carlo BeenakkerCarlo Beenakker
80.6k9193295
edited Apr 10 at 6:16
answered Apr 9 at 17:58
Carlo BeenakkerCarlo Beenakker
80.6k9193295
answered Apr 9 at 17:58
Carlo BeenakkerCarlo Beenakker
80.6k9193295
answered Apr 9 at 17:58
Carlo BeenakkerCarlo Beenakker
80.6k9193295
80.6k9193295
2
$begingroup$
If I understand correctly, Carnot refers to Carnot cycles, and therefore to the French physicist Sadi Carnot (1796-1832).
$endgroup$
– YCor
Apr 9 at 21:19
4
$begingroup$
Certainly, that’s him.
$endgroup$
– Carlo Beenakker
Apr 9 at 21:34
add a comment |
2
$begingroup$
If I understand correctly, Carnot refers to Carnot cycles, and therefore to the French physicist Sadi Carnot (1796-1832).
$endgroup$
– YCor
Apr 9 at 21:19
4
$begingroup$
Certainly, that’s him.
$endgroup$
– Carlo Beenakker
Apr 9 at 21:34
2
2
$begingroup$
If I understand correctly, Carnot refers to Carnot cycles, and therefore to the French physicist Sadi Carnot (1796-1832).
$endgroup$
– YCor
Apr 9 at 21:19
$begingroup$
If I understand correctly, Carnot refers to Carnot cycles, and therefore to the French physicist Sadi Carnot (1796-1832).
$endgroup$
– YCor
Apr 9 at 21:19
4
4
$begingroup$
Certainly, that’s him.
$endgroup$
– Carlo Beenakker
Apr 9 at 21:34
$begingroup$
Certainly, that’s him.
$endgroup$
– Carlo Beenakker
Apr 9 at 21:34
add a comment |
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