Carnot-Caratheodory metric












11












$begingroup$


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.






reference-request ho.history-overview sub-riemannian-geometry heisenberg-groups







share|cite|improve this question











$endgroup$

















    11












    $begingroup$


    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.






    reference-request ho.history-overview sub-riemannian-geometry heisenberg-groups







    share|cite|improve this question











    $endgroup$















      11












      11








      11





      $begingroup$


      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.






      reference-request ho.history-overview sub-riemannian-geometry heisenberg-groups







      share|cite|improve this question











      $endgroup$




      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.






      reference-request ho.history-overview sub-riemannian-geometry heisenberg-groups




      reference-request ho.history-overview sub-riemannian-geometry heisenberg-groups






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 7 hours ago









      YCor

      29k486140




      29k486140










      asked 10 hours ago









      Piotr HajlaszPiotr Hajlasz

      10.4k43976




      10.4k43976






















          1 Answer
          1






          active

          oldest

          votes


















          9












          $begingroup$

          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 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.






          share|cite|improve this answer











          $endgroup$









          • 1




            $begingroup$
            If I understand correctly, Carnot refers to Carnot cycles, and therefore to the French physicist Sadi Carnot (1796-1832).
            $endgroup$
            – YCor
            7 hours ago






          • 3




            $begingroup$
            Certainly, that’s him.
            $endgroup$
            – Carlo Beenakker
            6 hours ago












          Your Answer





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          1 Answer
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          active

          oldest

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          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          9












          $begingroup$

          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 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.






          share|cite|improve this answer











          $endgroup$









          • 1




            $begingroup$
            If I understand correctly, Carnot refers to Carnot cycles, and therefore to the French physicist Sadi Carnot (1796-1832).
            $endgroup$
            – YCor
            7 hours ago






          • 3




            $begingroup$
            Certainly, that’s him.
            $endgroup$
            – Carlo Beenakker
            6 hours ago
















          9












          $begingroup$

          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 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.






          share|cite|improve this answer











          $endgroup$









          • 1




            $begingroup$
            If I understand correctly, Carnot refers to Carnot cycles, and therefore to the French physicist Sadi Carnot (1796-1832).
            $endgroup$
            – YCor
            7 hours ago






          • 3




            $begingroup$
            Certainly, that’s him.
            $endgroup$
            – Carlo Beenakker
            6 hours ago














          9












          9








          9





          $begingroup$

          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 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.






          share|cite|improve this answer











          $endgroup$



          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 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.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited 10 hours ago

























          answered 10 hours ago









          Carlo BeenakkerCarlo Beenakker

          80k9190293




          80k9190293








          • 1




            $begingroup$
            If I understand correctly, Carnot refers to Carnot cycles, and therefore to the French physicist Sadi Carnot (1796-1832).
            $endgroup$
            – YCor
            7 hours ago






          • 3




            $begingroup$
            Certainly, that’s him.
            $endgroup$
            – Carlo Beenakker
            6 hours ago














          • 1




            $begingroup$
            If I understand correctly, Carnot refers to Carnot cycles, and therefore to the French physicist Sadi Carnot (1796-1832).
            $endgroup$
            – YCor
            7 hours ago






          • 3




            $begingroup$
            Certainly, that’s him.
            $endgroup$
            – Carlo Beenakker
            6 hours ago








          1




          1




          $begingroup$
          If I understand correctly, Carnot refers to Carnot cycles, and therefore to the French physicist Sadi Carnot (1796-1832).
          $endgroup$
          – YCor
          7 hours ago




          $begingroup$
          If I understand correctly, Carnot refers to Carnot cycles, and therefore to the French physicist Sadi Carnot (1796-1832).
          $endgroup$
          – YCor
          7 hours ago




          3




          3




          $begingroup$
          Certainly, that’s him.
          $endgroup$
          – Carlo Beenakker
          6 hours ago




          $begingroup$
          Certainly, that’s him.
          $endgroup$
          – Carlo Beenakker
          6 hours ago


















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