In the setting of step 2 sub-Finsler Carnot groups with strictly convex norms, we prove that all infinite geodesics are lines. It follows that for any other homogeneous distance, all geodesics are lines exactly when the induced norm on the horizontal space is strictly convex. As a further consequence, we show that all isometric embeddings between such homogeneous groups are affine. The core of the proof is an asymptotic study of the extremals given by the Pontryagin Maximum Principle.


  1. Carnot groups
  2. isometries
  3. isometric embeddings
  4. geodesics
  5. sub-Riemannian geometry
  6. sub-Finsler geometry

MSC codes

  1. 30L05
  2. 53C17
  3. 49K21
  4. 22E25

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Information & Authors


Published In

cover image SIAM Journal on Control and Optimization
SIAM Journal on Control and Optimization
Pages: 447 - 461
ISSN (online): 1095-7138


Submitted: 27 June 2019
Accepted: 18 November 2019
Published online: 12 February 2020


  1. Carnot groups
  2. isometries
  3. isometric embeddings
  4. geodesics
  5. sub-Riemannian geometry
  6. sub-Finsler geometry

MSC codes

  1. 30L05
  2. 53C17
  3. 49K21
  4. 22E25



Funding Information

Academy of Finland https://doi.org/10.13039/501100002341 : 288501

Funding Information

H2020 European Research Council https://doi.org/10.13039/100010663 : 713998

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