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On the classification of prolongations up to Engel homotopy

Author: Álvaro del Pino
Journal: Proc. Amer. Math. Soc. 146 (2018), 891-907
MSC (2010): Primary 58A30
Published electronically: August 1, 2017
MathSciNet review: 3731719
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Abstract: In (Casals, Pérez, del Pino, and Presas, preprint) it was shown that Engel structures satisfy an existence $ h$-principle, and the question of whether a full $ h$-principle holds was left open. In this note we address the classification problem, up to Engel deformation, of Cartan and Lorentz prolongations. We show that it reduces to their formal data as soon as the turning number is large enough.

Somewhat separately, we study the homotopy type of the space of Cartan prolongations, describing completely its connected components in the overtwisted case.

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Álvaro del Pino
Affiliation: Universidad Autónoma de Madrid and Instituto de Ciencias Matemáticas – CSIC, C. Nicolás Cabrera, 13–15, 28049, Madrid, Spain

Keywords: Engel structure, $h$--principle, Cartan prolongation, Lorentzian prolongation
Received by editor(s): October 29, 2016
Received by editor(s) in revised form: February 15, 2017, and March 16, 2017
Published electronically: August 1, 2017
Additional Notes: The author was supported by the Spanish Research Projects SEV–2015–0554, MTM2013–42135, and MTM2015–72876–EXP and a La Caixa–Severo Ochoa grant
Communicated by: Ken Bromberg
Article copyright: © Copyright 2017 American Mathematical Society