Curvature and the equivalence problem in sub-Riemannian geometry

Erlend Grong

Archivum Mathematicum (2022)

  • Volume: 058, Issue: 5, page 295-327
  • ISSN: 0044-8753

Abstract

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These notes give an introduction to the equivalence problem of sub-Riemannian manifolds. We first introduce preliminaries in terms of connections, frame bundles and sub-Riemannian geometry. Then we arrive to the main aim of these notes, which is to give the description of the canonical grading and connection existing on sub-Riemann manifolds with constant symbol. These structures are exactly what is needed in order to determine if two manifolds are isometric. We give three concrete examples, which are Engel (2,3,4)-manifolds, contact manifolds and Cartan (2,3,5)-manifolds. These notes are an edited version of a lecture series given at the 42nd Winter school: Geometry and Physics, Srní, Czech Republic, mostly based on [8] and other earlier work. However, the work on Engel (2,3,4)-manifolds is original research, and illustrate the important special case were our model has the minimal set of isometries.

How to cite

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Grong, Erlend. "Curvature and the equivalence problem in sub-Riemannian geometry." Archivum Mathematicum 058.5 (2022): 295-327. <http://eudml.org/doc/298909>.

@article{Grong2022,
abstract = {These notes give an introduction to the equivalence problem of sub-Riemannian manifolds. We first introduce preliminaries in terms of connections, frame bundles and sub-Riemannian geometry. Then we arrive to the main aim of these notes, which is to give the description of the canonical grading and connection existing on sub-Riemann manifolds with constant symbol. These structures are exactly what is needed in order to determine if two manifolds are isometric. We give three concrete examples, which are Engel (2,3,4)-manifolds, contact manifolds and Cartan (2,3,5)-manifolds. These notes are an edited version of a lecture series given at the 42nd Winter school: Geometry and Physics, Srní, Czech Republic, mostly based on [8] and other earlier work. However, the work on Engel (2,3,4)-manifolds is original research, and illustrate the important special case were our model has the minimal set of isometries.},
author = {Grong, Erlend},
journal = {Archivum Mathematicum},
keywords = {sub-Riemannian geometry; equivalence problem; frame bundle; Cartan connection; flatness theorem},
language = {eng},
number = {5},
pages = {295-327},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Curvature and the equivalence problem in sub-Riemannian geometry},
url = {http://eudml.org/doc/298909},
volume = {058},
year = {2022},
}

TY - JOUR
AU - Grong, Erlend
TI - Curvature and the equivalence problem in sub-Riemannian geometry
JO - Archivum Mathematicum
PY - 2022
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 058
IS - 5
SP - 295
EP - 327
AB - These notes give an introduction to the equivalence problem of sub-Riemannian manifolds. We first introduce preliminaries in terms of connections, frame bundles and sub-Riemannian geometry. Then we arrive to the main aim of these notes, which is to give the description of the canonical grading and connection existing on sub-Riemann manifolds with constant symbol. These structures are exactly what is needed in order to determine if two manifolds are isometric. We give three concrete examples, which are Engel (2,3,4)-manifolds, contact manifolds and Cartan (2,3,5)-manifolds. These notes are an edited version of a lecture series given at the 42nd Winter school: Geometry and Physics, Srní, Czech Republic, mostly based on [8] and other earlier work. However, the work on Engel (2,3,4)-manifolds is original research, and illustrate the important special case were our model has the minimal set of isometries.
LA - eng
KW - sub-Riemannian geometry; equivalence problem; frame bundle; Cartan connection; flatness theorem
UR - http://eudml.org/doc/298909
ER -

References

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