Curvature and the equivalence problem in sub-Riemannian geometry

Erlend Grong

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arXiv · 2206.15123

Curvature and the equivalence problem in sub-Riemannian geometry

Erlend Grong

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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 \href{https://conference.math.muni.cz/srni/}{42nd Winter school: Geometry and Physics}, Snrí, Check Republic, mostly based on 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.

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Erlend Grong. 2022-10-24. Curvature and the equivalence problem in sub-Riemannian geometry. https://arxiv.org/abs/2206.15123

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