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Curvature and the equivalence problem in sub-Riemannian geometry. (English) Zbl 07655750

Summary: 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 [E. Grong, “Canonical connections on sub-Riemannian manifolds with constant symbol”, arXiv:2010.05366] 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.

MSC:

53C17 Sub-Riemannian geometry
58A15 Exterior differential systems (Cartan theory)
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