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On Jacobi fields and a canonical connection in sub-Riemannian geometry. (English) Zbl 1424.53072
A sub-Riemannian structure is a triple $$(M,D,g)$$ where $$M$$ is a smooth $$n$$-dimensional manifold, $$D$$ is a smooth, nonholonomic distribution of $$TM$$ and $$g$$ is a smooth scalar product on $$D$$. Riemannian structures are obtained as a particular case, considering $$D=TM$$. The sub-Riemannian structures are strongly related with the driftless control affine systems and geodesics curves in the sub-Riemannian geometry are the optimal solutions in control theory. In this interesting paper the authors study the Jacobi fields and a canonical nonlinear Ehresman connection in sub-Riemannian geometry. They prove that the coefficients of the Jacobi equation define curvature-like invariants and these coefficients can be interpreted as the curvature of a canonical Ehresmann connection associated to the metric. In the Riemannian case this reduces to the classical, linear, Levi-Civita connection. Finally, some of properties of this nonlinear connection are given.

##### MSC:
 53C17 Sub-Riemannian geometry 53B21 Methods of local Riemannian geometry 53B15 Other connections
##### Keywords:
sub-Riemannian geometry; curvature; connection; Jacobi fields
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