What are some open problems in sub-Riemannian geometry?
I am interested especially in problems concerning connections and curvature, but any contribution is welcomed.
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Finding subriemannian geodesics of homogeneous spaces is one problem of practical importance. See http://hrl.harvard.edu/publications/khaneja02sub-riemannian.pdf
Andrei Agrachev recently posted an article on the arXiv: http://arxiv.org/abs/1304.2590 entitled "Some open problems" that you might find interesting. He discusses several open problems in control theory and sub-Riemannian geometry that are of considerable interest.