In the Riemannian world all directions in space are available to you. One can move along an arbitrary curve or calculate the scalar product between any two vectors. This makes Riemannian manifolds similar to Euclidean spaces, and indeed, Euclidean spaces provide excellent local approximations. In a sub-Riemannian manifold it is possible to measure distances only along certain directions, which forces us to consider only special curves, known as the horizontal curves, whose tangent vectors have finite length. As a result, sub-Riemannian manifolds exhibit a variety of different phenomena not present in the Riemannian case. Wave fronts of SR-manifolds are not compact and geodesics can loose optimality in arbitrary small time (the injectivity radius is always zero), there might be abnormal curves, which can be local minimizers for any possible metric; and the best approximation to a sub-Riemannian manifold is a nilpotent Lie group.
Nevertheless there is a constant search for generalizations of familiar objects and notions from the Riemannian geometry that still hold in the SR-geometry. For example, the concept of curvature in SR-geometry is quite evasive. In order to define it, one should first choose a model space (the zero curvature space) and then construct invariants that measure how different the given manifold is from the zero curvature model. One possible candidate is the tangent cone which is a nilpotent Lie group. In this case there are nice ways to define curvature, but then usually one loses all its nice geometric interpretations, like the divergence of geodesics for negative curvature! In a joint article with D. Barilari and A. Lerario, I studied the analog of the scalar curvature for 3D contact structures using expansion of the volume of small balls, and we saw, for example, that the isoperimetric problems on surfaces of constant curvatures K are exactly the constant scalar curvature K contact structures. If you are more interested in geodesics and minimality, you should check the articles mentioned in the optimal control section. Or if you would like to see an application check this joint work with R. Duits, A. Mashtakov, E. Bekkers and Yu. Sachkov on one can use SR-geometry for early stage diabetes diagnosis.