A sub-Riemannian manifold is a triple, where is a differentiable manifold, is a completely non-integrable "horizontal" distribution and is a smooth section of positive-definite quadratic forms on .
Any sub-Riemannian manifold carries the natural intrinsic metric, called the metric of Carnot–Carathéodory, defined as
where infimum is taken along all horizontal curves such that, .
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