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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“There is then creative reading as well as creative writing. When the mind is braced by labor and invention, the page of whatever book we read becomes luminous with manifold allusion. Every sentence is doubly significant, and the sense of our author is as broad as the world.”
—Ralph Waldo Emerson (1803–1882)