In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle and specified by a one-form, both of which satisfy a 'maximum non-degeneracy' condition called 'complete non-integrability'. From the Frobenius theorem, one recognizes the condition as the opposite of the condition that the distribution be determined by a codimension one foliation on the manifold ('complete integrability').
Contact geometry is in many ways an odd-dimensional counterpart of symplectic geometry, which belongs to the even-dimensional world. Both contact and symplectic geometry are motivated by the mathematical formalism of classical mechanics, where one can consider either the even-dimensional phase space of a mechanical system or the odd-dimensional extended phase space that includes the time variable.
Read more about Contact Geometry: Applications, Contact Forms and Structures, Legendrian Submanifolds and Knots, Reeb Vector Field, Some Historical Remarks
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“Yet for all that, there is nothing in me of a founder of a religion—religions are affairs of the rabble; I find it necessary to wash my hands after I have come into contact with religious people.”
—Friedrich Nietzsche (1844–1900)
“I am present at the sowing of the seed of the world. With a geometry of sunbeams, the soul lays the foundations of nature.”
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