Smooth manifolds | Differential geometry

Almost-contact manifold

In the mathematical field of differential geometry, an almost-contact structure is a certain kind of geometric structure on a smooth manifold. Such structures were introduced by Shigeo Sasaki in 1960. Precisely, given a smooth manifold an almost-contact structure consists of a hyperplane distribution an almost-complex structure on and a vector field which is transverse to That is, for each point of one selects a codimension-one linear subspace of the tangent space a linear map such that and an element of which is not contained in Given such data, one can define, for each in a linear map and a linear map by This defines a one-form and (1,1)-tensor field on and one can check directly, by decomposing relative to the direct sum decomposition thatfor any in Conversely, one may define an almost-contact structure as a triple which satisfies the two conditions * for any * Then one can define to be the kernel of the linear map and one can check that the restriction of to is valued in thereby defining (Wikipedia).

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Direct sum | Linear map | Linear subspace | Differential geometry | Tangent space | Tensor field | Kernel (linear algebra) | Distribution (differential geometry) | Vector field