Contactspanning

Contactspanning is an informal term used in the field of contact geometry to describe a spanning property of a contact structure on a manifold. It refers to the idea that a collection of contact-related submanifolds or trajectories can generate or cover the contact distribution at every point of the manifold.

In more concrete terms, consider a contact manifold (M, ξ) with a contact form α such that ξ = ker α and let R be the Reeb vector field associated with α. A family F of submanifolds {S_i} is said to be contact-spanning if, for every point p in M, the tangent spaces T_p S_i at p (for those submanifolds that pass through p) together with the Reeb direction R_p generate the tangent space in a way that projects to span the contact distribution ξ_p. Equivalently, if the submanifolds are Legendrian, their tangent spaces lie in ξ, and a finite set {S_i} is contact-spanning when the union of their tangent spaces at p covers ξ_p for all p.

Examples are typically found in three-dimensional or low-dimensional manifolds where families of Legendrian curves or surfaces intersect or foliate the space in a way that provides a global generating set for the contact structure.

Applications of contactspanning concepts include constructing global sections of ξ, analyzing Reeb dynamics, and informing surgery or decomposition arguments in contact topology. The idea also connects with broader themes in the subject, such as open book decompositions and the Giroux correspondence, which relate contact structures to compatible topological decompositions.

Notes: the term is not universally standardized and appears mainly in informal discussions or specific research contexts. See also contact geometry, Legendrian submanifold, Reeb vector field, open book decomposition, and Giroux correspondence.