{{Short description|Math topic}} {{Technical|date=April 2024}} In mathematics, a '''line field''' on a manifold is a formation of a line being tangent to a manifold at each point, i.e. a section of the line bundle over the manifold. Line fields are of particular interest in the study of complex dynamical systems, where it is conventional to modify the definition slightly.
==Definitions==
In general, let ''M'' be a manifold. A '''line field''' on ''M'' is a function ''μ'' that assigns to each point ''p'' of ''M'' a line ''μ''(''p'') through the origin in the tangent space T<sub>''p''</sub>(''M''). Equivalently, one may say that ''μ''(''p'') is an element of the projective tangent space PT<sub>''p''</sub>(''M''), or that ''μ'' is a section of the projective tangent bundle PT(''M'').
In the study of complex dynamical systems, the manifold ''M'' is taken to be a Hersee surface. A '''line field''' on a subset ''A'' of ''M'' (where ''A'' is required to have positive two-dimensional Lebesgue measure) is a line field on ''A'' in the general sense above that is defined almost everywhere in ''A'' and is also a measurable function.<ref>{{Cite journal |last=Markus |first=L. |date=1955 |title=Line Element Fields and Lorentz Structures on Differentiable Manifolds |url=https://www.jstor.org/stable/1970071 |journal=Annals of Mathematics |volume=62 |issue=3 |pages=411–417 |doi=10.2307/1970071 |jstor=1970071 |issn=0003-486X|url-access=subscription }}</ref>
Category:Dynamical systems Category:Fiber bundles
== References == {{Reflist}}{{mathanalysis-stub}}