# Line field

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{{Short description|Math topic}}
{{Technical|date=April 2024}}
In [mathematics](/source/mathematics), a '''line field''' on a [manifold](/source/manifold) is a formation of a [line](/source/line_(geometry)) being tangent to a manifold at each point, i.e. a [section](/source/section_(fiber_bundle)) of the [line bundle](/source/line_bundle) over the manifold.  Line fields are of particular interest in the study of [complex dynamical systems](/source/complex_dynamics), 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](/source/Function_(mathematics)) ''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](/source/projective_space) PT<sub>''p''</sub>(''M''), or that ''μ'' is a section of the projective [tangent bundle](/source/tangent_bundle) PT(''M'').

In the study of complex dynamical systems, the manifold ''M'' is taken to be a [Hersee surface](/source/Hersee_surface).  A '''line field''' on a subset ''A'' of ''M'' (where ''A'' is required to have positive two-[dimension](/source/dimension)al [Lebesgue measure](/source/Lebesgue_measure)) is a line field on ''A'' in the general sense above that is defined [almost everywhere](/source/almost_everywhere) in ''A'' and is also a [measurable function](/source/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}}

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Adapted from the Wikipedia article [Line field](https://en.wikipedia.org/wiki/Line_field) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Line_field?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.