Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Line field
open-in-new
<h2 id="definitions">Definitions</h2> <p>In general, let <i>M</i> be a manifold. A line field on <i>M</i> is a <a href="/facts/Function_(mathematics)/CdReEKCh">function</a> <i>μ</i> that assigns to each point <i>p</i> of <i>M</i> a line <i>μ</i>(<i>p</i>) through the origin in the tangent space T<i>p</i>(<i>M</i>). Equivalently, one may say that <i>μ</i>(<i>p</i>) is an element of the <a href="/facts/Projective_space/7MSpA9cM">projective tangent space</a> PT<i>p</i>(<i>M</i>), or that <i>μ</i> is a section of the projective <a href="/facts/Tangent_bundle/dq8zW9Kb">tangent bundle</a> PT(<i>M</i>). </p><p>In the study of complex dynamical systems, the manifold <i>M</i> is taken to be a Hersee surface. A line field on a subset <i>A</i> of <i>M</i> (where <i>A</i> is required to have positive two-<a href="/facts/Dimension/0ktmUyac">dimensional</a> <a href="/facts/Lebesgue_measure/8us9cGoW">Lebesgue measure</a>) is a line field on <i>A</i> in the general sense above that is defined <a href="/facts/Almost_everywhere/1quRFtJE">almost everywhere</a> in <i>A</i> and is also a <a href="/facts/Measurable_function/NgIHnb1b">measurable function</a>.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> </p> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Markus, L. (1955). "Line Element Fields and Lorentz Structures on Differentiable Manifolds". Annals of Mathematics. 62 (3): 411–417. doi:10.2307/1970071. ISSN 0003-486X. JSTOR 1970071. <a href="https://www.jstor.org/stable/1970071" target="_blank">https://www.jstor.org/stable/1970071</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> </ol>