Parallelizable manifold

<h2 id="examples">Examples</h2>
<ul><li>An example with 
  
    
      
        n
        =
        1
      
    
    {\displaystyle n=1}
  
 is the <a href="/facts/Circle/9fy5ggKn">circle</a>: we can take <i>V</i>1 to be the unit tangent vector field, say pointing in the anti-clockwise direction. The <a href="/facts/Torus/J9SkaV1W">torus</a> of dimension 
  
    
      
        n
      
    
    {\displaystyle n}
  
 is also parallelizable, as can be seen by expressing it as a <a href="/facts/Cartesian_product/BfqBWzdL">cartesian product</a> of circles. For example, take 
  
    
      
        n
        =
        2
        ,
      
    
    {\displaystyle n=2,}
  
 and construct a torus from a square of <a href="/facts/Graph_paper/NKJBaxCS">graph paper</a> with opposite edges glued together, to get an idea of the two tangent directions at each point.  More generally, every <a href="/facts/Lie_group/a9jCQyjF">Lie group</a> <i>G</i> is parallelizable, since a basis for the tangent space at the <a href="/facts/Identity_element/9nss3xHY">identity element</a> can be moved around by the action of the translation group of <i>G</i> on <i>G</i> (every translation is a diffeomorphism and therefore these translations induce linear isomorphisms between tangent spaces of points in <i>G</i>).</li>
<li>A classical problem was to determine which of the <a href="/facts/Sphere/jywYLNM2">spheres</a> <i>S</i><i>n</i> are parallelizable. The zero-dimensional case <i>S</i>0 is trivially parallelizable. The case <i>S</i>1 is the circle, which is parallelizable as has already been explained. The <a href="/facts/Hairy_ball_theorem/pSX6zDBz">hairy ball theorem</a> shows that <i>S</i>2 is not parallelizable. However <i>S</i>3 is parallelizable, since it is the Lie group <a href="/facts/SU(2)/Qgy2orcu">SU(2)</a>. The only other parallelizable sphere is <i>S</i>7; this was proved in 1958, by <a href="/facts/Friedrich_Hirzebruch/5MMEInjc">Friedrich Hirzebruch</a>, <a href="/facts/Michel_Kervaire/MQdXewUT">Michel Kervaire</a>, and by <a href="/facts/Raoul_Bott/5JHkhPx8">Raoul Bott</a> and <a href="/facts/John_Milnor/LTf24paN">John Milnor</a>, in independent work. The parallelizable spheres correspond precisely to elements of unit norm in the <a href="/facts/Normed_division_algebra/CW1n5sa0">normed division algebras</a> of the real numbers, complex numbers, <a href="/facts/Quaternion/T3tnu7mX">quaternions</a>, and <a href="/facts/Octonion/nC2sIz3p">octonions</a>, which allows one to construct a parallelism for each. Proving that other spheres are not parallelizable is more difficult, and requires <a href="/facts/Algebraic_topology/2wA8yrjT">algebraic topology</a>.</li>
<li>The product of parallelizable <a href="/facts/Manifold/rXPERJtu">manifolds</a> is parallelizable.</li>
<li>Every <a href="/facts/Orientability/1SEQM2Pt">orientable</a> <a href="/facts/Closed_manifold/jMG6G6Bl">closed</a> <a href="/facts/3-manifold/48yr0Tzx">three-dimensional manifold</a> is parallelizable.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a></li></ul>
<h2 id="remarks">Remarks</h2>
<ul><li>Any parallelizable <a href="/facts/Manifold/rXPERJtu">manifold</a> is <a href="/facts/Orientability/1SEQM2Pt">orientable</a>.</li>
<li>The term <i><a href="/facts/Framed_manifold/CQq7nQsJ">framed manifold</a></i> (occasionally <i><a href="/facts/Rigged_manifold/CQq7nQsJ">rigged manifold</a></i>) is most usually applied to an embedded manifold with a given trivialisation of the <a href="/facts/Normal_bundle/E44swhHQ">normal bundle</a>, and also for an abstract (that is, non-embedded) manifold with a given stable trivialisation of the <a href="/facts/Tangent_bundle/dq8zW9Kb">tangent bundle</a>.</li>
<li>A related notion is the concept of a π-manifold.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> A smooth manifold 
  
    
      
        M
      
    
    {\displaystyle M}
  
 is called a π-manifold if, when embedded in a high dimensional euclidean space, its normal bundle is trivial. In particular, every parallelizable manifold is a π-manifold.</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Chart_(topology)/pjoya29t">Chart (topology)</a></li>
<li><a href="/facts/Differentiable_manifold/aSnbXKcV">Differentiable manifold</a></li>
<li><a href="/facts/Frame_bundle/Gugt3qyL">Frame bundle</a></li>
<li><a href="/facts/Kervaire_invariant/vQRoPxz1">Kervaire invariant</a></li>
<li><a href="/facts/Orthonormal_frame_bundle/Gugt3qyL">Orthonormal frame bundle</a></li>
<li><a href="/facts/Principal_bundle/bbB2D94B">Principal bundle</a></li>
<li><a href="/facts/Connection_(mathematics)/KURsCmIu">Connection (mathematics)</a></li>
<li><a href="/facts/G-structure/NmGVT6v0">G-structure</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/Richard_L._Bishop/pvUt9kNG">Bishop, Richard L.</a>; Goldberg, Samuel I. (1968), <a href="https://archive.org/details/tensoranalysison00bish"><i>Tensor Analysis on Manifolds</i></a> (First Dover 1980 ed.), The Macmillan Company, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-64039-6</li>
<li><a href="/facts/John_Milnor/LTf24paN">Milnor, John W.</a>; <a href="/facts/Jim_Stasheff/kceG3XPX">Stasheff, James D.</a> (1974), <i>Characteristic Classes</i>, Princeton University Press</li>
<li><a href="/facts/John_Milnor/LTf24paN">Milnor, John W.</a> (1958), <a href="https://www.maths.ed.ac.uk/~v1ranick/papers/homsph.pdf"><i>Differentiable manifolds which are homotopy spheres</i></a> (PDF), mimeographed notes</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Bishop, Richard L.; Goldberg, Samuel I. (1968), Tensor Analysis on Manifolds, New York: Macmillan, p. 160 <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Milnor, John W.; Stasheff, James D. (1974), Characteristic Classes, Annals of Mathematics Studies, vol. 76, Princeton University Press, p. 15, ISBN 0-691-08122-0 <a href="0-691-08122-0" target="_blank">0-691-08122-0</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Benedetti, Riccardo; Lisca, Paolo (2019-07-23). "Framing 3-manifolds with bare hands". L'Enseignement Mathématique. 64 (3): 395–413. arXiv:1806.04991. doi:10.4171/LEM/64-3/4-9. ISSN 0013-8584. S2CID 119711633. <a href="https://www.ems-ph.org/doi/10.4171/LEM/64-3/4-9" target="_blank">https://www.ems-ph.org/doi/10.4171/LEM/64-3/4-9</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Milnor, John W. (1958), Differentiable manifolds which are homotopy spheres (PDF) <a href="https://www.maths.ed.ac.uk/~v1ranick/papers/homsph.pdf" target="_blank">https://www.maths.ed.ac.uk/~v1ranick/papers/homsph.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
</ol>

Parallelizable manifold open-in-new

Parallelizable manifold