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Difference between revisions of "Poincaré theorem"

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Let a vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073130/p0731301.png" /> be defined on a smooth closed two-dimensional Riemannian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073130/p0731302.png" /> (cf. [[Vector field on a manifold|Vector field on a manifold]]) and let it have a finite number of isolated singular points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073130/p0731303.png" />. Then
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073130/p0731304.png" /></td> </tr></table>
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here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073130/p0731305.png" /> is the index of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073130/p0731306.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073130/p0731307.png" /> (see [[Singular point, index of a|Singular point, index of a]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073130/p0731308.png" /> is the [[Euler characteristic|Euler characteristic]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073130/p0731309.png" />. This was established by H. Poincaré (1881).
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Let a vector field  $  X $
 +
be defined on a smooth closed two-dimensional Riemannian manifold  $  V $(
 +
cf. [[Vector field on a manifold|Vector field on a manifold]]) and let it have a finite number of isolated singular points  $  A _ {1} \dots A _ {k} $.  
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Then
  
 +
$$
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\sum  j ( X , A _ {i} )  =  \chi ( V) ;
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$$
  
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here  $  j ( X , A _ {i} ) $
 +
is the index of the point  $  A _ {i} $
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with respect to  $  X $(
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see [[Singular point, index of a|Singular point, index of a]]) and  $  \chi $
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is the [[Euler characteristic|Euler characteristic]] of  $  V $.
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This was established by H. Poincaré (1881).
  
 
====Comments====
 
====Comments====
 
This theorem has since been established for manifolds of all dimensions, [[#References|[a1]]].
 
This theorem has since been established for manifolds of all dimensions, [[#References|[a1]]].
  
An immediate consequence is that on a sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073130/p07313010.png" /> of even dimension there is no continuous vector field without a zero (singularity), the Poincaré–Brouwer theorem, also called the hairy ball theorem. This was established for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073130/p07313011.png" /> by Poincaré and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073130/p07313012.png" /> by L.E.J. Brouwer. On the other hand, for the odd-dimensional spheres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073130/p07313013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073130/p07313014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073130/p07313015.png" />, gives a continuous vector field with no zeros on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073130/p07313016.png" />. More generally one has that there exists a vector field without zero on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073130/p07313017.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073130/p07313018.png" />, [[#References|[a1]]].
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An immediate consequence is that on a sphere $  S  ^ {n} $
 +
of even dimension there is no continuous vector field without a zero (singularity), the Poincaré–Brouwer theorem, also called the hairy ball theorem. This was established for $  n = 2 $
 +
by Poincaré and for $  n> 2 $
 +
by L.E.J. Brouwer. On the other hand, for the odd-dimensional spheres $  v _ {2j-} 1 = - x _ {2j }  $,  
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$  v _ {2j} = x _ {2j-} 1 $,  
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$  j = 1 \dots ( n+ 1)/2 $,  
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gives a continuous vector field with no zeros on $  \{ {( x _ {1,} \dots , x _ {n+} 1 ) } : {\sum x _ {i}  ^ {2} = 1 } \} = S  ^ {n} $.  
 +
More generally one has that there exists a vector field without zero on a manifold $  M $
 +
if and only if $  \chi ( M) = 0 $,  
 +
[[#References|[a1]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.S. [P.S. Aleksandrov] Alexandroff,  H. Hopf,  "Topologie" , Chelsea, reprint  (1972)  pp. Chapt. XIV, Sect. 4.3</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.W. Hirsch,  "Differential topology" , Springer  (1976)  pp. Chapt. 6</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.S. [P.S. Aleksandrov] Alexandroff,  H. Hopf,  "Topologie" , Chelsea, reprint  (1972)  pp. Chapt. XIV, Sect. 4.3</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M.W. Hirsch,  "Differential topology" , Springer  (1976)  pp. Chapt. 6</TD></TR></table>

Revision as of 08:06, 6 June 2020


Let a vector field $ X $ be defined on a smooth closed two-dimensional Riemannian manifold $ V $( cf. Vector field on a manifold) and let it have a finite number of isolated singular points $ A _ {1} \dots A _ {k} $. Then

$$ \sum j ( X , A _ {i} ) = \chi ( V) ; $$

here $ j ( X , A _ {i} ) $ is the index of the point $ A _ {i} $ with respect to $ X $( see Singular point, index of a) and $ \chi $ is the Euler characteristic of $ V $. This was established by H. Poincaré (1881).

Comments

This theorem has since been established for manifolds of all dimensions, [a1].

An immediate consequence is that on a sphere $ S ^ {n} $ of even dimension there is no continuous vector field without a zero (singularity), the Poincaré–Brouwer theorem, also called the hairy ball theorem. This was established for $ n = 2 $ by Poincaré and for $ n> 2 $ by L.E.J. Brouwer. On the other hand, for the odd-dimensional spheres $ v _ {2j-} 1 = - x _ {2j } $, $ v _ {2j} = x _ {2j-} 1 $, $ j = 1 \dots ( n+ 1)/2 $, gives a continuous vector field with no zeros on $ \{ {( x _ {1,} \dots , x _ {n+} 1 ) } : {\sum x _ {i} ^ {2} = 1 } \} = S ^ {n} $. More generally one has that there exists a vector field without zero on a manifold $ M $ if and only if $ \chi ( M) = 0 $, [a1].

References

[a1] P.S. [P.S. Aleksandrov] Alexandroff, H. Hopf, "Topologie" , Chelsea, reprint (1972) pp. Chapt. XIV, Sect. 4.3
[a2] M.W. Hirsch, "Differential topology" , Springer (1976) pp. Chapt. 6
How to Cite This Entry:
Poincaré theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poincar%C3%A9_theorem&oldid=23496
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article