View source for Poincaré theorem

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

<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>

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).



====Comments====
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]]].

====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>