Poincaré theorem

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