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