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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} $. 
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|Singular point, index of a]]) and  $  \chi $
is the [[Euler characteristic|Euler characteristic]] of  $  V $. 
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  $  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 $, 
[[#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>