Lectures on String Theory

– 83 –
Summing up we obtain
∫
√
hR = −4π deg Ω ,
M
where deg Ω is defined as the difference of the number of zeros and the number of
poles of Ω:
deg Ω = # zeros − # poles .
It is now the Poincaré-Hopf theorem that states that deg Ω = 2g − 2, where g is the
genus of the Riemann surface. Thus, the Gauss-Bonnet theorem follows from the
Poincaré-Hopf theorem.
Finally we note that due to the identity
ɛ αβ ɛ γδ = h αγ h βδ − h αδ h βγ
the Euler characteristic can be rewritten in the form 16
χ(M) = 1
8π
∫
M
ɛ αβ ɛ γδ R αβγδ .
g
1 2
g
g + g
1
2
g=0
+
Fig. 7. If there are two surfaces of genera g 1 and g 2 then by removing from
each surface a half-sphere we can glue the resulting surfaces into a surface
of genus g 1 + g 2 and the Riemann sphere of genus zero.
To illustrate the Gauss-Bonnet theorem, we compute the topological invariant
eq.(5.2) for a sphere. We will take a model of a sphere which represent it as the
complex plane (including the point at infinity) with the metric
ds 2 =
4dzd¯z
(1 + |z| 2 ) 2 = 2eφ dzd¯z ,
16 This is a non-trivial characteristic class of the tangent bundle to M known as the Euler class.