[Note: Some diagrams below cannot be displayed correctly in the web version. Please use the PDF version for correct display.]
We prove a Gauss-Bonnet theorem for Seifert bundles over closed surfaces. We define an Euler characteristic \(\chi(\xi)=b+\sum \beta_ i/\alpha_ i\) associated to an oriented bundle \(\xi\) with Seifert invariants \((0,c;g/b;\alpha_ 1,\beta_ 1;...;\alpha_ N,\beta_ N)\), and a connection form \(\theta\) over the Seifert bundle \(\xi: M\to^{p}B\) where \(p*\alpha_{\theta}=d\theta =\Omega\), where \(\alpha_{\theta}\) is a two form over the base, we prove that \(\int_{B}\alpha_{\theta}=2\pi \cdot \chi (\xi).\)
The proof for an orientation base B and an orientable bundle \(\xi\) consists, after choosing \(\alpha_{\theta}\in H^ 2(B;{\mathbb{R}})\) as above and because of the exactness of the diagram \[ \begin{tikzcd}H^1(B;\mathbb{R}) \ar[r, "p^\ast"]\ar[d,"d" '] & H^1(M;\mathbb{R})\ar[d,"d"]\\ H^2(B;\mathbb{R})\ar[r,"p^\ast" '] & H^2(B;\mathbb{R} \end{tikzcd} \] in decomposing the base in \(N+2\) pieces \(p(T_ i)\), \(\bar D,\) \(p(\bar V_ 0)\), are composed by N disjoint small discs each of them containing the image of a singular fibre, \(p(T_ i)\), \(p(\bar V_ 0)\) is the difference between a disc containing \(\cup \dot p(T_ i)\) and \(\cup p(T_ i)\) and \(\bar D\) is the rest of B, we have by the definition of the Seifert invariants \[ \int_{B}\alpha_{\theta}=\int_{p(\bar V_ 0)}\alpha_{\theta}+\int_{\bar D}\alpha_{\theta}+\sum^{N}_{i=1}\int_{p(T_ i)}\alpha_{\theta}= \]
\[ =\int_{p(\bar V_ 0)}\alpha_{\theta}+\int_{-M_ 0}\theta +2\pi b+\sum^{N}_{i=1}(\int_{Q_ i\quad}\theta +2\pi \beta_ i/\alpha_ i)= \]
\[ =\int_{+M_ 0-Q_ i-...-Q_ M}\theta +\int_{-M_ 0}\theta +2\pi b+\sum^{N}_{i=1}\int_{Q_ i\quad}\theta +2\pi \sum \beta_ i/\alpha_ i \] where \(\bar V_ 0\) is a trivialization in M with section \(\partial s\approx Q_ i\cup...\cup Q_ M\cup M_ 0\), \(Q_ i\in T_ i\), \(M_ 0\in p^{-1}(\bar D)\).