= Riemann bilinear relations for a compact surface
{c}
= Riemann bilinear relations
{c}
{synonym}
For a <compact Riemann surface> with a <symplectic basis> $a_i,b_i$ of its first <homology group>, cutting along that basis and integrating a primitive gives
$$
\int_C\omega\wedge\eta=\sum_i\left(\int_{a_i}\omega\int_{b_i}\eta-\int_{b_i}\omega\int_{a_i}\eta\right)
$$
for closed smooth one-forms. For a <holomorphic differential form> $\omega$ and a <differential of the third kind> $\eta$, the corresponding boundary calculation gives
$$
\sum_i\left(\int_{a_i}\omega\int_{b_i}\eta-\int_{b_i}\omega\int_{a_i}\eta\right)=2\pi i\sum_P\operatorname{res}_P(\eta)\int_{P_0}^P\omega.
$$
The paths and representatives of the cycles are chosen in the same cut surface. Normalized <holomorphic differential forms> consequently have a symmetric period matrix with positive definite imaginary part.
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