Extremal Conformal Structures on Projective Surfaces

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  1. More generally, representation into a real split simple Lie group.

  2. Indices in round brackets are symmetrised over and indices in square brackets are anti-symmetrised over, for instance, we write \(S_{(ij)}=\frac{1}{2}\left(S_{ij}+S_{ji}\right)\) and \(S_{[ij]}=\frac{1}{2}\left(S_{ij}-S_{ji}\right)\) so that \(S_{ij}=S_{(ij)}+S_{[ij]}\).

  3. Recall that a differential form \(\alpha\) is said to be semibasic for the projection \(P\to \mathrm{C}(\Sigma)\) if the interior product \(X\hspace{3pt}\rule[0.5pt]{2mm}{0.5pt}\rule[0.5pt]{0.5pt}{4.5pt}\hspace{3pt}\alpha\) vanishes for every vector field \(X\) tangent to the fibres of \(P\to \mathrm{C}(\Sigma)\).