Minimal Lagrangian Connections on Compact Surfaces


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  1. Recall that a \(1\)-form \(\alpha \in \Omega^1(M)\) is semibasic for the projection \(\pi : M \to N\) if \(\alpha\) vanishes on vector fields that are tangent to the \(\pi\)-fibres.

  2. For a matrix \(S=(S_{ij})\) we denote by \(S_{(ij)}\) its symmetric part and by \(S_{[ij]}\) its anti-symmetric part, so that \(S_{ij}=S_{(ij)}+S_{[ij]}\).

  3. We define \(\varepsilon=(\varepsilon_{ij})\) by \(\varepsilon_{ij}=-\varepsilon_{ji}\) with \(\varepsilon_{12}=1\) and \(\varepsilon^{ij}\) denote the components of the transpose inverse of \(\varepsilon\).