Flat affine Lagrangian surfaces in \( \mathbf C^2\) (Q1023417)

An immersion \(f: M^n\rightarrow \mathbb C^n\) is called affine Lagrangian if \(Jf_*(TM)\) is transversal (not necessary orthogonal) to \(f_*(TM)\). It is called flat if its induced connection is flat. In this paper the case \(n=2\) is considered, namely a local characterization of flat affine Lagrangian surfaces in \(\mathbb C^{2}\) is given. The author distinguishes some geometric properties of such surfaces and studies subclasses depending on the properties, getting explicit parametrizations for some subclasses. She also studies the question whether to a given flat nowhere minimal affine Lagrangian surface in \(\mathbb C^2\) it is possible to adopt a Kähler structure (definite or indefinite) on \(\mathbb C^{2}\) relative to which the surface is metric Lagrangian. A surface for which there exist such a Kähler structure on \(\mathbb C^{2}\) is called metrizable. The author shows that the metrizability of flat affine Lagrangian surfaces is rather exceptional.