A characteristic class for totally real surfaces in the Grassmannian of two-planes in four-space (Q1080004)

Let \(Q_ 2=G_{4,2}({\mathbb{R}})\) be the non-degenerate quadric in \({\mathbb{P}}^ 3({\mathbb{C}})\) identified with the real Grassmann manifold of planes in \({\mathbb{R}}^ 4\). Let G denote the Grassmann bundle associated to the real tangent space \(TQ_ 2\) and let G' denote the sub-bundle of G corresponding fibre-wisely to the totally real planes in \(TQ_ 2\). If M is a totally real surface embedded into \(Q_ 2\), then its Gauss map \(t: M\to G\) factorizes through G'. Due to the topology of G', the manifold M carries by pull-back a form \(\alpha_ M\in H^ 1_{comp}(M,{\mathbb{Z}}_ 2)\) which reflects its position in G'. This cohomology class is analogous to the Maslov-Arnold index attached to Lagrangian submanifolds of \({\mathbb{R}}^{2n}.\) The author presents an invariant construction of the class \(\alpha_ M\) in terms of the twistor space. As an application of these tools, the author studies the geometry of algebraic curves in \({\mathbb{P}}^ 3({\mathbb{C}})\), via the twistor map \({\mathbb{P}}^ 3({\mathbb{C}})\to S^ 4\).