Para-complex geometry and cyclic Higgs bundles

We introduce para-complex and pseudo-Riemannian geometric methods for the study of representations of surface groups in SL (2m+1, R). For m=1 our techniques allow to recover several known results for Hitchin representations without any reference to convex projective geometry or hyperbolic affine spheres. In particular, we describe analytically the Guichard-Wienhard domain of discontinuity in the flag variety and the corresponding concave foliated flag structure of Nolte-Riestenberg. In higher rank, we obtain a one-to-one correspondence between stable cyclic SL (2m+1, R) -Higgs bundles (not necessarily in the Hitchin component) and a special class of surfaces, which we call isotropic P-alternating, in the para-complex hyperbolic space H^2m_. As a result, we give a geometric interpretation to the holomorphic differential q₂₌+₁ in the Hitchin base in terms of harmonic sequences for immersions in para-complex manifolds.