The cross-ratio Manifold: a model of centro-affine geometry
Résumé
Every non-degenerated Lagrangian immersion in a para-Kähler manifold carries a natural Codazzi structure. For $n\geq1$, we construct a $2n$-dimensional para-Kähler manifold M such that every centro-affine hypersurface immersion $f : M \to R^{n+1}$, equipped with its affine metric, is isometric to a Lagrangian immersion $\tilde f : M \to M$. On the other hand, every Lagrangian immersion $\tilde f : M \to M$ corresponds to a unique homothetic family of centroaffine hypersurface immersions $f : M \to R^{n+1}$. The Codazzi structure defined by the affine connection and the affine metric of $f$ coincides with the Codazzi structure generated by the Lagrangian immersion $\tilde f$. The construction is compatible with the duality defined by the conormal map. The immersion $f$ is a proper affine sphere if and only if the Lagrangian immersion $\tilde f$ is minimal. The velocity field of the affine normal flow generated by f and that of the mean curvature flow generated by $\tilde f$ are related. The pseudo-Riemannian metric and the symplectic form on M are generated in the infinitesimal limit by a realvalued symmetric function ( .;. ) on M $\times$M. The manifold M is constructed as a subset of the product $\mathbb{R}P^n\times \mathbb{R}P_n$ of the real projective space and its dual, and the function ( .;. ) is defined by the projective cross-ratio.