Equiaffine immersions and pseudo-Riemannian space forms

Abstract

We introduce an explicit construction that produces immersions into the pseudosphere $\mathbb{S}^{n,n+1}$ and the pseudohyperbolic space $\mathbb{H}^{n+1,n}$ starting from equiaffine immersions in $\mathbb{R}^{n+1}$, and conversely. We describe how these immersions interact with a para-Sasaki metric defined on $\mathbb{H}^{n+1,n}$ via a principal $\mathbb{R}$-bundle structure over a para-K\"ahler manifold $\mathbb H_\tau^n$, called the para-complex hyperbolic space. In the case where the immersion in $\mathbb{R}^{n+1}$ is an $n$-dimensional hyperbolic affine sphere, we obtain spacelike maximal immersions in $\mathbb{H}^{n+1,n}$ that satisfy a transversality condition with respect to the principal $\mathbb{R}$-bundle structure. As a first application, given a strictly convex subset $\Omega \subset \mathbb{RP}^n$, we define a boundary set $\overline{\Lambda}_\Omega$ in the partial flag variety of lines and hyperplanes in $\mathbb{R}^{n+1}$, and prove the existence and uniqueness of a spacelike, Lagrangian, maximal $n$-submanifold in $\mathbb H^n_\tau$ with boundary $\overline{\Lambda}_\Omega$. We also discuss its implications in the case of $\mathbb H^{n+1,n}$. As a second application, we show that the Blaschke lift of the hyperbolic affine sphere, introduced by Labourie for $n=2$, into the symmetric space of $\mathrm{SL}(n+1,\mathbb{R})$ is a harmonic map.