Equiaffine immersions and pseudo-Riemannian space forms

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. 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, we show that, given a certain boundary set $\Lambda_\Omega \subset \partial_\infty \mathbb{H}^{n+1,n}$, associated with a properly convex subset $\Omega \subset \mathbb{RP}^n$ and homeomorphic to an $(n-1)$-sphere, there exists an $n$-dimensional maximal spacelike submanifold in $\mathbb{H}^{n+1,n}$ whose boundary is precisely $\Lambda_\Omega$. 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.