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-Kähler 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 ${\mathrm{\Lambda }}_{\mathrm{\Omega }}\subset {\mathrm{\partial }}_{\mathrm{\infty }}{\mathbb{H}}^{n+1,n}$, associated with a properly convex subset $\mathrm{\Omega }\subset {\mathbb{R}\mathbb{P}}^n$ 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 ${\mathrm{\Lambda }}_{\mathrm{\Omega }}$. 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 $\mathrm{S}\mathrm{L}(n+1,\mathbb{R})$ is a harmonic map.