Equiaffine immersions and pseudo-Riemannian space forms
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911311254781952 |
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| author | Rungi, Nicholas |
| author_facet | Rungi, Nicholas |
| contents | 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 $Λ_Ω\subset \partial_\infty \mathbb{H}^{n+1,n}$, associated with a properly convex subset $Ω\subset \mathbb{RP}^n$ 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 $Λ_Ω$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_09569 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Equiaffine immersions and pseudo-Riemannian space forms Rungi, Nicholas Differential Geometry Geometric Topology Symplectic Geometry 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 $Λ_Ω\subset \partial_\infty \mathbb{H}^{n+1,n}$, associated with a properly convex subset $Ω\subset \mathbb{RP}^n$ 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 $Λ_Ω$. 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. |
| title | Equiaffine immersions and pseudo-Riemannian space forms |
| topic | Differential Geometry Geometric Topology Symplectic Geometry |
| url | https://arxiv.org/abs/2512.09569 |