Pseudo-Hyperbolic Manifolds Overview

• Pseudo-hyperbolic manifolds are defined as pseudo-Riemannian spaces with constant negative curvature that generalize classical hyperbolic geometry to include indefinite signatures.
• Their models—such as the quadric, half-space, and affine-sphere constructions—offer explicit geometric realizations, facilitating studies of maximal submanifolds and angle structures.
• Group actions and boundary compactifications in these spaces bridge differential geometry, representation theory, and higher Teichmüller theory, leading to novel insights in moduli problems.

A pseudo-hyperbolic manifold is a pseudo-Riemannian manifold of constant negative sectional curvature, generalizing classical hyperbolic geometry to indefinite signature. Such spaces—commonly denoted $H^{p,q}$—admit rich geometric, topological, and analytic structures spanning projective, affine, and representation-theoretic frameworks. The theory encompasses ambient model geometries, the half-space realization, submanifold theory, boundary and compactification, angle structures in dimension 3, and links to higher Teichmüller theory and complex hyperbolicity.

1. Ambient Models and Geometric Structure

The pseudo-hyperbolic space $H^{p,q}$ is defined as the quadric

$$\widetilde{H}^{p,q} = \left\{ X \in \mathbb{R}^{p,q+1} : \langle X, X \rangle = -1 \right\}$$

in $\mathbb{R}^{p+q+1}$, with $\langle \cdot, \cdot \rangle$ the symmetric bilinear form of signature $(p,q+1)$ (Seppi et al., 2021). The metric induced from this ambient space on $H^{p,q}$ has constant sectional curvature $-1$ in every nondegenerate 2-plane (Seppi et al., 2023). $H^{p,q}$ is naturally interpreted also in projective space: $$H^{p,q} = \left\{ [X] \in \mathbb{RP}^{p+q} : \langle X, X \rangle < 0 \right\}$$ with its boundary at infinity $\partial_\infty H^{p,q}$ identified as the projectivized null cone $\{ [X] : \langle X, X \rangle = 0 \}$ (Seppi et al., 2021). The isometry group is $\mathsf{PO}(p, q + 1)$, acting by projective linear transformations (Seppi et al., 2023).

2. The Half-Space Model: Realization and Isometry Group

The half-space model for $H^{p,q}$ is

$$\mathcal{H}^{p,q} = \{ (x, y, z) \in \mathbb{R}^{p-1} \times \mathbb{R}^q \times (0, \infty) \}$$

with pseudo-Riemannian metric

$$g = \frac{1}{z^2} \left( dx_1^2 + \cdots + dx_{p-1}^2 - dy_1^2 - \cdots - dy_q^2 + dz^2 \right)$$

of signature $(p,q)$ (Seppi et al., 2021). This model embeds isometrically into the quadric model, omitting a degenerate totally geodesic hyperplane. The boundary at infinity matches the space of degenerate hyperplanes, and geodesics can be classified by causality: spacelike, timelike, and lightlike, each described explicitly in terms of the half-space coordinates.

The isometry group acts via dilations, orthogonal transformations in $O(p-1, q)$, horizontal translations, and inversion: $$G = \{ (x, y, z) \mapsto \lambda (A(x, y) + (x_0, y_0), z) : \lambda > 0,\, A \in O(p-1, q),\, (x_0, y_0) \in \mathbb{R}^{p-1} \times \mathbb{R}^q \}$$ together with the inversion $\mathcal{J}$ defined by rational functions in $(x, y, z)$ (Seppi et al., 2021). The full isometry group thus admits an Iwasawa-type decomposition.

3. Maximal Spacelike Submanifolds: Classification and Boundary

A spacelike immersion $f : M^p \to H^{p,q}$ yields a Riemannian metric on $M$; $f$ is maximal if its mean curvature vanishes. The fundamental theorem in (Seppi et al., 2023) establishes a bijection: $$\partial_\infty : \{ \text{complete maximal spacelike } p\text{-submanifolds } M \subset H^{p,q} \} \longleftrightarrow \{ \text{non-negative spheres } S^{p-1} \subset \partial_\infty H^{p,q} \}$$ where a non-negative sphere is an embedded $S^{p-1} \subset \partial_\infty H^{p,q}$ satisfying combinatorial positivity conditions on triples of points. For such spheres, there is a unique complete maximal spacelike $p$-submanifold with prescribed boundary at infinity. The proof relies on compactness, uniqueness via maximum principles and Hessian analysis, and existence via implicit function theorem and Fredholm theory for the Jacobi operator.

Key equations include the Gauss, Codazzi, and Jacobi identities, with the induced curvature and normal bundle structure determined by the ambient curvature and the second fundamental form. These maximal submanifolds are central to the study of domains of discontinuity for Anosov representations and higher Teichmüller spaces (Seppi et al., 2023).

4. Angle Structures on Pseudo 3-Manifolds

In dimension 3, pseudo-hyperbolic manifolds intersect the theory of triangulated pseudo 3-manifolds $(M, \mathcal{T})$ (Ge et al., 17 Feb 2025). A pseudo 3-manifold is the quotient $M = X / \Phi$ of disjoint simplices $X$ under affine face-pairing maps $\Phi$. An ideal triangulation occurs when every vertex link is a closed surface of genus $\geq 1$.

Normal surfaces are represented by vectors $$(x_1, \dots, x_{3n}; y_1, \dots, y_{4n}) \in \mathbb{R}^{7n}$$ subject to compatibility equations for internal faces. Prescribed area–curvature assignments $A, \kappa$ lead to generalized angle structures: assignments of dihedral angles $\alpha(e_{ik})$ to each edge, with triangle area and edge curvature constraints. Existence is characterized by comparison inequalities between two Euler-type functionals $\chi^*$ and $\chi^{(A, \kappa)}$ on the space of normal disks.

The main existence theorem (Theorem 1, Corollary 2 in (Ge et al., 17 Feb 2025)) asserts that $(M, \mathcal{T})$ admits an angle structure with $(A, \kappa)$ iff, for every normal surface $s$ with at least one quad, $\chi^*(s) < \chi^{(A, \kappa)}(s)$. For compact hyperbolic 3-manifolds with totally geodesic boundary, Kojima's cell decomposition and subsequent subdivision yield a truncated ideal triangulation supporting a flat semi-angle structure. Perturbation produces a genuine angle structure, and as a consequence, these manifolds admit ideal triangulations carrying honest angle structures with negative triangle areas.

Topologically, angle structures with $A \leq 0$, $\kappa \leq 0$ enforce strong restrictions: the absence of essential spheres, tori, disks, or annuli, and $0$-efficient ideal triangulations.

5. Equiaffine and Affine-Sphere Immersions into Pseudo-Hyperbolic Spaces

Immersions into $H^{n+1, n}$ based on equiaffine geometry in $\mathbb{R}^{n+1}$ connect affine spheres to pseudo-hyperbolic metrics (Rungi, 10 Dec 2025). Using the neutral bilinear form $\hat{g}$ on $V = \mathbb{R}^{n+1} \oplus (\mathbb{R}^{n+1})^*$, the quadric

$H^{n+1, n} = \{ u \in V : \hat{g}(u, u) = -1 \}$

serves as the pseudo-hyperbolic space of index $(n+1, n)$. The para-Sasaki metric arises by quotienting $H^{n+1,n}$ by a natural $\mathbb{R}$-action, resulting in a para-Kähler base, with para-contact structure preserved.

Given an equiaffine immersion $f: M^n \to \mathbb{R}^{n+1}$, the map

$\sigma^-(p) = (-\xi_p, \nu_p)\in H^{n+1, n}$

produces a horizontal, $\phi$-anti-invariant immersion. If $f$ is a hyperbolic affine sphere ($S = -\mathrm{Id}$, $h$ positive-definite), $\sigma^-$ is a spacelike maximal immersion, transverse to the principal $\mathbb{R}$-bundle structure.

For a properly convex subset $\Omega \subset \mathbb{RP}^n$, the associated boundary set $\Lambda_\Omega \subset \partial_\infty H^{n+1,n}$, homeomorphic to $S^{n-1}$, is realized as the boundary of a unique $n$-dimensional maximal, spacelike submanifold via the affine sphere construction (Rungi, 10 Dec 2025), generalizing the Plateau problem in this setting.

The Blaschke lift of a hyperbolic affine sphere into the symmetric space $\mathrm{SL}(n+1, \mathbb{R})/\mathrm{SO}(n+1)$ is harmonic; further, its composition with the inclusion into the $\mathrm{SO}_0(n+1, n+1)$ symmetric space produces equivariant harmonic maps, linking affine, pseudo-hyperbolic, and representation-theoretic geometry.

6. Pseudo-Kobayashi Hyperbolicity

For quasi-projective base spaces $V$ of smooth families of minimal projective manifolds with maximal variation, pseudo-hyperbolicity is manifested as Kobayashi hyperbolicity modulo a proper subvariety $Z \subsetneq V$ (Deng, 2018). The Kobayashi pseudo-distance $d_V$ is positive off $Z$, and every nonconstant entire curve $f:\mathbb{C} \to V$ is contained within $Z$.

The analytic proof constructs Viehweg–Zuo Higgs bundles, employs curvature estimates on Finsler metrics, and applies the Ahlfors–Schwarz lemma to relate negative curvature to the Kobayashi metric. The result has direct implications for the algebraic degeneracy of entire curves and the Brody hyperbolicity of moduli spaces. The phenomenon of pseudo-hyperbolicity motivates further study of the Kobayashi exceptional locus in moduli-theoretic terms and relates to Lang’s conjecture for log-general type quasi-projective varieties.

7. Boundary at Infinity, Compactification, and Group Actions

The boundary at infinity $\partial_\infty H^{p,q}$ is pivotal for understanding compactification, limiting behavior of geodesics, and representation-theoretic quotients. In the half-space model, $\partial_\infty H^{p,q}$ is obtained by projectivizing the null cone, with explicit parametrizations in terms of boundary coordinates (Seppi et al., 2021). Topologically, it encodes spaces of degenerate hyperplanes and admits identification with Einstein universes or products, e.g., $S^n$, $S^{n-1} \times S^1$.

Group actions—including convex-cocompact Anosov representations—act on both $H^{p,q}$ and the boundary, producing domains of discontinuity and homogeneous bundles related to maximal submanifolds (Seppi et al., 2023). The interplay between the geometry of $H^{p,q}$, maximal submanifolds, and group actions underpins higher Teichmüller theory and the theory of geometric structures on manifolds.

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In summary, pseudo-hyperbolic manifolds provide a framework for extending classical hyperbolic concepts to indefinite signature, integrating differential geometric, topological, algebraic, and representation-theoretic aspects. Central constructs include the quadric and half-space models, maximal submanifold theory, angle structures for triangulated 3-manifolds, boundary analysis, equiaffine and affine-sphere constructions, and various notions of hyperbolicity and compactification, each with precise algebraic and analytic characterizations.