Definition

A manifold of constant negative sectional curvature $-1/\rho^2$ (curvature $-1$ for the standard normalisation). Two standard models, isometric to each other:

• Poincaré disk: the open unit disk $\mathbb{D}^2 = \{z \in \mathbb{R}^2 : \|z\| < 1\}$ with metric $ds^2 = \dfrac{4(dx^2+dy^2)}{(1-\|z\|^2)^2}$.
• Upper half-plane: $\mathbb{H} = \{(x,y) : y>0\}$ with metric $ds^2 = (dx^2+dy^2)/y^2$.

Geodesics in the Poincaré disk are arcs of circles meeting the boundary $\partial\mathbb{D}^2$ orthogonally (or diameters, as a degenerate case).

Intuition

In flat space, a tree with branching factor $> 1$ “sprawls”: the number of nodes at depth $r$ grows polynomially with $r$ (or the tree simply doesn’t embed at low distortion). In hyperbolic space, volume itself grows exponentially with radius, so exponentially many nodes can be placed at (approximately) equal distance from a root — hyperbolic space has “room” for a tree that flat space doesn’t.

Properties

• Exponential ball growth. In $n$-dimensional Euclidean space, the volume of a ball of radius $r$ grows as $\Theta(r^n)$ (polynomial). In $n$-dimensional hyperbolic space, it grows as $\Theta(e^{(n-1)r})$ (exponential in $r$) — a qualitatively different growth rate, not just a constant-factor difference.
• Negative Gaussian curvature everywhere, constant at $-1/\rho^2$. By Gauss–Bonnet, a hyperbolic triangle’s angle sum is less than $\pi$, by an amount proportional to its area: $\alpha+\beta+\gamma = \pi - \frac{A}{\rho^2}$.
• Conformal but not isometric to the Euclidean disk. The Poincaré metric $ds^2_{\text{hyp}} = \lambda(z)^2\, ds^2_{\text{Euc}}$ with conformal factor $\lambda(z) = \frac{2}{1-\|z\|^2} \to \infty$ as $\|z\|\to 1$ — angles are preserved (conformal), but lengths blow up near the boundary, so the boundary circle $\partial\mathbb{D}^2$ sits at infinite hyperbolic distance from the center despite being at finite Euclidean distance.
• Isometry group. Isometries of $\mathbb{D}^2$ are exactly the Möbius transformations preserving the disk, $z \mapsto e^{i\phi}\frac{z-a}{1-\bar a z}$ for $a \in \mathbb{D}^2$, $\phi \in \mathbb{R}$ — analogous to $O(2)$ rotations/reflections for the circle, but a much larger (non-compact) group.
• Trees embed with arbitrarily low distortion. A tree with bounded branching factor and $N$ nodes requires $\Omega(\log N)$ Euclidean dimensions to embed with low distortion, but embeds in the 2-dimensional Poincaré disk with distortion $\to 1$ as the scale factor increases — the exponential volume growth matches the exponential growth in the number of tree nodes at each depth.

Open question: is BERT’s syntactic subspace actually hyperbolic?

Chen, Fu, Xu, Xie, Tan, Chen & Jing (2021, “Probing BERT in Hyperbolic Spaces,” ICLR) fit a Poincaré-ball probe (Hyperbolic (Poincaré) probing) against the same dependency-tree recovery task Tree Metric Embedding targets, and find it consistently outperforms an equivalent Euclidean Structural probing (metric/distance probe) probe on BERT/ELMo (e.g. BERTBASE layer 7 UUAS 79.8%→83.7%) — but the paper is explicit that this does not establish genuine negative curvature: “we cannot conclude that the syntactic subspaces are indeed hyperbolic.” Per Exercise 6 above, probe-outperformance on a distance-recovery task is exactly the kind of evidence this page already flags as weaker than the exponential-volume-growth signature that would directly confirm hyperbolic structure — no paper in this map’s corpus has yet run that more direct test on a real pretrained LLM’s residual stream. This remains the open item this page previously flagged: a probing-based paper now exists (unlike the pre-LLM WordNet hyperbolic-embedding work, which trains embeddings from scratch rather than probing a pretrained model), but it stops short of confirming the structure itself — logged as a citation here, not as a manifolds-hyperbolic Observation (see the syntax-tree-metric-embedding Observation, filed instead under Tree Metric Embedding, for the specific, textually-supported claim this paper does support).

Key evidence: post-hoc hyperbolic projection reveals a depth ordering absent in Euclidean space, without any hyperbolic training

Madan, Pujol Vidal, Møgelmose, Escalera, Nasrollahi, Taylor & Moeslund (2026) take frozen slot embeddings from three existing object-centric slot-attention systems (SPOT, VideoSAUR, SlotContrast — none trained with any hyperbolic objective) and apply Post-hoc Lorentz projection: mapping the Euclidean slots onto the Lorentz hyperboloid at fixed curvature, with zero re-training. Coarse, scene-level slots consistently land farther from the hyperboloid origin than fine, object-level slots — a hierarchy-consistent depth ordering invisible in the native Euclidean space — and hyperbolic projection measurably reduces cross-level distance-distribution overlap relative to Euclidean (e.g. 0.62→0.37 on VideoSAUR at $c=0.2$). Notably, the direction of the depth ordering is “the opposite of what a supervised hyperbolic model would produce,” which the authors attribute to the complete absence of hyperbolic supervision — a useful contrast case to the BERT-hyperbolic-probing open question above (that probe is supervised to recover tree distance) and the HierNet caveat below (which finds no advantage at all for a supervised hyperbolic model over Euclidean on a controlled benchmark). See object-centric-slot-embeddings-show-hierarchy- consistent-depth-ordering-under-post-hoc-hyperbolic-projection.

Caveat: hyperbolic embeddings don’t beat Euclidean ones at recovering known hierarchy, in a controlled head-to-head test

Shen, Farhadi & Kusupati (2023, “Are ‘Hierarchical’ Visual Representations Hierarchical?”) build HierNet, a purpose-made 12-dataset benchmark from ImageNet’s BREEDs subset with known ground-truth class hierarchies at multiple depths, then directly compare MERU (a hyperbolic version of CLIP) against ordinary CLIP by clustering each model’s embeddings (hyperbolic distance for MERU, Euclidean for CLIP) and scoring the discovered clusters against the true hierarchy via adjusted mutual information, purity, and hierarchical optimal-transport distance. Across all 12 datasets, MERU and CLIP are “almost indistinguishable” on every metric — hyperbolic geometry provides no measurable advantage at recovering the very hierarchical structure it is theoretically supposed to encode better. (A parallel comparison of Matryoshka Representations against fixed and PCA-reduced embeddings finds a real but narrow benefit, mostly on fine-grained/high-dimension settings, not a general hierarchy-capturing advantage either.) This is a purely comparative benchmark result with no causal intervention and no new geometric-shape claim of its own — logged here as a citation and caveat on this page’s own exponential-volume-growth motivation for hyperbolic embeddings, not filed as a manifolds-hyperbolic Observation (consistent with how the BERT-hyperbolic-probing open question above is handled): a rigorous negative result about whether the theoretical promise pays off in practice is different in kind from finding a new instance of the structure, which is what this map’s Observations otherwise catalog.

Exercises

Base

1. In 2-dimensional Euclidean space, how does the circumference of a circle of radius $r$ grow with $r$? Contrast with hyperbolic space, where the circumference of a hyperbolic circle of radius $r$ is $2\pi\sinh(r)$ (for curvature $-1$).

2. A hyperbolic triangle has angle sum $\pi - 0.5$ (in a space of curvature $-1$). What is its area?

Middle

3. Show that as a point $z \in \mathbb{D}^2$ approaches the boundary along a radius (i.e. $z = t\cdot u$ for fixed unit vector $u$, $t \to 1^-$), the hyperbolic distance from the origin to $z$ diverges, using $ds^2 = \frac{4\,d\ell^2}{(1-\|z\|^2)^2}$ along the radius.

4. Explain, using the exponential-volume-growth property, why a balanced binary tree of depth $k$ (with $2^k$ leaves) can be embedded in the 2-dimensional Poincaré disk with all leaves at approximately the same radius, whereas embedding the same tree in the Euclidean plane with all leaves equidistant from the root forces leaves to crowd arbitrarily close together as $k$ grows.

Pro

5. Prove that the Poincaré disk and the upper half-plane model are isometric, using the Möbius map $w = i\dfrac{1+z}{1-z}$ sending $\mathbb{D}^2 \to \mathbb{H}$.

6. A dataset’s activations for a hierarchical concept are probed for hyperbolic structure by checking whether pairwise cosine similarity (an $S^{d-1}$-native, not hyperbolic-native, quantity) correlates with tree distance. Explain why a high correlation here is weaker evidence for genuinely hyperbolic geometry than checking the exponential-volume-growth signature directly, and propose a more direct test.