---
created: 2023-03-14T16:33
updated: 2024-04-13T20:57
aliases:
  - hyperbolic geometry
  - hyperbolic
  - hyperbolic plane
  - hyperbolic disc
  - Poincare model
  - hyperbolic 3-manifolds
  - hyperbolic curve
  - hyperbolic curves
  - hyperbolic surface
  - hyperbolic surfaces
  - hyperbolic structures
  - hyperbolic manifolds
---

---

- Tags: 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [lattice](lattice.md)
	- [orthogonal group](Unsorted/orthogonal%20group.md)

---

# hyperbolic geometry

As a [symmetric space](symmetric%20space.md):

![](2024-04-13-13.png)

![](2023-04-03-9.png)
![](2023-04-03-10.png)

Can realize $$\HH^n = {\Orth_{1, n} \over \Orth_1 \times \Orth_n}.$$ and as a special case $$\HH^1 = {\SL_2(\RR)\over \SO_2}$$ since $\SO^+_{1, 2} \cong \PSL_2(\RR) \implies \Orth_{1, 2}\cong \SL_2(\RR)$.


Hyperbolic metrics:

![](attachments/2023-03-14metric.png)

Orientation-preserving isometries:
$\Isom(\DD) = \PSU_{1, 1}(\CC)$ (contains [Fuchsian groups](Unsorted/Fuchsian%20group.md)) and $\Isom(\HH) = \PSL_2(\RR)$.

General isometies: $\PSU_{1, 1}(\CC)\semidirect C_2$, similar to how $\Isom(\EE^2) = \Orth_2(\RR) = \SO_2(\RR)\semidirect C_2$.


# hyperbolic curves


![](2023-04-26-3.png)

![](attachments/Pasted%20image%2020220526001811.png)

![](attachments/Pasted%20image%2020220430141323.png)


# hyperbolic surface

![](attachments/Pasted%20image%2020220502181322.png)