2 notation; added 68 characters in body

This is a response to Will Sawin's comment "What algebraic structure would you place on $\mathbb{H}^2$?"

On $\mathbb{R}^n$ there is a vector space structure where

1) The metric is given by $d(\mathbf{u},\mathbf{v})=\|\mathbf{u}-\mathbf{v}\|$

and

2) given three points $U,V,W$ the angle $U\hat WV$ satisfies $\cos \theta = \frac{(-W+U)\cdot(-W+V)}{\|-W+U\|\|-W+V\|}$

Analogously, for some models of $\mathbb{H}^n$ with points identified with a subset of $\mathbb{R}^n$ there is a vector-like structure but with a noncommutative, nonassociative binary operation $\oplus$ where

1) The metric is given by $d(\mathbf{u},\mathbf{v})=\|\mathbf{u} \ominus \mathbf{v} \|$

and

2) given three points $U,V,W$ the angle $U\hat WV$ satisfies $\cos \theta = \frac{(-W\oplus U)\cdot(-W\oplus V)}{\|-W\oplus U\|\|-W\oplus frac{(\ominus W\oplus U)\cdot(\ominus W\oplus V)}{\|\ominus W\oplus U\|\|\ominus W\oplus V\|}$

$\| \|$ and $\cdot$ are the vector norm and dot product inherited from $\mathbb{R}^n$.

$\ominus a$ denotes the left inverse of a.

$a\ominus b$ denotes $a\oplus (\ominus b)$.

Note the use of the trig function "cos" even though this hyperbolic geometry.

For the Beltrami-Klein model the binary operation $\oplus$ is given by the formula for relativistic velocity-addition:.

1

This is a response to Will Sawin's comment "What algebraic structure would you place on $\mathbb{H}^2$?"

On $\mathbb{R}^n$ there is a vector space structure where

1) The metric is given by $d(\mathbf{u},\mathbf{v})=\|\mathbf{u}-\mathbf{v}\|$

and

2) given three points $U,V,W$ the angle $U\hat WV$ satisfies $\cos \theta = \frac{(-W+U)\cdot(-W+V)}{\|-W+U\|\|-W+V\|}$

Analogously, for some models of $\mathbb{H}^n$ with points identified with a subset of $\mathbb{R}^n$ there is a vector-like structure where

1) The metric is given by $d(\mathbf{u},\mathbf{v})=\|\mathbf{u} \ominus \mathbf{v} \|$

and

2) given three points $U,V,W$ the angle $U\hat WV$ satisfies $\cos \theta = \frac{(-W\oplus U)\cdot(-W\oplus V)}{\|-W\oplus U\|\|-W\oplus V\|}$

$\| \|$ and $\cdot$ are the vector norm and dot product inherited from $\mathbb{R}^n$.

Note the use of the trig function "cos" even though this hyperbolic geometry.

For the Beltrami-Klein model the binary operation $\oplus$ is given by the formula for relativistic velocity-addition: