$\DeclareMathOperator{\arccosh}{arccosh} \DeclareMathOperator*{\argmin}{arg\,min} \DeclareMathOperator{\Exp}{Exp} \newcommand{\geo}[2]{\gamma_{\overset{\frown}{#1,#2}}} \newcommand{\geoS}{\gamma} \newcommand{\geoD}[2]{\gamma_} \newcommand{\geoL}[2]{\gamma(#2; #1)} \newcommand{\gradM}{\nabla_{\M}} \newcommand{\gradMComp}[1]{\nabla_{\M,#1}} \newcommand{\Grid}{\mathcal G} \DeclareMathOperator{\Log}{Log} \newcommand{\M}{\mathcal M} \newcommand{\N}{\mathcal N} \newcommand{\mat}[1]{\mathbf{#1}} \DeclareMathOperator{\prox}{prox} \newcommand{\PT}[3]{\mathrm{PT}_{#1\to#2}#3} \newcommand{\R}{\mathbb R} \newcommand{\SPD}[1]{\mathcal{P}(#1)} \DeclareMathOperator{\Tr}{Tr} \newcommand{\tT}{\mathrm{T}} \newcommand{\vect}[1]{\mathbf{#1}}$

# The Hyperbolic Space $\mathbb H^n$

The Riemannian manifold called the hyperbolic space $\mathbb H^n$ can be isometrically embedded into $\mathbb R^{n+1}$ using the Minkowski inner product. The Minkowski inner product is given by

The hyperbolic space is give by the set

together with the Minkowski inner product as Riemannian metric.

This yields a manifold of constant curvature $-1$.

There are several different representations of this space that are useful for vizualisation. However, one can compute in the embedding most easily, since the formulae are quite similar to those of the second model space (of constant curvature $+1$) the sphere $\mathbb S^n$

### Functions

The orthonormal basis in a tangent space
1
% Xi = TpMONB(x,y)


This function computes an ONB where the $\xi_1=\log_x y$ belongs to the ONB and hence it diagonalizes the curvature tensor along $\geo{x}{y}$, i.e. the eigenvalues $-1$ for all $\xi_k$, $k>1$ and zero for the first value. Note that the basis is orthonormal with respect to the inner product in the tangent space, i.e. the Minkowski metric.

The distance on the hyperpolic space
1
% d = dist(x,y) distance between points x and y on Hn


The distance on the sphere $\mathbb H^n$ is defined for any $x,y\in\mathbb S^n$ by

where $\langle\cdot,\cdot\rangle_M$ denotes the Minkowski metric.

The inner product on the hyperbolic space
1
% ds = dot(x,xi,nu) inner product of two tangent vectors in TxHn


Since the manifold $\mathbb H^n$ is isometrically embedded into $\mathbb R^{n+1}$ with the Minkowski metric, the inner product of twio tangent vectors $\xi,\nu\in T_x\mathbb H^n$ at $x\in\mathbb H^n$ is given by

The exponential map on the hyperbolic space
1
% exp(x,xi) exponential map at the point(s) x towards xi in T_xHn


The exponential map on the hyperbolic space $\mathbb H^n$ is given for $x\in\mathbb H^n$ and $\xi\in T_x{\mathbb H^n}$ by the formula

where $\langle\cdot,\cdot\rangle_M$ denotes the Minkowski metric.

The logarithmic map on the hyperbolic space
1
% xi = log(x,y) logarithmic map the point(s) x from y


The logarithmic map on the sphere $\mathbb H^n$ is defined for any $x,y\in\mathbb S^n$ by

where $\langle\cdot,\cdot\rangle_M$ denotes the Minkowski metric.

The parallel transport the hyperbolic space
1
% eta = parallelTransport(x,y,xi) parallel transport xi along g(.,x,y)


This function parallel transports a vector $\xi\in T_x\mathbb H^n$ along a geodesic $\geo{x}{y}$ by

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