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

# The parallel transport the hyperbolic space

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

$P_{x\to y}(\xi) = \xi - \frac{\langle \log_xy,\xi\rangle_x}{d_{\mathbb H^n}(x,y)} \bigl( \log_xy+\log_yx \bigr)$

This formula is similar to parallel transport on the sphere, since the components orthogonal to the direction of the geodesic are also unchanged here due to constant curvature and we have only to change the component pointing „along the geodesic“ (with respect to the inner product in $T_x\mathbb H^n$).

### Matlab Documentation

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% eta = parallelTransport(x,y,xi) parallel transport xi along g(.,x,y)
%
% INPUT
%    x : a point( set) on P(m)
%    y : a point( set) on P(m)
%   xi : a tangential vector( set, one) at (each) X
%
% OUTPUT
%  eta : tangential vector( set, one) at (each) Y
%
% ---
% Manifold-Valued Image Restoration Toolbox 1.
%  R. Bergmann ~ 2018-03-05 | 2018-03-05