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

# The parallel transport the sphere

This function parallel transports a vector $\xi\in T_x\mathbb S^n$ along a geodesic $\geo{x}{y}$ (where uniqueness is determined) by the logarithmic map implementation) by

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

This formula is taken from [1,2] and can be interpreted as follows: all components of $\xi$ that share no part with the direction $\log_xy$ are left unchanged. For the remaining part (second term) we have to perform a correction.

### Matlab Documentation

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% eta = parallelTransport(x,y,xi) transport xi from TxM parallel to TyM
%
% INPUT
%   x,y : two (sets of) points on the manifold
%   xi  : a (set of) vectors from TxM
%
% OUTPUT
%   eta : the parallel transported vectors in TyM
% ---
% Manifold-valued Image Restoration Toolbox 1.2
% R. Bergmann | 2018-03-01