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

# The differential of the logarithmic map

This function evaluates for $F(x)=\log_yx$ with fixed $y\in\M$ the differenital $D_xF(x)[\eta].$

It is calculated a corresponding Jacobi field JacoiField.

Let $\xi_k$, $k=1,\ldots,d$ be an orthonormal basis in $T_x\M$ and $\Xi_k(t)\in T_{\geo{x}{y}(t)}\M$ its parallel transported frame, i.e. we have $\Xi(0) = \xi_k$.

We know by the inverse function theorem

And we have from the exponential map we know that there exists a unique Jacobi field along \geo{y}{x}(t) with

and inversion yields

Setting $t=1$ and $\nu=\log_xy$, and $J_k(1)=\xi_k$

we obtain the same form as above and we obtain for the boundary conditions $J_k(0) = 0$, $J_k(1)=\xi$. Note that this Jacobi field is aloing the geodesic $\geo{y}{x}$, i.e. the reversed geodesic, hence this differential is evaluated similarily to the geodesic differential with respect to its end point.

where $d_\geoS = D_\M(x,y)$ is the length of the geodesic. and the differential in total reads

### Matlab Documentation

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%   nu = DyLog(x,y,eta) - Adjoint of the Derivative of Log
%       with respect to y.
%   INPUT
%      x   : base point of the logarithm
%      y   : argument of the logarithm
%     eta  : (in TyM) direction to take the derivative at.
%
%    OUTPUT
%     nu   : ( in TxM ) - the adjoint of DxLog with respect to eta
% ---
% MVIRT R. Bergmann, 2017-12-04