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

# The adjoint differential of the exponential map

This function evaluates for $F(\xi)=\exp_x\xi$ with fixed $x\in\M$ the adjoint differenital $D^*_xF(\xi)[\eta]$

It is calculated a corresponding adjoint Jacobi field AdjJacoiField. Since the weights are the same as for the differential DxiExp, we refer to that page for details.

### Matlab Documentation

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% nu = AdjDxExp(x,xi,eta) - Adjoint of the Derivative of Exp with
%   respect to the tangential vector xi
%   INPUT
%      x   : base point of the exponential
%      xi  : direction of the exponential
%     eta  : (in TExp(x,xi)M) direction to take the Adjoint derivative at.
%
%    OUTPUT
%     nu   : ( in TxM, more precisely TTxM ) - the adjoint of DxExp with respect to eta
% ---
% MVIRT R. Bergmann, 2017-12-04