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

# The adjoint differential of the start point of a geodesic

This function evaluates for $F(x)=\geo{x}{y}(t)$ with fixed $y\in\M$ and $t\in\mathbb R$ the adjoint differenital $D^*_xF(x)[\eta]$

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

### Matlab Documentation

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% AdjDxGeo(x,y,t,eta) Adjoint of the Derivative of geo(x,y,t) wrt x.
%
%    For a function f: M \mapsto R and fixed y,t we have for the
%    gradient of g(x) = f(geo(x,y,t)) that
%    <grad g, nu>_x = <grad f, DxGeo(.,y,t)(x)[nu]>_g(x,y,t)
%    hence with the Adjoint we obtain
%    This function hence only requires eta=grad f to computed
%    the chain rule.
%
%    INPUT
%      x   : start point of a geodesic, g(x,y,0)=x
%      y   : end point of a geodesic, g(x,y,1) = y
%      t   : [0,1] a point on the geodesic to be evaluated,
%            may exceed [0,1] to leave the segment between x and y
%     eta  : (in Tg(t,x,y)) direction to take the Adjoint derivative at.
%
%    OUTPUT
%     xi   : ( in TxM ) - the adjoint of DxGeo with respect to eta
% ---
% MVIRT R. Bergmann, 2017-12-04