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

# The adjoint differential of the end point of a geodesic

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

It is calculated a corresponding adjoint Jacobi field AdjJacoiField. Since we can rewrite the problem as computing the differential of the reverted geodesic $\geo{y}{x}(1-t)$, this differential is computed using the adjoint differential of the start point of a geodesic.

### Matlab Documentation

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%
%    For a function f: M \mapsto R and fixed x,t we have for the
%    gradient of g(y) = f(geo(x,y,t)) that
%    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, geo(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(x,y,t)) direction to take the Adjoint derivative at.
%
%    OUTPUT
%     xi   : ( in TyM ) - the adjoint of DyGeo with respect to eta
% ---
% MVIRT R. Bergmann, 2017-12-04