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

Evaluates the adjoint $J^*$ of the Jacobi field along the geodesic $\geo{x}{y}$ at $t$ with initial conditions $\eta$.

For the geodesic $\gamma_{x,y}(t)$ from $x\in\M$ to $y\in\M$ a Jacobi field $J\in\mathcal X(\geo{x}{y})$ is a smooth vector field along $\geo{x}{y}$ that can be evaluated with the function JacobiField(x,y,t,eta), where $\eta$ denotes one of the two initial conditions, i.e. $\eta\in T_x\M$.

We interpret in the following the Jacobi field $J_{x,y,t}(\eta)$ for fixed $x,y,t$ as a function of $\eta$ and obtain $J_{x,y,t}\colon T_x\M \to T_{\geoL{t}{x,y}}\M$. We introduce the short notation $z=\geoL{t}{x,y}$ The adjoint is then meant in the sense that for any $\xi\in T_{z}\M$ it holds

and hence $J^*_{x,y,t}\colon T_z\M \to T_x\M$. The adjoint can be computed employing the same weights as for the Jacobi field. If the weights of the Jacobi field represent a differential, this yields the adjoint differential.

### Matlab Documentation

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%    along the geodesic geo(x,y) at point t, where the
%    'weight'-function f(k,t,d) determines the boundary
%    conditions of the field and hence the meaning of eta,
%
% INPUT
%    x   : a point on the manifold Sn (or a vector of points)
%    y   : a point on the manifold Sn (or a vector of points)
%    t   : a value from [0,1] indicating the point geo(x,y,t)
%           (or a vector of values)
%    eta : an initial condition of the Jacobi field, where the
%           following weights determine the type of initial
%           condition (given at g(t; x,y).
%
% OUTPUT
%    xi : tangent vectors in TxM.
%
% OPTIONAL
%    'weights' : [@(k,t,d) = (k==0)*t
%       + (k>0)*sin(sqrt(k)*t*d)/sin(sqrt(k)*d)
%       + (k<0)*sinh(sqrt(-k)*d*t)/sinh(sqrt(-k)*d)
%       provides the weight depending on the eigenvalue (k) of
%       the curvature tensor coresponding to the ONB basis
%       vector, the position t along the Jacobi field and d the
%       length of the geodesic.
%       For the standard value, eta is the Jacobi field at 0
%       and the second boundary condition is J(1)=0, i.e. the
%       Jacobifield corresponds to D_x\gamma_{xy}(t)[\eta]
%
% ---
% R. Bergmann | MVIRT | 2017-12-01