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

# The differential of the exponential

This function evaluates for $F(\xi)=\exp_x\xi$ with fixed $x\in\M$ the differenital $D_\xi F(\xi)[\eta], \eta\in T_x\M.$

It is calculated a corresponding Jacobi field JacoiField. Note that the tangent space is its own tangent space, i.e. $T_\xi(T_x\M) = T_x\M$.

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 $\Xi(0) = \xi_k$.

With the geodesic variation

We obtain $D_\xi F(\xi)[\xi_k] = \tfrac{D}{\partial s}\Gamma(s,1)\big|_{s=0}$ which is also the Jacobi field and $J_k(t) = \tfrac{D}{\partial s}\Gamma(s,1)\big|_{s=0}$ with boundary conditions $J_k(0) = 0$ and $J'(0) = \xi_k$.

For $\xi_k$ being an orthonormal basis diagonalizing the curvature tensor on a symmetric Riemannian manifold (see Jacobi field) we obatin slving the corresponding ODE depending on $\kappa$ the coefficients

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

### Matlab Documentation

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% nu = DxExp(x,xi,eta) - Derivative of Exp w.r.t. xi
%   INPUT
%      x   : base point of the exponential
%      xi  : direction of the exponential
%     eta  : (in Txi(TxM)=TxM direction to take the derivative at.
%
%    OUTPUT
%     nu   : ( in TExp(x,xi)M) ) - the adjoint of DxExp with respect to eta
% ---
% MVIRT R. Bergmann, 2017-12-04