$ \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 differential of the logarithmic map

This function evaluates for with fixed the differenital

It is calculated a corresponding Jacobi field JacoiField.

Let , be an orthonormal basis in and its parallel transported frame, i.e. we have .

We know by the inverse function theorem

And we have from the exponential map we know that there exists a unique Jacobi field along $$\geo{y}{x}(t) with

and inversion yields

Setting and , and

we obtain the same form as above and we obtain for the boundary conditions , . Note that this Jacobi field is aloing the geodesic $\geo{y}{x}$, i.e. the reversed geodesic, hence this differential is evaluated similarily to the geodesic differential with respect to its end point.

where is the length of the geodesic. and the differential in total reads

Matlab Documentation

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%   nu = DyLog(x,y,eta) - Adjoint of the Derivative of Log
%       with respect to y.
%   INPUT
%      x   : base point of the logarithm
%      y   : argument of the logarithm
%     eta  : (in TyM) direction to take the derivative at.
%
%    OUTPUT
%     nu   : ( in TxM ) - the adjoint of DxLog with respect to eta
% ---
% MVIRT R. Bergmann, 2017-12-04

See also