$ \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 exponential

This function evaluates for with fixed the differenital

It is calculated a corresponding Jacobi field JacoiField. Note that the tangent space is its own tangent space, i.e. .

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

With the geodesic variation

We obtain which is also the Jacobi field and with boundary conditions and .

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

where 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

See also