$ \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 start point of a geodesic

This function evaluates for with fixed and 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 .

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 solving 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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% xi = DxGeo(x,y,t,eta) - Compute the Derivative D_xGeo(t; x,y)[eta]
%    i.e. of geo(x,y,t) with respect to the start point x.
%
%    For a function f: M \mapsto R and fixed y,t we have for the
%    gradient of g(x) = f(geo(x,y,t)) that
%    <grad g, nu>_x = <grad f, DxGeo(.,y,t)(x)[nu]>_g(x,y,t)
%    hence with the Adjoint we obtain
%    grad g = AdjDxGeo(.,y,t)(x)[grad f].
%    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(t;x,y)M) direction to take the Adjoint derivative at.
%
%    OUTPUT
%     xi   : ( in TxM ) - the adjoint of DxGeo with respect to eta
% ---
% MVIRT R. Bergmann, 2017-12-04

See also