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

The differential of the start point of a geodesic

This function evaluates for $F(x)=\geo{x}{y}(t)$ with fixed $y\in\M$ and $t\in\mathbb R$ the differenital $D_xF(x)[\eta].$

It is calculated a corresponding Jacobi field JacoiField. 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 $D_xF(x)[\xi_k] = \tfrac{D}{\partial s}\Gamma(s,t)\big|_{s=0}$ which is also the Jacobi field and $J_k(t) = \tfrac{D}{\partial s}\Gamma(s,t)\big|_{s=0}$ with boundary conditions $J_k(0) = \xi_k$ and $J(1) = 0$.

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

where $d_\geoS = D_\M(x,y)$ 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
%    hence with the Adjoint we obtain
%    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