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

# Jacobi field

For the geodesic $\geo{x}{y}(t)$ to $y\in\M$ a Jacobi field $J\in\mathcal X(\geo{x}{y})$ is a smooth vector field along $\geo{x}{y}$ fulfilling the ODE

where $\frac{D}{\partial t}$ denotes the (partial) covariant derivative, $R\colon T\M\times T\M\times T\M \to T\M$ is the curvature tensor and $\Gamma\colon (-\varepsilon,\varepsilon)\times[0,1]\to\M$ is a geodesic variation, , i.e. for any $s\in%-\varepsilon,\varepsilon)$ the function $\Gamma(s,\cdot)$ is a geodesic; and we have $J=\tfrac{\partial\Gamma}{\partial s}\big|_{s=0}$.

This function evaluates certain special Jacobi fields given a weight function $\alpha_{k,c}(t)$, where $k\in\{1,\ldots,d\}$ refers to the index $k$ of the eigenvalue, $t\in(0,1)$ and $c=d_\M(x,y)$

By  there exists a geodesic variation for every Jacobi field. The Jacobi field itself is unique given initial conditions $J(t_0),\frac{D}{dt}J(t_0)\in T_{\gamma(t_0)}\M$ for some $t_0$.

Note that instead of initial conditions, the ODE might also be given with boundary conditions.

With an parallel transported orthonormal basis $\{\Xi_k(t)\}_{k=1}^{d}\subset T_{\gamma(t)}\M$ we can write

which on a symmetric Riemannian manifold simplifies the ODE characterizing a Jacobi field to $\alpha''(t) + G\alpha(t) = 0$, where $\alpha(t) = (\alpha_i(t))_{i=1}^d$ and $G = (\langle R(\Xi_i,\dot\gamma)\dot\gamma,\Xi_j\rangle_{\gamma})_{i,j=1}^d$.

If we choose a basis $\Xi_k(0)=\xi_k$ that diagonalizes the operator $\Xi \mapsto R(\Xi,\tfrac{\dot\gamma}{\lVert\dot\gamma\rVert_{\gamma}})\tfrac{\dot\gamma}{\lVert\dot\gamma\rVert_{\gamma}}$ then the matrix $G$ is a diagonal matrix and the system of ODEs simplifies to $d$ ODEs

where $\kappa_k$ is the eigenvalue corresponding to $\xi_k$. They can be interpreted as characterizing curvature. With this formulation they are independent of the length of the geodesic, because we use the unit speed geodesic $\tfrac{\dot\gamma}{\lVert\dot\gamma\rVert_{\gamma}}$. Using just $\dot\gamma$ introduces a factor of $d_\M^2(\gamma(0),\gamma(1))$ to the eigenvalues $\kappa_k$. We prefer to introduce the eigenvalues similar to the ONB with respect to unit vectors in the other arguments of the curvature tensor.

Finally, the parameter $\eta\in T_x\M$ specifies one of the initial conditions depending on the geodesic variation $\Gamma$ (i.e. the weights $\alpha$ herein) involved; see the corresponding differentials for details.

Note that for fixed $x,y\in\M$ and

### Matlab Documentation

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% xi = JacobiField(x,y,t,eta) - evaluate a Jacobi field
%    along the geodesic geo(x,y) at point t, where the
%    'weight'-function f(k,t,d) determines the boundary
%    conditions of the field and hence the meaning of eta,
%
% INPUT
%    x   : a point on the manifold Sn (or a vector of points)
%    y   : a point on the manifold Sn (or a vector of points)
%    t   : a value from [0,1] indicating the point geo(x,y,t)
%           (or a vector of values)
%    eta : an initial condition of the Jacobi field, where the
%           following weights determine the type of initial
%           condition.
%
% OPTIONAL
%    'weights' : [@(k,t,d) = (k==0)*t
%       + (k>0)*sin(sqrt(k)*t*d)/sin(sqrt(k)*d)
%       + (k<0)*sinh(sqrt(-k)*d*t)/sinh(sqrt(-k)*d)
%       provides the weight depending on the eigenvalue (k) of
%       the curvature tensor coresponding to the ONB basis
%       vector, the position t along the Jacobi field and d the
%       length of the geodesic.
%       For the standard value, eta is a tangential vector at 0
%       and the second boundary condition is J(1)=0, i.e. the
%       Jacobifield corresponds to D_x\gamma_{xy}(t)[\eta]
%
% ---
% Manifold-valued Image Restoration Toolbox 1.3
% R. Bergmann | MVIRT | 2017-12-01