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

# Notation

This is a list of mathematical notations used throughout the MVIRT pages sorted alphabetically

$\lVert\cdot\rVert_x$
norm in the tangent space $T_x\M$, $x\in\M$, induced by the inner product.
$\langle\cdot,\cdot\rangle_x$
inner product in the tangent space $T_x\M$ of $x\in\M$, where the index $\cdot_x$ might be ommited if it is clear from the context.
$\frac{D}{dt}$
covariant derivative
$D_xf$
differential of the function $f$ with respect to its variable $x$, e.g. if $f\colon\M \to \M$ then $x\in\M$ and $D_xf\colon T\M \to T\M$
$D_xf(y)$
differential of the function $f$ evaluated at $y$, e.g. if $f\colon\M \to \M$ then $x,y\in\M$ and $D_xf(x)\colon T_y\M \to T_{f(y)}\M$. One might also use $y=x$, i.e. after derivation with respect to $x$ we again use $x$ as the variable of the differential
$D_xf(y)[\xi]$
directional derivative of the function $f$ with respect to $x$ evaluated at $y$ with respect to the direction $\xi\in T_y\M$, e.g. if $f\colon\M \to \M$ then $D_xf(y)[\xi] \in T_{f(y)}\M$.
$d_{\M}(\cdot,\cdot)$
distance on the manifold $\M$, i.e. for two points $x,y\in\M$ the value $d_{\M}(x,y)$ is the length of a shortest geodesic connecting $x$ and $y$.
$f,g,F,G$
functions $f\colon\M\to\mathcal N$ defined on the manifold and mapping into (a possibily different) manifold. Capital letters might be used to indicate the main objective, while for general theory or atoms of the main objetive, small letters are used.
$\gamma$, $\gamma_{x,y}$, $\gamma_{x;\xi}$
A geodesic is a function with vanishing covariant derivative $\frac{D}{dt}\gamma(t)=0$ with either fixed start point $\gamma_{x,y}(0)=x\in\M$ and end point $\gamma_{x,y}(1)=y\in\M$ or given start point $\gamma_{x;\xi}(0)=x\in\M$ and initial velocity $\frac{D}{dt}\gamma_{x;\xi}(0)=\xi\in T_x\M$, where $\frac{D}{dt}$ denotes the covariant derivative. Sometimes geodesic only refers to a length minimizing geodesic.
$\nabla_{\M}f(x)$
the gradient of $f\colon\M \to \mathbb R$ defined by the property $\langle \nabla_{\M}f(x), \xi \rangle_x = D_xf(x)[\xi] \quad\text{ for all }\xi\in T_x\M$
$\M$
A complete $d$-dimensional Riemannian manifold.
$\vect{A},\vect{B}$
Matrices are denoted in bold capital letters. The only exception is, when we speak of points on a manifold or in the tangent plane, where we empahsize the character of being a point on the manifold or tangent plane, respectively, while they are (also) represented as matrices.
$\operatorname{prox}_{\lambda f}$
proximal map of a function $f\colon\M \to\mathbb R\cup\{+\infty\}$ is given by $\operatorname{prox}_{\lambda f}(x) = \operatorname{arg min}_{y\in \M}\Bigl\{\frac{1}{2\lambda} d_{\M}^2(x,y)+f(x)\Bigr\}$
$T\M$
the tangent bundle, $T\M \mathrel{≔}\dot\cup_{x\in\M}T_x\M$, see also tangent space
$T_x\M$
the tangent space at $x\in\M$
$\xi,\nu,\eta,\zeta,\xi_x$
tangent vectors from the tangent space $T_x\M$ at $x$. The index $\cdot_x$ is ommited, if the point is clear from the context.
$x,y,z,x_i$
points on the manifold $\M$
$X,Y,Z,X_i$
tangent vector fields, i.e. $X\colon\M \to T\M$, $X_x=X(x)\in T_x\M$
$\vect{a},\vect{b}$
Vectors are denotes in bold letters. The only exception is, when we speak of points on a manifold or in the tangent plane, where we empahsize the character of being a point on the manifold or tangent plane, respectively, while they are also represented as vectors.