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

# The Euclidean Space $\mathbb R^n$

This class is merely a class for checks on real valued data, which is of course just a special case of manifold-valued data. This means, that all functions are realized by quite easy functions, which we will list here shortly for completeness but without special own pages and examples

distance
The distance is the usual euclidean norm dist(x,y,p) = norm(x,y,p)
dot
the inner product is the classical inner product dot(x,y) = <x,y>. note that the first dimension (manifold dimension) still collapses if the have multidimensional data on the Eucliean space
exponential map
the exponential map (following the direction of a vector) is addition exp(x,xi) = x+xi
logarithmic map
The question of “how to get from x to y” is the vector pointing fotm x to y and hence log(x,y) = y-x.
parallel transport
The parallel transport of a vector is the identity parallelTransport(x,y,xi) = xi
tangential orthonormal basis
since $\mathbb R^n$ is its own tangential space, the tangential ONB TpMONB(x,y) is the classical ONB eye(n). Furthermore all curvature values $\kappa_i$, $i=1,\ldots,n$ are zero, since the $\mathbb R^n$ is flat.