$
\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 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 is its own tangential space, the tangential ONB
`TpMONB(x,y)`

is the classical ONB `eye(n)`

. Furthermore all curvature values , are zero, since the is flat.

### See also