$ \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 inner product on the hyperbolic space

Since the manifold is isometrically embedded into with the Minkowski metric, the inner product of twio tangent vectors at is given by

Matlab Documentation

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% ds = dot(x,xi,nu) inner product of two tangent vectors in TxHn
%
% INPUT
%     x  : base point (optional because all TXM are equal)
%    xi  : a first tangent vector( set)
%    nu  : a secod tangent vector( set)
%
% OUTPUT
%     ds : the corresponding value(s) of the inner product of (each triple) V,W at X
%
% ---
% Manifold-Valued Image Restoration Toolbox 1.1
% R. Bergmann ~ 2015-10-20

See also