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

The inner product on the sphere

Since the manifold $\mathbb S^n$ is isometrically embedded into $\mathbb R^{n+1}$ we obtain the inner product of two tangential vectors $\xi,\nu\in T_x\mathbb S^n\subset\mathbb R^{n+1}$ from the embedding space.

Matlab Documentation

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% ds = dot(x,xi,nu) inner product of two tangent vectors in T_xSn
%
% INPUT
%     x  : a point(Set) in P(n)
%     xi  : a first tangent vector( set) to (each) x
%     nu  : a secod tangent vector( set) to (each) x
%
% OUTPUT
%     ds : the corresponding value(s) of the inner product of
%     (each triple) xi,nu at x
%
% ---
% MVIRT 1.0 ~ J. Persch 2016-06-13