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\section{Objective}
\paragraph{Objective}\label{Objective}
The objective is that
given a natural number \(n\) and a real number \(\kappa\),
one can construct
\begin{APAenumerate}
    \item a set \(M\)
    \item a Lie group \(M\) with operation \(\otimes_M\),
    \item an \(n\)-dimensional \(C^\infty\)-manifold \(M\) with chart \(\varphi_i \subset M\to\R^n\),
    \item a Riemannian manifold \(M\) with inner product \(g_p\in T_pM\times T_pM\to\R\)
\end{APAenumerate}
(to be determined)
such that
\begin{APAitemize}
    \item group action is distance preserved.
    \item model is continuous with respect to \(\kappa\) (and smooth with respect to each basis).
    \item for all two-dimensional linear subspaces of the manifold, the sectional curvature is \(\kappa\).
\end{APAitemize}

\begin{conjecture}\label{GeometricGroupStructure}
    From \cref{Group:Sphere,Group:Euclidean,Group:Hyperbolic},
    \begin{APAitemize}
        \item For \(\kappa>0\), \(M\cong O\left(n+1\right)\).
        \item For \(\kappa=0\), \(M\cong E\left(n\right)\).
        \item For \(\kappa<0\), \(M\cong O_{+}\left(n,1\right)\).
    \end{APAitemize}
\end{conjecture}
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