## 3 The Singular Pseudo-Hyperbolic Space and the Foliation  
In the previous section, we have shown that the metric \(d\tau^{2}\) is invariant with respect to the \([\mathfrak{A}]\) action (where \(\mathfrak{A}\in\widehat{G}_{\tau}^{+}\)). Nevertheless, we can define another Lorentzian metric \(d\hat{h}^{2}\) invariant with respect to the \([\mathfrak{A}]\) action. However, this new metric \(d\hat{h}^{2}\) is no more defined on \(\mathbb{R}^{4}\) but only on the four-dimensional open ball \(B^{4}\subset\mathbb{R}^{4}\). Actually, \(d\hat{h}^{2}\) is defined to be such that \(d\hat{h}^{2}=\mathfrak{a}^{-1*}(d\tau^{2})\) with \([\mathfrak{A}]^{*}(d\hat{h}^{2})=d\hat{h}^{2}\). In addition, this metric \(d\hat{h}^{2}\) is totally discontinuous although bounded on the boundary \(\hat{B}\equiv\partial\overline{B}^{4}=S^{3}\) of \(\overline{B}^{4}\). Now, let \(I:(u^{i})\in\overline{B}^{4*}\longrightarrow(u^{i}/R^{2})\in\mathbb{R}^{4}-B^ {4}\) be the _inversion_ map _in_ the 3-sphere \(S^{3}\) where \(\overline{B}^{4*}\equiv\overline{B}^{4}-\{0\}\) and \(R^{2}\equiv\sum_{j=1}^{4}(u^{j})^{2}\). This inversion commutes with \([\mathfrak{A}]\), i.e., we have \(I\circ[\mathfrak{A}]=[\mathfrak{A}]\circ I\). Then, from \(I\), we can define the metric \(d\hat{h}^{2}\) on \(\mathbb{R}^{4}-\overline{B}^{4}\) such that \(d\hat{h}^{2}\equiv I^{-1*}(d\hat{h}^{2})\). Then, we obtain a metric \(dh^{2}\equiv(d\hat{h}^{2},d\hat{h}^{2})\) on \(B^{4}\cup(\mathbb{R}^{4}-\overline{B}^{4})\equiv\mathbb{R}^{4}-S^{3}\) which is invariant with respect to the \([\mathfrak{A}]\) action. We can partially only bypass this discontinuity issue on \(S^{3}\) defining a particular class of conformal metrics \(d\
u^{2}\) from \(dh^{2}\). For this purpose, we define first the functions \(q(u)\) and \(\Xi(q)\) such that  
\[q(u)\equiv 1+2\,\frac{(u^{1})^{2}}{1-\sum_{i=1}^{4}(u^{i})^{2}}\,,\qquad\Xi(q) \equiv\frac{C(q)}{q^{5/4}}\,, \tag{3}\]  
where \(C(q)\) is a continuous function such that \(C(+\infty)\
eq 0\) whenever \(u^{1}\
eq 0\) and bounded on \([1,+\infty]\). It is important to note that the function \(q\) is invariant with respect to homographie in \([\widehat{G}_{\tau}^{+}]\), i.e., \(q([\mathfrak{A}](u))=q(u)\) for any \(\mathfrak{A}\in\widehat{G}_{\tau}^{+}\). As a result, we can define metrics \(d\vartheta^{2}\equiv\Xi(q)^{2}\,d\hbar^{2}\) on \(\overline{B}^{4}-S^{2}\) with their associated metrics \(dv^{2}\equiv I^{-1*}(d\vartheta^{2})\) on \(\mathbb{R}^{4}-(B^{4}\cup S^{2})\). As a consequence, we obtain metrics \(dv^{2}\equiv(dv^{2},d\vartheta^{2})\) on \(\mathbb{R}^{4}-S^{2}\) where \(S^{2}\) is the spatial 2-sphere, i.e., such that \(u^{1}=0\) and \(\sum_{k=2}^{4}(u^{k})^{2}=1\). These two metrics are invariant with respect to \(\widehat{G}^{+}_{\tau}\). In particular, we have \([\mathbb{A}]^{*}(d\vartheta^{2})\equiv\Xi(q([\mathfrak{A}](u)))^{2}\,[\mathbb{ A}]^{*}(d\hbar^{2})=d\vartheta^{2}\) for any \(\mathfrak{A}\in\widehat{G}^{+}_{\tau}\). Then, we call _singular pseudo-hyperbolic space_\(\mathcal{H}^{1,3}_{s}\) the space \(\mathbb{R}^{4}-S^{2}\) equipped with the metric field \(dv^{2}\). Then, we deduce: (1) an invariance of \(\mathcal{H}^{1,3}_{s}\) with respect to homographic action of group \(\widehat{G}^{+}_{\tau}\), and (2) a class of conformal metrics \(dv^{2}\) projectively invariant and singular (discontinuous) only on the spacelike equatorial sphere \(S^{2}\subset S^{3}\). But also, in addition, we obtain a particular foliation from the group action \(\widehat{G}^{+}_{\tau}\) on \(\mathcal{H}^{1,3}_{s}\).