1.5.1 Tangent Space Angles and Path-Connectedness in the Symmetric Cone Group

In the last chap­ter, we de­rived the sym­met­ric conic group$\ S_{3\mathbb{C}}^{\Delta}$ and con­sid­ered some of its sub­groups. The $S_{3\mathbb{C}}^{\Delta}$ group con­tains many dif­fer­ent geo­met­ric spaces that are path-con­nect­ed on its topo­log­i­cal sur­face of tan­gent spaces, mean­ing we can move freely from one tan­gent space to an­oth­er on the sur­face of the sphere. To demon­strate path-con­nect­ed­ness, we use the ex­am­ple of moving from the from the ${\mathfrak{T}(X}_{1})$ gen­er­a­tor's tan­gent space $\in U(3)$ to all el­e­ments of the $SU(2)$ sub­group in the pos­i­tive $X_{1}$-plane. This is ac­com­plished by using the el­e­ments $\in S_{3\mathbb{C}}^{conic}$ to form matrix prod­ucts with ${\mathfrak{T}(X}_{1})$,

The $\overrightarrow{\mathcal{u}_{i}}$ el­e­ments acting to trans­form the ${\mathfrak{T}(X}_{1})$ tan­gent space are char­ac­ter­ized by their axis number de­ter­mi­nants. Having de­fined the sym­me­try group, makes path-con­nect­ed­ness of all the el­e­ments in the tan­gent space pos­si­ble. An­oth­er ex­am­ple serves to demon­strate this, where we use an el­e­ment from the $X_{3,k}$ sub­group with a cone number de­ter­mi­nant to easily arrive at $\mathcal{u}_{2} \in SU(2)$, by the prod­uct

This is a process that works by taking ad­van­tage of the fact that every el­e­ment in the sym­met­ric group has an in­verse, per the de­f­i­n­i­tion of a group, and these trans­for­ma­tion re­la­tions can be for­mal­ized as the fol­low­ing trans­form

In the ex­pres­sion above, the matrix $\mathfrak{T}(A)$ rep­re­sents the start­ing tan­gent space vector matrix, $\overrightarrow{B}$ is the path-con­nect­ing tan­gent vector matrix, and $C$ is the des­ti­na­tion tan­gent space vector matrix en­abled by the trans­form. Using the matrix prop­er­ties of the in­verse of $\mathfrak{T}(A)$, it is simple to find our path-con­nect­ing matrix $\overrightarrow{B}$, since it will be equal to the prod­uct of the in­verse of our start­ing tan­gent space and the des­ti­na­tion tan­gent space

The path-con­nect­ed­ness of the $S_{3\mathbb{C}}^{conic}$ group el­e­ments is so­lid­i­fied by the fol­low­ing the­o­rem.

The­o­rem (1.5.5) – Any el­e­ment $\in A^{\Delta},$ rep­re­sent­ing the set of all per­mu­ta­tions of the sym­met­ric conic group $S_{3\mathbb{C}}^{\Delta}$, is path-con­nect­ed to every other el­e­ment by way of an­oth­er el­e­ment $\in A^{\Delta}$ $\left( A_{i}^{\Delta} \circ A_{j}^{\Delta} \rightarrow A_{k}^{\Delta} \right)$.

Proof – $S_{3\mathbb{C}}^{\Delta}$ is a group under com­po­si­tion where $A_{i}^{\Delta}$ and $A_{j}^{\Delta}$ are bi­jec­tive and the com­po­si­tion, $A_{i}^{\Delta} \circ A_{j}^{\Delta} \rightarrow A_{k}^{\Delta}$ is thus bi­jec­tive. For every per­mu­ta­tion $A_{i}^{\Delta}$ there exists a bi­jec­tive func­tion ${A_{i}^{\Delta}}^{- 1}$, for which ${A_{i}^{\Delta}}^{- 1} \circ A_{i}^{\Delta} \rightarrow \epsilon$. Since $S_{3\mathbb{C}}^{\Delta}$ is a sym­met­ric group, its bi­jec­tive set of per­mu­ta­tion el­e­ments $A^{\Delta}$ con­tains all pos­si­ble com­bi­na­tions of the set of de­fined cone and axis num­bers, $\left\{ \pm 1, \pm i, \pm ( - 1)^{\left( \frac{1}{4} \right)}, \pm ( - 1)^{\left( \frac{3}{4} \right)} \right\}$. Given that the in­verse of our start­ing tan­gent space exists ${A_{i}^{\Delta}}^{- 1},$ bi­jec­tiv­i­ty is sat­is­fied, im­ply­ing the el­e­ment $A_{j}^{\Delta}$ , which path-con­nects two el­e­ments $A_{i}^{\Delta},A_{k}^{\Delta}$ in the $S_{3\mathbb{C}}^{\Delta}$ tan­gent space, is also path-con­nect­ed and equal to

$\square$