1.3.2 The Space of Angles $\in \mathbb{C}$

Con­sid­er­ing geo­met­ric man­i­folds as open sets re­sid­ing in a higher-di­men­sion­al angle space, as op­posed to a higher-di­men­sion­al Eu­clid­ean space, af­fords pre­cise clar­i­ty re­gard­ing how these geome­tries in­ter­re­late. Real-valued dis­tances, like the $x$ and $y$ co­or­di­nate axes in the two-di­men­sion­al Carte­sian plane, typ­i­cal­ly rep­re­sent our first for­ma­tive ex­po­sure to geo­met­ric spaces. When we give pri­or­i­ty to an angle space $\in \mathbb{C}$, in­stead of the Carte­sian space geom­e­try $\in \mathbb{R}$, a more in­flu­en­tial mean­ing and depth is ren­dered con­cern­ing geo­met­ric spaces. When con­sid­er­ing the real-valued ex­pres­sion of the circle

and its trigono­met­ric form

we have so­lu­tions for Eq. 1.3.4 for any chosen angle $\alpha\in \mathbb{R}$. The single angle ar­gu­ment in the trigono­met­ric ex­pres­sion of the circle im­plies the two di­men­sions of the Carte­sian co­or­di­nate system. These trigono­met­ric sine and cosine func­tions are de­pen­dent on $\angle\alpha$. At the same time, these func­tions are in­ter­de­pen­dent, and the give and take that de­fines their cor­re­lat­ed values en­sures the circle's dis­tance metric holds in­vari­ant. We con­sid­er the ex­po­nen­tial series for a single angle value $X$

which can be sep­a­rat­ed into odd and even el­e­ments to define the $\cosh(X)$ and $\sinh(X)$ func­tions

Math­e­mat­i­cal­ly, it has been known since the time of Euler, al­low­ing $X$ to be imag­i­nary-valued leads to the for­mu­la we all hold dear

In plain sight, with the ex­am­ples above, is an exact cor­re­spon­dence be­tween the unit circle and the unit hy­per­bo­la. This cor­re­spon­dence boils down to their re­spec­tive ori­en­ta­tions in the angle-ar­gu­ment space $\in \mathbb{C}$. These ex­po­nen­tial series and their trigono­met­ric func­tions are sub­sets of the same higher-di­men­sion­al com­plex angle space.

For the two-di­men­sion­al ex­am­ple, the com­plex angle space rep­re­sents a com­plex unit circle nested in the space of the angle vari­ables. Through the ex­po­nen­tial map­ping, each point on its bound­ary rep­re­sents a unique man­i­fold, ad­mit­ting angles $\in \mathbb{C}$. We can refer to this as the Angle Space Ring, shown in Figure 1.3.6, where com­plex-valued angles are point rep­re­sen­ta­tions on its bound­ary. The ring, by way of the ex­po­nen­tial map­ping, gives rise to unique ex­po­nen­tial func­tions, where the figure's ex­am­ples are ori­ent­ed along the imag­i­nary-axis $\left( e^{\left( \frac{i\pi X}{2} \right)},e^{\left( \frac{3i\pi X}{2} \right)} \right)$, the real-axis $\left( e^{(2i\pi X)},e^{(i\pi X)} \right)$, and the as­ymp­tot­ic axes $\left( e^{\left( \frac{i\pi X}{4} \right)},e^{\left( \frac{3i\pi X}{4} \right)},e^{\left( \frac{5i\pi X}{4} \right)}\ ,e^{\left( \frac{7i\pi X}{4} \right)} \right)$, all rep­re­sent­ing points on the Angle Space Ring.

Along pos­i­tive real and imag­i­nary axes, the ex­po­nen­tial map­ping of the points $e^{(2i\pi X)},e^{\left( \frac{i\pi X}{2} \right)}$ is straight­for­ward, giving rise to the subset man­i­folds $e^{iX}$ and $e^{X}$. For subset man­i­folds, the angle ar­gu­ments are rep­re­sent­ed by points on the ring be­tween axes, taking on com­plex-valued angles of the form $\angle = X\left( e^{i\beta} \right):X,\beta\in \mathbb{R}$. Using the ex­am­ple for the point lo­cat­ed on the right at $e^{\left( \frac{i\pi X}{4} \right)},$ $\beta = \frac{i\pi}{4}$ ra­di­ans, re­sult­ing in a man­i­fold de­fined by the ex­po­nen­tial map­ping

Com­plex-Angle Space is the for­mal­ly de­fined, as fol­lows:

De­f­i­n­i­tion (1.3.9) – The Com­plex-Angle Space is a com­plex $n$-di­men­sion­al angle space $\mathrm{B}_{n}$, which is the set that con­tains all the subset Lie groups G, with angles of the form $X^{n}\left( e^{i\beta_{n}} \right)$, rep­re­sent­ing the tan­gent space of the Lie group $G = \exp\left( X^{n}\left( e^{i\beta_{n}} \right) \right)$, $X^{n} \in \mathbb{R}^{n}$, and all tan­gent spaces $X^{n}\left( e^{i\beta_{n}} \right)$ lay on the bound­ary $\partial \mathrm{B}_{n}.$

De­f­i­n­i­tion (1.3.10) – Let $G(X),G(Y) \subset \mathrm{B}$ be ar­bi­trary Lie groups, rep­re­sent­ing closed sub­sets of the Com­plex-Angle Space $\mathbf{\mathrm{B}}_{\mathbf{n}}.$ $\boldsymbol{\Omega}:\mathbf{G}\left( \mathbf{X} \right)\rightarrow \mathbf{G}\left( \mathbf{Y} \right)$ is a bi­jec­tive and con­tin­u­ous map­ping that takes the Lie group ${G(X}^{n}) \in \mathrm{B}_{n}$ to $G\left( Y^{n} \right) \in \mathrm{B}_{n}$,

We define the func­tion map­ping $\Omega:G\left( X^{n} \right)$, for a chosen com­plex angle trans­for­ma­tion $e^{i\beta_{n}}$

for $X^{n} \bullet e^{i\beta_{n}}$, rep­re­sent­ing the el­e­ment-to-el­e­ment mul­ti­plica­tive prod­uct of the two $n \times n$ ma­tri­ces.

For the $n = 1$ di­men­sion­al case, the map­ping $\Omega:G\left( X^{n} \right) \rightarrow {G(Y}^{n})$ is straight­for­ward, ap­pre­ci­at­ed for three-di­men­sion­al map­ping ex­am­ple

for ${G(X}_{1}^{3\mathbb{C}})$, a gen­er­a­tor from the $D_{3\mathbb{C}}$ di­he­dral group, and ${G(X}_{1}^{3\mathbb{R}})$, a gen­er­a­tor from the $D_{3\mathbb{R}}$ group. We set $\beta = \frac{- \pi}{2}$, while re­call­ing the func­tion $X \circ \beta$ rep­re­sents the re­sult­ing matrix formed by the el­e­ment-to-el­e­ment mul­ti­pli­ca­tion prod­ucts $(x_{j,k} \bullet \beta_{j,k}$), find­ing

The di­he­dral forms of the Lie groups that we have been work­ing with belong to the gen­er­al linear group $GL\left( 3,\mathbb{C} \right) \in \mathbb{C}^{3}$

The $\Omega(X)$ map­ping is bi­jec­tive, and given that these groups are all con­tin­u­ous, the topo­log­i­cal spaces $G\left( X^{n} \right),{G(Y}^{n})$ are home­o­mor­phic to one an­oth­er.

The­o­rem (1.3.14) – For any two topo­log­i­cal spaces of the form $G(X),G(Y)$, lo­cat­ed on the bound­ary $\partial \mathrm{B}$ in com­plex-angle space, there is a ho­mo­mor­phism be­tween $G(X)$ and $G(Y)$.

Proof – The map­ping $\Omega:G(X) \rightarrow G(Y)$ of any two func­tions of the form $G(X),G(Y)$ lo­cat­ed on the bound­ary $\partial B$ in com­plex-angled space is a bi­jec­tive (one-to-one) and con­tin­u­ous map­ping, and there is an in­verse map­ping $\Omega^{- 1}:G(Y) \rightarrow G(X)$ for any map­ping $\Omega:G(X) \rightarrow G(Y)$, that is also bi­jec­tive and con­tin­u­ous. The map­ping,

maps each el­e­ment of $X^{n}$ to $Y^{n}$ bi­jec­tive­ly. For every map­ping $G\left( X^{n} \circ e^{i\beta_{n}} \right)$ there is an in­verse map­ping,

by way of com­plex con­ju­ga­tion $\overline{e^{i\beta_{n}}} = e^{i( - \beta)_{n}}$, which maps $G\left( Y^{n} \right)$ back to $G\left( X^{n} \right)$.

To frame the math­e­mat­ics in simple terms, one can choose an angle in com­plex space to rotate one man­i­fold to an­oth­er through an angle space trans­for­ma­tion. Con­cern­ing the di­he­dral form's Lie groups, these can be rep­re­sent­ed on the sphere in com­plex-angle space, which is anal­o­gous to the two-di­men­sion­al rep­re­sen­ta­tion from Figure 1.3.6. Using the $U(3)$ group ex­am­ple, we find it can be ro­tat­ed to the hy­per­bol­ic form by the map­ping,

by set­ting $\beta_{1},\beta_{2},\beta_{3} = \frac{- \pi}{2}$, then eval­u­at­ing the al­ge­bra map­ping

A geo­met­ric vi­su­al­iza­tion of the angle trans­for­ma­tion from above can be shown by con­nect­ing points rep­re­sent­ing the per­mu­ta­tion gen­er­a­tors from the $G^{3\mathbb{C}}$ group$\ \left( e^{iX_{1}},e^{iX_{2}},e^{iX_{3}} \right)$, with cir­cu­lar arcs on the com­plex-angled space sphere. For $\beta_{1},\beta_{2},\beta_{3} = \frac{- i\pi}{2}$, a clock­wise ro­ta­tion occurs for each $G^{3\mathbb{C}}$per­mu­ta­tion gen­er­a­tor by $\frac{- i\pi}{2}$ ra­di­ans. By taking $e^{iX_{j}} \rightarrow e^{X_{j}}$, seen in Fig­ures 1.3.7-8, the imag­i­nary-valued gen­er­a­tors lo­cat­ed on the ver­tices of the tri­an­gle have each ro­tat­ed by $\frac{- i\pi}{2}$ in their re­spec­tive planes, and arrive at their co­in­cid­ing real-valued gen­er­a­tor forms.