(pp. 1-32)

Let

\mathrm{M(2,\mathbb{C}) = \left \{[_{c\; d}^{a\: b}\]\; |\; a,b,c,d\:\; \epsilon \; \mathbb{C}\left. \right \}}

\mathrm{SU(2,\mathbb{C}) = \left \{ A\;\; \epsilon\:\; M(2,\mathbb{C})\; |\; A\bar{A^{t}} = 1,\; det(A) = 1\left. \right \}}.

\mathrm{S^{3}=\left \{ \right.(a,b)\; \epsilon \; \mathbb{C}^{2}\; |\; a\bar{a}+b\bar{b}=\; 1\left. \right \}.}

There is a natural identification of S³ with\mathrm{SU(2,\mathbb{C})}:

\mathrm{S^{3}\; \overset{\cong }{\rightarrow}\; SU(2,\mathbb{C})}

\mathrm{(a,b)\; \rightarrow \; \begin{bmatrix} \; \: \: a\; \; \; b \\-\bar{b}\; \; \; \bar{a} \end{bmatrix}.}

(1) We shall identify S³ and\mathrm{SU(2,\mathbb{C})}by this fixed diffeomorphism.

(2) We shall consider S³ to have a fixed orientation throughtout our discussion.

By differentiating the defining equations for S³, we obtain the tangent bundle of S³:

\mathrm{TS^{3}=\left [\left [\begin{bmatrix} \; \; \mathrm{a}\; \; \; \mathrm{b}\\ -\bar{\mathrm{b}}\; \; \; \bar{\mathrm{a}} \end{bmatrix},\; \begin{bmatrix} \; \; \mathrm{u}\; \; \; \mathrm{v} \\ -\bar{\mathrm{v}}\; \; \; \bar{\mathrm{u}} \end{bmatrix}\right ]\begin{bmatrix}\mathrm{a\bar{u}+\bar{a}u+b\bar{v}+\bar{b}v=0\\ \mathrm{a\bar{a}+b\bar{b}=1 \end{bmatrix}.}}}

In particular, the Lie algebra of S³ is given as:

\mathrm{S=SU(2,\mathbb{C})=T_{I}(S^{3})}

\mathrm{s}=\left [\begin{bmatrix} \mathrm{is}\; \; \; \; \mathrm{v} \\ \mathrm{-\bar{v}-is } \end{bmatrix} \left [ \mathrm{s}\; \epsilon \; \mathrm{R},\; \mathrm{v}\; \epsilon \; \mathbb{C} \right ].

The action of S³ on itself by left translations provides a trivialization of TS³:

\mathrm{s^{3}\; x\; S\; \overset{\cong }{\rightarrow}\; TS^{3}}

\mathrm{(A,X)\rightarrow (A,AX)}.

We identify TS³ with S³ x S by the natural trivialization provided above....