NOTE

The equator ${\displaystyle \mathbb{E}}$ of ${\displaystyle S{U}_2}$ is the latitude defined by the equation ${\displaystyle \mathrm{Tr}(P)=0,P\in S{U}_2}$. What's more, ${\displaystyle \mathbb{E}\cong S^2}$.

In the context of the special unitary group $SU(2)$, the term "equator" can be interpreted geometrically, often in relation to the group's manifold structure.

Geometric Interpretation of $SU(2)$

The group $SU(2)$ consists of all $2\times 2$ unitary matrices with determinant 1. These matrices can be written in the form:

$$U=\left(\begin{array}{cc}\alpha & -{\beta }^{\ast }\\ \beta & {\alpha }^{\ast }\end{array}\right),\text{where}\alpha ,\beta \in \mathbb{C},|\alpha {|}^2+|\beta {|}^2=1.$$

Topologically, $SU(2)$ is homeomorphic to the 3-sphere $S^3$.

The Equator of $S^3$

Given that $SU(2)\cong S^3$, the "equator" of $SU(2)$ would correspond to the equator of the 3-sphere $S^3$. A 3-sphere can be embedded in 4-dimensional space ${\mathbb{R}}^4$ as:

$$S^3=\{(z_1,z_2)\in {\mathbb{C}}^2:|z_1{|}^2+|z_2{|}^2=1\}.$$

The equator of $S^3$ is typically a 2-dimensional sphere $S^2$ embedded in $S^3$. In terms of $SU(2)$, this can be seen as a subset of matrices that form a lower-dimensional manifold within $SU(2)$.

Parameterization and Example

To find an explicit description of the "equator," consider the unit quaternions, which are another representation of $SU(2)$. A quaternion $q=a+bi+cj+dk$ is a unit quaternion if $a^2+b^2+c^2+d^2=1$.

The "equator" can be described by fixing one of the components, say $a=0$, which leaves:

$$b^2+c^2+d^2=1.$$

This describes a 2-sphere $S^2$ within the 3-sphere $S^3$.

Explicit Matrix Form

In terms of the matrix representation, if we fix the real part of $\alpha$ to 0, the remaining part describes the equator. For example, if we set $\alpha =i{\beta }_1$ with $\beta ={\beta }_1+i{\beta }_2$, then the matrix becomes:

$$U=\left(\begin{array}{cc}i{\beta }_1& -{\beta }_1-i{\beta }_2\\ {\beta }_1+i{\beta }_2& -i{\beta }_1\end{array}\right),\text{where}{\beta }_1^2+{\beta }_2^2=\frac{1}{2}.$$

This subset of matrices represents a lower-dimensional manifold within $SU(2)$, analogous to the equator in $S^3$.

Conclusion

The equator of $SU(2)$ can be understood as a 2-dimensional sphere $S^2$ within the 3-sphere $S^3$, which corresponds to fixing one parameter and letting the others vary such that the total still lies on the 3-sphere. This provides a geometric way to visualize and understand the structure of $SU(2)$ and its "equatorial" subset.