The isomorphisms between $S^5$ and $SU(3)/SU(2)$?

Question

What is the precise isomorphisms between the coset $SU(3)/SU(2)$ and the five-sphere $S^5$?

Answer 1

The word "isomorphism" is not really appropriate here, because $SU(3)/SU(2)$ doesn't inherit any algebraic structure (in particular, it's not a Lie group). But there's a diffeomorphism $\phi\colon SU(3)/SU(2)\to S^5$, constructed as follows.

Let $p_0=(0,0,1)$ be the "north pole" in $S^5\subseteq \mathbb C^3$. The isotropy group of $p_0$ in $SU(3)$ is $SU(2)$, identified with the following block-diagonal subgroup of $SU(3)$: $$ \left\{ \left( \begin{matrix} A & 0\\ 0 & 1 \end{matrix} \right): A\in SU(2) \right\} $$

Define a map $\Phi\colon SU(3)\to S^5$ by $\Phi(g) = g\cdot p_0$. This is constant on left cosets of $SU(2)$, so it descends to the quotient to yield a map $\phi\colon SU(3)/SU(2)\to S^5$, which sends the coset $g\cdot SU(2)$ to the point $g\cdot p_0$. The fact that this is a diffeomorphism follows from the standard theory of quotient manifolds. (See my Introduction to Smooth Manifolds, 2nd ed., Chapter 21.)