Question

I've been thinking about $SU(n)$-invariant metrics on the odd-dimensional spheres $S^{2n-1} \simeq SU(n)/SU(n-1)$. For $S^1$, all such metrics are in correspondence with the positive reals. For $S^{3} \simeq SU(2)$, the tangent space is parallelizable, and so, such metrics are in correspondence with metrics on $R^2$. Does such a characterization exist for $S^5$, or higher.

Answer 1

Assume $n>2$. The $SU(n)$-invariant metrics on the sphere $S^{2n-1}$ are in bijection with $SU(n-1)$ invariant metrics on $T_x S^{2n-1}$ where $SU(n-1)$ is realized as the stabilizer of some $x\in S^{2n-1}$. This action is the sum of the defining $SU(n-1)$-module $V$ and the trivial real 1-dimensional module $T$. The space of invariant real valued bilinear forms on this is 2-dimensional and is spanned by the invariant Hermitian metric on the defining representation and some metric on the trivial representation.

Notice that if $(\cdot,\cdot)$ is invariant and $v\in V, t\in T$, then $(v,t)=0$ since there is an $A\in SU(n-1)$ such that $Av=-v$. This is what makes the case $n>2$ different from the case $n=2$.

Answer 2

Classification of $SU(n+1)$-homogeneous metrics on $S^{2n+1}$:

First, note that for any $n\geq1$, the round metric on the sphere $S^{2n+1}$ can be scaled by $t^2$ in the direction of the Hopf fibers $S^1\to S^{2n+1}\to \mathbb C P^n$, giving rise to a one-parameter family $g_t$ of $SU(n+1)$-homogeneous metrics (so that $g_1$ is the original round metric).

Prop. Any $SU(n+1)$-homogeneous metric on $S^{2n+1}$ is isometric to some $g_t$, $t>0$.

In other words, they are always parameterized by (positive) real numbers. Alternatively, these $SU(n+1)$ homogeneous metrics on $S^{2n+1}$ are isometric to distance spheres in the complex projective space $\mathbb C P^{n+1}$.

Both of the above statements follow from a more general result, namely W. Ziller's classification of all homogeneous metrics on spheres, see [Homogeneous Einstein metrics on spheres and projective spaces. Math. Ann. 259 (1982), no. 3, 351–358].

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A few more details:

An explicit formula for any $G$-homogeneous metric on the homogeneous space $G/H$, in terms of an $Ad(H)$-invariant decomposition $g= h\oplus p$ of the Lie algebra of $G$ is: $$\langle ,\rangle=h|_{p_0} + \sum_{i=1}^r\alpha_i B|_{p_i},$$ where $p= p_0\oplus\dots\oplus p_r$ is a decomposition so that the $H$ representations on $p_i$ are not equivalent, $H$ acts trivially on $p_0$ and irreducibly on $p_i$, $i=1,\dots r$; $h|_{p_0}$ is any inner product on $p_0$; $B$ is a bi-invariant metric on $G$ and $\alpha_i>0$ are real parameters. Any $G$-homogeneous metric is defined by choosing these parameters. In the case of $SU(n+1)$, this decomposition is $p=p_0\oplus p_1$ and $\dim p_0=1$, so (up to renormalization) there is only one parameter, $\alpha_1$ to be chosen (that I called $t$ above).

EDIT. Since I claim the metrics are parameterized by one positive real number (and another answer above claims there must be two real parameters), a clarification is in order here. The point is that, indeed, there are two parameters ($h|_{p_0}$ and $\alpha_1$ in the notation above), nevertheless it is always possible to divide the entire metric by the first one (the number that determines the metric $h$ on the 1-dim space $p_0$), which leaves us with just one parameter. Treating the family as having the $2$ parameters is slightly ambiguous because lots of metrics will be simple rescaling of other ones. In my description above, they are pairwise non-conformal.