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. 



Invariant Metrics on the Sphere
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 $n=2$.


