SO(5)-invariant metrics on the 4-sphere Are there any examples of Riemannian metrics on $S^{4} \subset \mathbb{R}^{5}$ that are not 
SO(5)-invariant? Or are all 
metrics on the 4-sphere SO(5)-invariant? Hope my question is not too trivial :).
Dmitri
 A: You can inherit the metric from $\Bbb R^5$. The canonical metric is rotationally invariant, but you can easily construct any metric that is not, in fact, any non-trivial metric is likely to not be $\mathrm{SO}(5)$ invariant.
A: One thing that is true is that the only metric (up to scaling) on $S^4$ which is homogeneous is the round metric, which is $SO(5)$ invariant.  In fact, more generally, on any even dimensional sphere, there is a unique (up to scaling) metric which is homogeneous - the usual round metric.
On the other hand, this is not true for odd dimensional spheres (other than $S^1$).
On spheres of dimension $2n-1$, there is an $SU(n)\subseteq SO(2n)$ invariant metric which is not $SO(2n)$ invariant.  This comes from shrinking or enlarging the metric in the direction of the $(S^1)$- Hopf fibers.  For spheres of dimension $4n+1$, these deformations account for all homogeneous metrics.
On spheres of dimension $4n-1$, there is also an $Sp\left(\frac{n}{2}\right)\subseteq SU(n)\subseteq SO(2n)$ invariant metric which is neither $SU(n)$ invariant nor $SO(2n)$ invariant.  This metric is obtained by shrinking or enlarging the metric in the direction of the $S^3$-Hopf fibers.  One can then individually adjust the metric in the $S^1$-Hopf fiber directions as well.  These two changes together account for all homogeneous metrics on $S^{4n-1}.$ 
