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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

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2 Answers 2

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.

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In fact, a generic metric has no symmetries whatsoever. –  Jason DeVito Nov 29 '12 at 20:55

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 $S^{4n-1}.$

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