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Are there any examples of Riemannian metrics on $S^{4} \in \mathbb{R}^{5}$ 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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 This is a question on definition of metrics. If you consider $\mathbb S^4\subset \mathbb R^5$ then the metric induced on $\mathbb S^4$ from $\mathbb R^5$ is $SO(5)$ invariant iff your sphere a sphere of constant radius like $\sum_i (x_i)^2=r^2$. For a generic sphere in $\mathbb R^5$ the group of isometries is trivial. For example for the sphere $x_1^2+2x_2^2+3x_3^2+4x_4^2+5x_5^2$ with metric induced from $\mathbb R^5$ its group of isometries is finite. I would say your question is more appropriate for math.stackexchange.com – Dmitri Nov 29 at 9:32
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 Dmitri, yes there is a metric on it which is SO(5) invariant. For example take the metric $(dx_1)^2+2(dx_2)^2+3(dx_3)^2+...$ on $\mathbb R^5$ and restrict it to your unite sphere. – Dmitri Nov 29 at 10:16
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 sorry, I meant there is a metric that is NOT $SO(5)$ invariant. Basically you can construct such metrics easily. Just write down any formula apart from THE most standard one and you will get your non-invariant metric. Unless you don't state some additional conditions on your metric the answer to your question is trivially YES. – Dmitri Nov 29 at 10:19
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 Maybe it will help you to answer the same question replacing $S^4$ by $S^2$ and $SO(5)$ by $SO(3)$. Again the answer will be trivially YES. There space of all possible metrics on $S^2$ up to isometry is infinite-dimensional. At the same time metrics on $S^2$ that have $SO(3)$ symmetry are all proportional to each other, i.e. there is only one such metric up to scaling. – Dmitri Nov 29 at 11:51
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 Thet's so funny! There are two distinct users, both with nickname "Dmitri": one is the OP and the other is the user who first commented! I was a bit puzzled when I saw Dmitri telling Dmitri "I would say your question is more appropriate for math.stackexchange"! – Qfwfq Nov 29 at 14:38
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closed as too localized by Andreas Blass, unknown (google), Deane Yang, Allen Knutson, Misha Nov 29 at 13:58

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