I want to describe all pseudo-Riemannian metrics (symmetric covariant non degenerate 2-tensor fields) invariant by $O(3)$ on different manifolds ($\mathbb R^3$, the sphere $S^2$, and $]-\epsilon,\epsilon[\times S^2$ in priority.
The difficulty depends on the given space, for instance for the sphere I think I have the idea : Since the sphere is a reductive $O(3)$-homogeneous space $(S^n=\frac{O(n+1)}{O(n)})$, all metrics on the sphere invariant by $O(3)$ are equal to the first fundamental form up to a scalar factor.
For $\mathbb R^3$ it starts to be more challenging (for me at least) since I don't have the same highly simplifying theorems. I first wanted to just classify all metrics on $\mathbb R^3$ and get rid of those that are not invariant by $O(3)$. However "classifiying all metrics on $\mathbb R^3$" is actually really hard. An other way would be to say : if $q$ is a metric on $\mathbb R^3$, then take a vector $v$, you have a $g \in O(3)$ such that $gv=||v||e_1$ with $e_1$ a canonical unitary vector of $\mathbb R^3$ (because $O(3)$ acts transitively on a sphere). If you want your metric $q$ to be invariant under $O(3)$, then $q(v,v)=q(gv,gv)=||v||^2q(e_1,e_1)$, so all metrics are equal to the euclidian one up to a scalar again.
Then for $]-\epsilon,\epsilon[\times S^2$ it looks even harder to find some tricks.
An easier thing to start may be to look at $]-\epsilon,\epsilon[\times S^1$.
Thus I would like to know if there is a general method for this problem, and if not, at least some methods (even methods using supplementary hypothesis that we would make if needed are interesting to know about).
$\newcommand\R{\mathbb{R}}$Let’s assume you want Riemannian metrics on $\R^3$ that are invariant under the standard action of $O(3)$. Let $g$ be such a metric. It’s restriction to any sphere centered at $0$ has to be a constant times the standard metric. From here it is not hard to show that the metric in spherical coordinates has to be of the form $$g = (a(r))^2\,dr^2 + (b(r))^2h $$ where $h$ is the standard metric on the sphere.