Good day.

Let (M,g) be an n-dimensional Riemannian manifold (complete, if you wish), and suppose that there exists an n-1 dimensional Abelian group acting by isometries on M. Or locally, near a point p in M, let A be an Abelian Lie algebra of Killing vector fields of M. Local coordinates can be chosen so that A is generated by d/dx_1,..., d/dx_{n-1}, and g=g(x_n).

In many cases it turns out that A is a codimension one Lie subalgebra of the full isometry algebra, which is bigger. Can anyone provide a counterexample, please? That is, a Riemannian n-manifold of which the full isometry group is Abelian and n-1 dimensional. I did not manage to investigate this thoroughly, but intuitively it seems that (though g(x_n) is arbitrary) you can find one more Killing vector which involves d/dx_n as well ( similar to reparameterization of a one dimensional Riemannian manifold to make the metric constant). If this or the opposite is obvious for experts or can be found readily somewhere in the literature, I would not like to spend hours thinking on it. Otherwise, I will delve into it.

Thank you in advance.