What group preserves a metric up to a constant?

Question

The group $SO(d,2)$ preserves the Minkowski space $\mathbb{R}^{d-1,1}$ up to a function $\Omega(x)^2$, that depends on the position co-ordinates.

$$ds^2 \rightarrow \Omega(x)^2 ds^2.$$

What group preserves the metric just up to a constant factor say $\lambda$?

$$ds^2 \rightarrow \lambda^2 ds^2~?$$

I can see that the usual Lorentz and scale transformations will do. But, are there any other non-trivial transformations (like special conformal for the conformal group)? And what group are we talking about?

Answer 1

If we consider the global conformal group $${\rm Conf}(p,q)~\cong~O(p\!+\!1,q\!+\!1)/\{\pm {\bf 1} \}\tag{A}$$ in a space with signature $(p,q)$, restriction to $x$-independent scale factor $\Omega(x)^2$ would exclude the special conformal transformations, i.e. we are left with the product of the Poincare group and the group of dilations.

See also this related Phys.SE post.