I tried to google the following but couldn't find an answer that helped - so I hope I might find some here - the question is short and very basic (I guess) :
what does it mean when someone writes that a Riemannian metric $g$ on $\mathbb{R}^n$ is such that $g(x)$ and $g^{-1}(x)$ are uniformly bounded ?
From the theory that I know (unfortunately not much so far) I understand that for each $x$ $g$ is a symmetric positive definite $n \times n$ matrix (given a local coordinate system), and as such it is a linear transformation of $\mathbb{R}^n$. So, does the uniform bound mean that there exists $M > 0$ such that \begin{equation} \|g(x)\| \leq M \quad \forall x \quad ? \end{equation} (Here $\| . \|$ is understood to be some norm on $GL(n, \mathbb{R})$).
Thanks for your help!