The distance metric for a space $\mathcal{W}$ described by Riemannian metric tensor (see this paper) $G(\mathbf{w}){\in}\mathbb{R}^{\text{dim}(\mathcal{W})\times\text{dim}(\mathcal{W})}$ is
\begin{align*} d_{\mathbf{w}}(\mathbf{w}{+}\delta\mathbf{w})^{2}=\delta\mathbf{w}^{\top}G(\mathbf{w})\delta\mathbf{w},\quad\quad \mathbf{w}{\in}\mathcal{W}. \end{align*}
Do inhabitants of $\mathcal{W}$ get to 'see' $G(\mathbf{w})$ or are they only privy to the results of $d_{\mathbf{w}}(\cdot)$? I am trying to figure out if inhabitants to a space see different size rulers (measuring devices) if $\mathcal{W}$ is stretched? i.e. i) do measuring devices stretch but $d_{\mathbf{w}}(\cdot)$ gives different values depending on the intrinsic curvature of the space? Or ii) do measuring devices remain the same length and the inhabitants see space expanding thus can record stretched lengths? How does this link with extrinsic curvature?