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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?

Henno Brandsma
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rnoodle
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  • Older question: https://math.stackexchange.com/questions/2953020/intrinsic-curvature-and-is-the-space-described-by-polar-coordinates-euclidean – rnoodle Oct 12 '18 at 19:32
  • Centuries ago we Earthlings figured out how to measure large distances on a big curved surface. – William Elliot Oct 12 '18 at 22:58
  • @William Elliot - But we are 3D beings standing on a roughly 2D surface curved in 3D. I would instead like to know about the 2d beings on that surface. – rnoodle Oct 13 '18 at 00:11
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    Flatlanders can draw a square with three equal lines and two right angles and measure the fourth line to find out how warped they are. – William Elliot Oct 13 '18 at 00:54

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