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So the Theorema Egregium for a surface is remarkable because it expresses the Gaussian curvature K (a priori extrinsic) in terms of the metric tensor, which is intrinsic to the surface hence K is intrinsic as well after all. Here "intrinsic" means that a 2d creature living in the surface can compute it by making measurements solely in the surface.
So how does a creature computes the metric? I thought that it does so by computing the lengths of curves and then use:
[tex]
\left.\frac{d}{dt} L(\gamma)|_{[-\epsilon,t)}\right|_{t=0}=\frac{d}{dt}\int_{-\epsilon}^t||\gamma'(\tau)||d\tau=||\gamma'(0)||
[/tex]
But how does a creature in the surface measure lengths? It chooses some small stick lying in the 2d forest behind his 2d house to use as "unit of length", and then he measures. But then the length of curves he gets, and hence also the derivative of this length will in general be some multiple of the same derivative computed in the units of the 3d space in which the surface is embedded (i.e. the LHS above). So a creature can only measure the metric up to a constant (homethety). Am I wrong?
So how does a creature computes the metric? I thought that it does so by computing the lengths of curves and then use:
[tex]
\left.\frac{d}{dt} L(\gamma)|_{[-\epsilon,t)}\right|_{t=0}=\frac{d}{dt}\int_{-\epsilon}^t||\gamma'(\tau)||d\tau=||\gamma'(0)||
[/tex]
But how does a creature in the surface measure lengths? It chooses some small stick lying in the 2d forest behind his 2d house to use as "unit of length", and then he measures. But then the length of curves he gets, and hence also the derivative of this length will in general be some multiple of the same derivative computed in the units of the 3d space in which the surface is embedded (i.e. the LHS above). So a creature can only measure the metric up to a constant (homethety). Am I wrong?