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Gauss's Theorema Egregium states that Gaussian curvature is intrinsic to a surface, meaning that it can be "measured inside of the surface". However I can't make sense of what this really means. What are other quantities that can be measured inside a surface? Are there quantities that can only be measured outside a surface?

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Everything that can be expressed in terms of the first fundamental form (in a coordinate-independent way) gives an intrinsic quantity. An amazing property of the Gaussian curvature is that being defined as the determinant of the shape operator (i.e. using the "outside" geometry) it turns out to be expressible only as a combination of the components of the first fundamental form and their partial derivatives (see the Brioschi formula).

There are many things that you can make using the first fundamental form, for instance, we can measure lengths of smooth curves. These ares obviously intrinsic quantities.

Polynomial expressions involving components of the first fundamental form and all its partial derivatives of finite order are called natural if the form of the expression does not depend on the choice of (normal) coordinates. Examples are the Levi-Civita connection and the Riemann tensor (and of course the first fundamental form itself, which is also known as the intrinsic metric). It turns out that all possible natural tensors are obtained as iterated covariant derivatives of the Riemann curvature tensor and their contractions. Details see in the paper of D.B.A. Epstein "Natural tensors in Riemannian geometry", e.g. here.

On the other hand, the second fundamental form (which is an appearance of the shape operator) cannot be measured without an immersion: you need a unit normal field along the surface.

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Some intrinsic quantities depending on first fundamental form (FFF) are:

All coefficients of FFF, their partial derivatives w.r.t. u and v and and any combination thereof. E.g., Gauss curvature $K$, geodesic curvature, geodesic torsion, length, area, integral curvature, Christoffel symbols,isometry, Levi-Civita connection, tangential rotation, normal rotation $\psi$ in between principal planes etc. These are natural/intrinsic creatures of embedding.

All surfaces sharing common FFF can be bent or twisted amongst them. Not only $K$ but the above and all such remain unchanged.

On the other hand, the second fundamental form (SFF) cannot be measured without an immersion ... a field along the surface normal is a must. Normal curvature changes in isometry. coefficients of L,N and M change but not its determinant $(LN-M^2)$.

In the Euler identity: $ k_n = k_1 \cos^2 \psi + k_2 \sin^2 \psi $

it should be noted $\psi$ and product of $k_1 k_2=K$ are embeddable, but individually $ k_n, k_1 , k_2 $ are creatures of exterior immersion !

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