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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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