Theorema Egregium

Theorem
The Gaussian curvature of a surface does not change if one bends the surface without stretching it.

That is, Gaussian curvature can be determined entirely by measuring angles, distances and their rates on the surface itself, without further reference to the particular way in which the surface is embedded in the ambient $3$-dimensional Euclidean space.

Thus the Gaussian curvature is an intrinsic invariant of a surface.