Theorema Egregium

Statement

Gauss's Theoreman Egregium or Gauss's Remarkable Theorem states that the Gaussian curvature of a surface embedded in $\mathbb {R} ^{3}$ is invariant under isometries. In other words, the Gaussian curvature is intrinsic to the geometry of the surface and is independent of the manner in which the surface is embedded in $\mathbb {R} ^{3}$.