Gauss's Theorema Egregium

Gauss's theorema egregium states that the Gaussian Curvature of a surface embedded in 3-space may be understood intrinsically to that surface. ``Residents'' of the surface may observe the Gaussian Curvature of the surface without ever venturing into full 3-dimensional space; they can observe the curvature of the surface they live in without even knowing about the 3-dimensional space in which they are embedded.

In particular, Gaussian Curvature can be measured by checking how closely the Arc Length of small Radius Circles correspond to what they should be in Euclidean Space, $2\pi r$. If the Arc Length of Circles tends to be smaller than what is expected in Euclidean Space, then the space is positively curved; if larger, negatively; if the same, 0 Gaussian Curvature.

Gauß (effectively) expressed the theorema egregium by saying that the Gaussian Curvature at a point is given by $-R(v,w)v,w,$ where $R$ is the Riemann Tensor, and $v$ and $w$ are an orthonormal basis for the Tangent Space.

See also Christoffel Symbol of the Second Kind, Gauss Equations, Gaussian Curvature

References

Gray, A. ``Gauss's Theorema Egregium.'' §20.2 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 395-397, 1993.

Reckziegel, H. In Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, pp. 31-32, 1986.