Gauss's theorema egregium states that the Gaussian curvature of a surface embedded in three-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 three-dimensional space; they can observe the curvature of the surface they live in without even knowing about the three-dimensional space in which they are embedded.
In particular, Gaussian curvature can be measured by checking how closely the arc lengths of circles
of small radii correspond to what they should be in Euclidean space, 
. 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 
where 
is the Riemann tensor, and
and 
are an orthonormal basis for the tangent space.
