Gauss's Theorema Egregium
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 -R(v,w)v,w where R is the Riemann
tensor, and v and w are an orthonormal basis for the tangent space.
Excerpt from MathWorld
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