Gauss's Theorema Egregium
Gauss's theorema egregium states that the Gaussian Curvature of a surface embedded in 3-space may beunderstood intrinsically to that surface. ``Residents'' of the surface may observe the Gaussian Curvature of thesurface without ever venturing into full 3-dimensional space; they can observe the curvature of the surface they live inwithout 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 RadiusCircles correspond to what they should be in Euclidean Space, 
. If the Arc Length ofCircles tends to be smaller than what is expected in Euclidean Space, then the space is positivelycurved; if larger, negatively; if the same, 0 Gaussian Curvature.
Gauß 
(effectively) expressed the theorema egregium by saying that the Gaussian Curvature at a point isgiven by 
where 
is the Riemann Tensor, and 
and 
are an orthonormal basis for the TangentSpace.
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.
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