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
Theorem of the Day -- 28/09/2005
Deep Theorem
Qualitatively, a deep theorem is a theorem whose proof is long, complicated, difficult, or appears to involve branches of mathematics which are not obviously related to the theorem itself (Shanks 1993). Shanks (1993) cites the quadratic reciprocity theorem as an example of a deep theorem.
(Excerpt from Mathworld)
Comments:
This theorem is actually not a theorem, more like a definition, I wonder if it's actually possible to rigorously prove if a theorem is a deep theorem.
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