When Gauss defined the curvature of a surface as the rate of change of its normal direction, he made explicit use of the way the surface sits in space. Evidently, this is the extrinsic “curvature as bending” rather than the intrinsic “curvature as stretching”that we argued in Section 4.2 must be the basis of general relativity. It is altogether remarkable, then, that Gauss was able to prove that curvature is intrinsic. We begin this chapter by analyzing Gauss’s famous argument, the theorema egregium, that curvature can be determined from a knowledge of the metric tensor alone, without reference to the surface’s embedding in space.
KeywordsTangent Space Covariant Derivative Tangent Vector Coordinate Frame Tangent Plane
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