In the case of a surface, in @needham2021visual the author defines it as the local area magnification factor of the Gauss map. It is Gaussian curvature#Definition 2.
In page 134 he shows that if we draw a quadrilateral following principal directions then the area magnification is
$$ K=\kappa_1 \kappa_2, $$the product of the principal curvatures. It is the determinant of the shape operator.
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Author of the notes: Antonio J. Pan-Collantes
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