Multidimensional generalizations [of intrinsically defined curves]
Curvatures for surfaces and higher-dimensional objects can be defined in terms of the principal axes of approximating ellipsoids at each point. There are combinations of these curvatures—in 2D Gaussian curvature and mean curvature—which are independent of the coordinate system used. (Compare page 1049.) Given such curvatures, a surface can in principle be obtained by solving certain partial differential equations. But even in the case of zero mean curvature, which corresponds to minimal surfaces of the kind followed by an idealized soap film, this is already a mathematical problem of significant difficulty.
If one looks at projections of surfaces, it is common to see lines of discontinuity at which a surface goes, say, from having three sheets to one. Catastrophe theory provides a classification of such discontinuities—the simplest being a cusp. And as emphasized by René Thom in the 1960s, it is possible that some structures seen in animals may be related to such discontinuities.