A huge problem that many people initially have with differential geometry is over-representation. Manifolds are conventionally defined via absurdly large, Zorn’s-lemma’d maximal atlases, giving an enormous sea of possible coordinates which could represent an object. Maps with some property from one surface to another are often defined relative to local charts, involving many arbitrary choices of coordinates, covers, and so forth. Tensor fields might be built up as collections of numbers with absurd transformation rules. In that mass of symbols, where is the geometry?
Conversely, who would expect upon first inspection that the trace and the determinant of a matrix are coordinate-invariant quantities? What about the Riemann curvature tensor? Would you have seen that the Gaussian curvature is intrinsic, yet the mean curvature is extrinsic? Can you identify the Willmore energy as invariant under Mobius transformations, despite being constructed from Euclidean invariants? What if it was written in local coordinates?
This overabundance of representations in differential geometry is a mixed blessing. They give us a thousand ways to look at a problem, but 995 of those obscure the geometric features we are trying to understand. Ideally, we would like to have a language for geometry which is rich enough that we can efficiently describe and compute with geometric objects, yet weak enough that we cannot state non-geometric facts. This might be a bit much to ask in general, but many big advances in differential geometry have come through finding clever representations for certain classes of geometric objects which allow us to easily read off or insert geometric information. At Greg’s request, I’d like to share a modern version of one of the classical representation theorems in the geometry of surfaces.