Affine differential geometry

Summary

Affine differential geometry studies curves and surfaces through the properties that do not change under volume-preserving affine maps of space. Lengths and angles are not preserved by such maps, so the Euclidean curvature of a curve or surface is not an invariant here. Instead the theory builds its own invariants. Examples are the affine arc length of a curve and the affine normal and Blaschke metric of a hypersurface. It is an instance of the idea that each group of transformations has its own geometry.