Affine differential geometry

In differential geometry, affine differential geometry is the study of geometric invariants of a manifold under affine transformations, and equiaffine differential geometry is the study of geometric invariants of a manifold under volume-preserving affine transformations. Equivalently, affine differential geometry is the study of affine manifolds, which are smooth manifolds equipped with an affine connection (usually assumed torsion-free), and equiaffine differential geometry is the study of affine manifolds equipped with a nowhere-vanishing volume form.

There are many examples of affine manifolds, since any (pseudo-)Riemannian manifold is automatically an affine manifold, such as the sphere, the cylinder, the hyperboloid, etc. However, there are affine manifolds that are not induced by any (pseudo-)Riemannian manifold.

Intuitively, conformal geometry is affine geometry where angles become invariant, and affine geometry is conformal geometry where angles become flexible. Compared to Riemannian geometry, special affine geometry studies manifolds equipped with a volume form rather than a metric. Symplectic geometry is intermediate in rigidity between special affine geometry and Riemannian geometry.

The name affine differential geometry follows from Klein's Erlangen program.