Affine differential geometry

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

The basic difference between affine and Riemannian differential geometry is that affine differential geometry studies manifolds equipped with a volume form rather than a metric.

Notice that ∇ and h depend upon the choice of transverse vector field ξ.

This is a property of the hypersurface M and does not depend upon the choice of transverse vector field ξ.

Again, both S and τ depend upon the choice of transverse vector field ξ.

Notice that ω depends on the choice of transverse vector field ξ.

For tangent vectors X1,...,Xn let H := (hi,j) be the n × n matrix given by hi,j := h(Xi,Xj).

If M = Rn and h is the Euclidean scalar product then ν(X1,...,Xn) is always the standard Euclidean volume spanned by the vectors X1,...,Xn.

Since h depends on the choice of transverse vector field ξ it follows that ν does too.

In other words, if we parallel transport the vectors X1,...,Xn along some curve in M, with respect to the connexion ∇, then the volume spanned by X1,...,Xn, with respect to the volume form ω, does not change.

This means that the derivative of ξ, in a tangent direction X, with respect to D always yields a, possibly zero, tangent vector to M. The second condition is that the two volume forms ω and ν coincide, i.e. ω ≡ ν.

It can be shown[1] that there is, up to sign, a unique choice of transverse vector field ξ for which the two conditions that ∇ω ≡ 0 and ω ≡ ν are both satisfied.

[2] From its dependence on volume forms for its definition we see that the affine normal vector field is invariant under volume preserving affine transformations.

These transformations are given by SL(n+1,R) ⋉ Rn+1, where SL(n+1,R) denotes the special linear group of (n+1) × (n+1) matrices with real entries and determinant 1, and ⋉ denotes the semi-direct product.

The affine normal vector field for a curve in the plane has a nice geometrical interpretation.

[2] Let I ⊂ R be an open interval and let γ : I → R2 be a smooth parametrisation of a plane curve.

We assume that γ(I) is a non-degenerate curve (in the sense of Nomizu and Sasaki[1]), i.e. is without inflexion points.

In that case the affine normal line to the curve at γ(0) has the equation x = 2y.

A similar analogue exists for finding the affine normal line at elliptic points of smooth surfaces in 3-space.