Affine differential geometry is a type of differential geometry which studies invariants of volume-preserving affine transformations. 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.
Preliminaries
Here we consider the simplest case, i.e. manifolds of codimension one. Let be an n-dimensional manifold, and let ξ be a vector field on transverse to such that for all where ⊕ denotes the direct sum and Span the linear span.
For a smooth manifold, say N, let Ψ(N) denote the module of smooth vector fields over N. Let be the standard covariant derivative on Rn+1 where We can decompose DXY into a component tangent to M and a transverse component, parallel to ξ. This gives the equation of Gauss: where is the induced connexion on M and is a bilinear form. Notice that ∇ and h depend upon the choice of transverse vector field ξ. We consider only those hypersurfaces for which h is non-degenerate. This is a property of the hypersurface M and does not depend upon the choice of transverse vector field ξ. If h is non-degenerate then we say that M is non-degenerate. In the case of curves in the plane, the non-degenerate curves are those without inflexions. In the case of surfaces in 3-space, the non-degenerate surfaces are those without parabolic points.
We may also consider the derivative of ξ in some tangent direction, say X. This quantity, DXξ, can be decomposed into a component tangent to M and a transverse component, parallel to ξ. This gives the Weingarten equation: The type-(1,1)-tensor is called the affine shape operator, the differential one-form is called the transverse connexion form. Again, both S and τ depend upon the choice of transverse vector field ξ.
The first induced volume form
Let be a volume form defined on Rn+1. We can induce a volume form on M given by given by This is a natural definition: in Euclidean differential geometry where ξ is the Euclidean unit normal then the standard Euclidean volume spanned by X1,...,Xn is always equal to ω(X1,...,Xn). Notice that ω depends on the choice of transverse vector field ξ.
The second induced volume form
For tangent vectors X1,...,Xn let be the given by We define a second volume form on M given by where Again, this is a natural definition to make. 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.
Two natural conditions
We impose two natural conditions. The first is that the induced connexion ∇ and the induced volume form ω be compatible, i.e. ∇ω ≡ 0. This means that for all 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. A direct calculation shows that and so for all if, and only if, τ ≡ 0, i.e. for all 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.
The conclusion
It can be shown that there is, up to sign, a unique choice of transverse vector field ξ for which the two conditions that and are both satisfied. These two special transverse vector fields are called affine normal vector fields, or sometimes called Blaschke normal fields. 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 where SL(n+1,R) denotes the special linear group of matrices with real entries and determinant 1, and ⋉ denotes the semi-direct product. forms a Lie group.
The affine normal line
The affine normal line at a point is the line passing through p and parallel to ξ.
Plane curves
The affine normal vector field for a curve in the plane has a nice geometrical interpretation. Let be an open interval and let 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), i.e. is without inflexion points. Consider a point on the plane curve. Since γ(I) is without inflexion points it follows that γ(t0) is not an inflexion point and so the curve will be locally convex, i.e. all of the points γ(t) with for sufficiently small ε, will lie on the same side of the tangent line to γ(I) at γ(t0).