Affine differential geometry
Affine differential geometry, is a type of differential geometry in which the differential invariants are invariant under 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 in the affine case we introduce volume forms over a manifold instead of metrics.
Here we consider the simplest case, i.e. manifolds of codimension one. Let M ⊂ Rn+1 be an n-dimensional manifold, and let ξ be a vector field on Rn+1 transverse to M such that TpRn+1 = TpM ⊕ Span(ξ) for all p ∈ M, 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 D : Ψ(Rn+1) × Ψ(Rn+1) → Ψ(Rn+1) be the standard covariant derivative on Rn+1 where D(X, Y) = DXY. We can decompose DXY into a component tangent to M and a transverse component, parallel to ξ. This gives the equation of Gauss: DXY = ∇XY + h(X,Y)ξ, where ∇ : Ψ(M) × Ψ(M) → Ψ(M) is the induced connexion on M and h : Ψ(M) × Ψ(M) → R 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. Interestingly, 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.
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