Affine Differential Geometry - Preliminaries

Preliminaries

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 T_pRn+1 = T_pM ⊕ 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) = D_XY. We can decompose D_XY into a component tangent to M and a transverse component, parallel to ξ. This gives the equation of Gauss: D_XY = ∇_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.

We may also consider the derivative of ξ in some tangent direction, say X. This quantity, D_Xξ, can be decomposed into a component tangent to M and a transverse component, parallel to ξ. This gives the Weingarten equation: D_Xξ = −SX + τ(X)ξ. The type-(1,1)-tensor S : Ψ(M) → Ψ(M) is called the affine shape operator, the differential one-form τ : Ψ(M) → R is called the transverse connexion form. Again, both S and τ depend upon the choice of transverse vector field ξ.