Affine differential geometry

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. This is a property of the hypersurface M and does not depend upon the choice of transverse vector field ξ.[1] 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.

Let Ω : Ψ(Rⁿ⁺¹)ⁿ⁺¹ → R be a volume form defined on Rⁿ⁺¹. We can induce a volume form on M given by ω : Ψ(M)ⁿ → R given by ω(X₁,...,X_n) := Ω(X₁,...,X_n,ξ). This is a natural definition: in Euclidean differential geometry where ξ is the Euclidean unit normal then the standard Euclidean volume spanned by X₁,...,X_n is always equal to ω(X₁,...,X_n). Notice that ω depends on the choice of transverse vector field ξ.

For tangent vectors X₁,...,X_n let H := (h_i,j) be the n × n matrix given by h_i,j := h(X_i,X_j). We define a second volume form on M given by ν : Ψ(M)ⁿ → R, where ν(X₁,...,X_n) := |det(H)|^1⁄2. Again, this is a natural definition to make. If M = Rⁿ and h is the Euclidean scalar product then ν(X₁,...,X_n) is always the standard Euclidean volume spanned by the vectors X₁,...,X_n. Since h depends on the choice of transverse vector field ξ it follows that ν does too.

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 ∇_Xω = 0 for all X ∈ Ψ(M). In other words, if we parallel transport the vectors X₁,...,X_n along some curve in M, with respect to the connexion ∇, then the volume spanned by X₁,...,X_n, with respect to the volume form ω, does not change. A direct calculation[1] shows that ∇_Xω = τ(X)ω and so ∇_Xω = 0 for all X ∈ Ψ(M) if, and only if, τ ≡ 0, i.e. D_Xξ ∈ Ψ(M) for all X ∈ Ψ(M). 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. These two special transverse vector fields are called affine normal vector fields, or sometimes called Blaschke normal fields.[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. SL(n+1,R) ⋉ Rⁿ⁺¹ forms a Lie group.

The affine normal line at a point p ∈ M is the line passing through p and parallel to ξ.

Plane curves

Affine normal line for the curve γ(t) = (t + 2t²,t²) at t = 0.

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. Consider a point p = γ(t₀) 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,[3] i.e. all of the points γ(t) with t₀ − ε < t < t₀ + ε, for sufficiently small ε, will lie on the same side of the tangent line to γ(I) at γ(t₀).

Consider the tangent line to γ(I) at γ(t₀), and consider near-by parallel lines on the side of the tangent line containing the piece of curve P := {γ(t) ∈ R² : t₀ − ε < t < t₀ + ε}. For parallel lines sufficiently close to the tangent line they will intersect P in exactly two points. On each parallel line we mark the midpoint of the line segment joining these two intersection points. For each parallel line we get a midpoint, and so the locus of midpoints traces out a curve starting at p. The limiting tangent line to the locus of midpoints as we approach p is exactly the affine normal line, i.e. the line containing the affine normal vector to γ(I) at γ(t₀). Notice that this is an affine invariant construction since parallelism and midpoints are invariant under affine transformations.

Consider the parabola given by the parametrisation γ(t) = (t + 2t²,t²). This has the equation x² + 4y² − 4xy − y = 0. The tangent line at γ(0) has the equation y = 0 and so the parallel lines are given by y = k for sufficiently small k ≥ 0. The line y = k intersects the curve at x = 2k ± √k. The locus of midpoints is given by {(2k,k) : k ≥ 0}. These form a line segment, and so the limiting tangent line to this line segment as we tend to γ(0) is just the line containing this line segment, i.e. the line x = 2y. In that case the affine normal line to the curve at γ(0) has the equation x = 2y. In fact, direct calculation shows that the affine normal vector at γ(0), namely ξ(0), is given by ξ(0) = 2^1⁄3·(2,1).[4] In the figure the red curve is the curve γ, the black lines are the tangent line and some near-by tangent lines, the black dots are the midpoints on the displayed lines, and the blue line is the locus of midpoints.

• Affine geometry of curves
• Affine focal set
• Affine sphere