Affine differential geometry

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 $M\subset {ℝ}^{n+1}$ be an $n$-dimensional manifold, and let $\xi$ be a vector field on ${ℝ}^{n+1}$ transverse to $M$ such that $T_p{ℝ}^{n+1}=T_{p}M\oplus \text{Span}(\xi )$ for all $p\in M$ where $\oplus$ denotes the direct sum and $\text{Span}$ the linear span.

For a smooth manifold, say N, let Ψ(N) denote the module of smooth vector fields over N. Let Template:Nowrap be the standard covariant derivative on Rⁿ⁺¹ where Template:Nowrap We can decompose D_XY into a component tangent to M and a transverse component, parallel to ξ. This gives the equation of Gauss: Template:Nowrap where Template:Nowrap is the induced connexion on M and Template:Nowrap 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, D_Xξ, can be decomposed into a component tangent to M and a transverse component, parallel to ξ. This gives the Weingarten equation: Template:Nowrap The type-(1,1)-tensor Template:Nowrap is called the affine shape operator, the differential one-form Template:Nowrap 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 Template:Nowrap be a volume form defined on Rⁿ⁺¹. We can induce a volume form on M given by Template:Nowrap given by Template:Nowrap 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.

The second induced volume form

For tangent vectors X₁,...,X_n let Template:Nowrap be the Template:Nowrap given by Template:Nowrap We define a second volume form on M given by Template:Nowrap where Template:Nowrap 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.

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 Template:Nowrap for all Template:Nowrap 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 shows that Template:Nowrap and so Template:Nowrap for all Template:Nowrap if, and only if, τ ≡ 0, i.e. Template:Nowrap for all Template:Nowrap 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. Template:Nowrap

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 Template:Nowrap and Template:Nowrap 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 Template:Nowrap where SL(n+1,R) denotes the special linear group of Template:Nowrap matrices with real entries and determinant 1, and ⋉ denotes the semi-direct product. Template:Nowrap forms a Lie group.

The affine normal line

The affine normal line at a point Template:Nowrap is the line passing through p and parallel to ξ.

Plane curves

Affine normal line for the curve Template:Nowrap at Template:Nowrap

The affine normal vector field for a curve in the plane has a nice geometrical interpretation. Let Template:Nowrap be an open interval and let Template:Nowrap 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 Template:Nowrap 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 Template:Nowrap 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 Template:Nowrap 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 Template:Nowrap. This has the equation Template:Nowrap The tangent line at γ(0) has the equation Template:Nowrap and so the parallel lines are given by Template:Nowrap for sufficiently small Template:Nowrap The line Template:Nowrap intersects the curve at Template:Nowrap The locus of midpoints is given by Template:Nowrap 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 Template:Nowrap In that case the affine normal line to the curve at γ(0) has the equation Template:Nowrap In fact, direct calculation shows that the affine normal vector at γ(0), namely ξ(0), is given by Template:Nowrap 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.

Surfaces in 3-space

A similar analogue exists for finding the affine normal line at elliptic points of smooth surfaces in 3-space. This time one takes planes parallel to the tangent plane. These, for planes sufficiently close to the tangent plane, intersect the surface to make convex plane curves. Each convex plane curve has a centre of mass. The locus of centres of mass trace out a curve in 3-space. The limiting tangent line to this locus as one tends to the original surface point is the affine normal line, i.e. the line containing the affine normal vector.

See also

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