In differential geometry, affine differential geometry is the study of differential invariants of curves, surfaces, and higher-dimensional submanifolds under affine transformations. Its equiaffine form, often called equiaffine differential geometry, studies invariants under affine transformations that preserve a fixed volume form.

Classically, the subject concerns nondegenerate hypersurfaces in affine space. In contrast with Euclidean differential geometry, which uses a metric and a unit normal vector, equiaffine differential geometry uses the affine or Blaschke normal, the induced affine connection, the affine fundamental form, the affine shape operator, and related invariants such as the cubic form.

The name affine differential geometry reflects Klein's Erlangen program, in which geometries are studied through the invariants of transformation groups.

Affine geometry in the plane

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. Many properties are only invariant under an area-preserving affine transformation. These properties are equi-affine, not affine. However, equiaffine properties are essentially affine properties with a fixed length scale, which allows us to convert between equiaffine geometry and affine geometry. For example, since the equiaffine arclength of a strictly convex and smooth curve well-defined, the affine arclength-ratio between two strictly convex and smooth curves is also well-defined.

For the rest of the section, we only consider curves that have continuous 4th derivatives.

Affine normal

An affine normal line of a parabola is its axis.The affine normal line through a strictly convex point on a curve is the axis of the osculating parabola. Equivalently, it is constructed by midpoints.

. , and its curving is determined by higher-order coefficients, etc. By an affine transformation of the parabola, the axis of the parabola can be matched to the curve.

The envelope curve of the affine normals is the affine focal set. The affine focal set of a parabola is the point at infinity of its axis. Similarly, for both the ellipse and the hyperbola, the affine focal set degenerates to a single point. In general, the affine focal set is a single point iff the curve is a conic section.

Note that in affine geometry, the focus of a conic section is not a meaningful concept, since it is not invariant under equiaffine transformation. Therefore, when the word "focus" is used, it refers to the affine focal set.

In general

Similarly, for a smooth surface in 3-space, consider its Dupin indicatrix at a point as we shift its tangent plane. If the point is an elliptic point, then the tangent plane cuts a series of ellipses, and the affine normal line is spanned by the centers of the ellipses. Similarly if it is a hyperbolic point. This generalizes to a hypersurface in n-space.

Line elements

We define a line element is only defined up to a multiplication by a nonzero real number.

, there exists an equiaffine transformation mapping one to the other. Thus, many constructions using tangent lines in Euclidean differential geometry can be translated to constructions in affine differential geometry using osculating parabolas instead.

Through any 4 lines in general position.

is unique up to a sign.

.

span a parallelogram.

We will subsequently call this the equiaffine parameterization.

Equiaffine arclength

is merely conventional to make the equations appear simpler.

Therefore, we can speak of an equiaffine arclength on a strictly convex curve by integrating over the infinitesimal triangles formed by its successive tangent lines, which then allows us to define an equiaffine parameterization.

.

Additivity on a parabola

.

We see that the equiaffine arclength is invariant under subdivisions of the same parabola, meaning the equiaffine arclength converges to a well-defined limit under increasingly fine subdivisions.

.

is an equiaffine parameterization.

Equiaffine curvature

The equiaffine arclength construction relies on approximating a curve as a sequence of osculating parabolas. Obviously, osculating parabolas of a parabola are just the parabola itself. However, for a generic strictly convex curve, its osculating parabolas change over the arclength. This can be used to define curvature in equiaffine geometry.

, so it has the natural parametrization rotated clockwise by a right angle.

, so it has the natural parametrization is the equiaffine curvature.

. This shows that the parabola is analogous to the straight line, since it has zero curvature.

is its area.

The Euclidean curvature of a curve at a point is obtained by finding the unique degree-3 osculating circle. Similarly, the affine curvature is obtained by finding the unique degree-5 osculating conic section.

Where the equiaffine curvature reaches a generic local maximum or minimum, the corresponding focal point reaches a semicubical cusp.

Affine geometry in space

.

Normal form

Or in cylindrical coordinates.

plane, with the two axes of the Dupin indicatrix directions. In other words, locally the surface is a paraboloid modified by a monkey saddle.

is a well-defined quantity in equiaffine geometry.

. This is the corresponding equiaffine sectional curvature. For each unit vector is the Transon plane corresponding to the line element. As the unit vector rotates, it sweeps out a 1-parameter family of Transon planes. The envelope surface of these planes is a deltoid curve, and has an in-ellipse. This is related to the focal surface of the elliptical umbilic.

. On the deltoid cone, 6 points are special: the 3 cusps, and the 3 tangents to its in-ellipse. They correspond to 6 Transon planes, which correspond to the 3 Darboux tangents (for the cusps), and the 3 Segre tangents.

. The intersection looks like an umbrella with 3 pairs of spines, which are the 3 Darboux tangents.

Blaschke metric

is well-defined in equiaffine geometry. This is the Blaschke metric, or Berwald–Blaschke metric.

Given a curve on the surface, integration of the Blaschke metric along it produces the same result as the previously constructed equiaffine arclength. Thus, the metric structure of equiaffine geometry of the surface is the same as the metric structure of a Riemannian geometry, though this does not reduce equiaffine geometry to Riemannian geometry (discussed below).

.

The immersed manifold perspective

, regarded as an affine manifold.

Affine fundamental form

In the case of surfaces in Euclidean space, we have two fundamental forms: the first fundamental form and the second fundamental form to two subspaces: the tangent plane of the surface, and the line perpendicular to the tangent plane.

is the affine fundamental form.

into a Riemannian manifold again.

There is also an affine shape operator.

is the Blaschke connection. Abstractly speaking, this is an immersion of affine manifolds.

vector at that point. When this is used, we have an immersion of equiaffine manifolds.

Many equations in the differential geometry of surfaces, such as the Gauss–Codazzi equations, has direct analogs in affine geometry using the affine shape operator and the affine fundamental form.

At certain points, the shape operator may be an identity. Such points are umbilical points. An affine manifold where all points are umbilical points is an affine sphere.

Blaschke structure

is in general different, the cubic form is a quadric.

. Then it produces a conjugate affine connection.

The abstract manifold perspective

Like how Riemannian geometry is the generalization of the Euclidean differential geometry of curves and surfaces, affine differential geometry is the generalization of the affine differential geometry of curves and surfaces.

Here we consider the simplest case, i.e. manifolds of codimension be an transverse denotes the direct sum 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 inflexionss. In the case of surfaces in 3-space, the non-degenerate surfaces are those without parabolic pointss.

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.

Structures

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For convenience, we always assume the manifold is path-connected, since if it is not, then we can study each component separately.

Flat affine geometry

regarded as an affine geometry. It is flat, that is, with curvature zero.

There is only one k-simplex, one k-ellipsoid, and one k-parallelopiped.

There is no length of a line segment, but the length ratio of two directed line segments on the same line is well-defined. Similarly, the separation between two parallel planes is not well-defined, but the ratio of directed separations is well-defined.

The centroid of a compact subset is well-defined.

Geodesics

An affine manifold has an affine connection (by default assumed to be torsion-free. It allows defining many structures that appears also in Riemannian geometry. First, covariant derivative or parallel transport, and is essentially equivalent to the connection itself. This then allows geodesics. Every geodesic has an affine parameterization that is unique up to an affine change of variables.

The affine parameterization. Note that it is meaningless in general to speak of a constant-speed curve when the curve is not a geodesic, because we cannot compare the length-ratio of two vectors not lying on the same affine line.

A geodesic is complete is geodesically complete is essentially geodesically incomplete.

defines the exponential map.

Killing

An affine Killing vector field is the Lie derivative with the Lie bracket makes for a Lie algebra.

A vector field is complete to itself.

is affine Killing complete iff every affine Killing vector field is complete.

. It is a finite-dimensional Lie group. Its Lie algebra consisting of the affine Killing complete vector fields.

Morphisms

An affine diffeomorphism is a diffeomorphism that preserves the affine connection. A local affine diffeomorphism is an affine diffeomorphism restricted on only an open subset.

An affine covering map restricts to an affine diffeomorphism.

An affine manifold is (locally) symmetric. For example, any (locally) symmetric pseudo-Riemannian space is automatically (locally) symmetric as an affine manifold.

Intuitively, a locally symmetric manifold is one that, if one stands at any point of it, and looks out from it, one cannot discover where one is (unless one looks so far that global geometry comes to play). Similarly, a symmetric manifold is one that, if one stands at any point of it, and looks out from it, one cannot discover where one is, no matter how far one looks.

, it is totally geodesic. For such manifolds, the affine connection on the bigger manifold is an affine connection on the submanifold, so the submanifold is an affine manifold in a natural way.

.

Immersion

. In the Riemannian case, where a second fundamental form. In the affine case, we can similarly define affine fundamental forms, though there is no longer a unique natural choice.

can be interpreted as a choice of transverse directions to the immersion. There is no natural choice unlike in Riemannian geometry, since perpendicularity is undefined.

An affine fundamental form, plus a correction in a transverse direction.

is an affine immersion iff there exists such an affine fundamental form.

An affine immersion is totally geodesic iff its affine fundamental form is zero.

Coordinates

Given a local coordinate chart spanning the tangent bundle. The Christoffel symbolsIn Riemannian geometry, the normal coordinates causes the Christoffel symbols of the Levi-Civita connection.

This construction requires torsion-freeness, because normal coordinates exist iff the torsion is zero. This is one reason affine geometry usually assumes that the affine connection is torsion-free.

Instead of using coordinates, one can also use the vielbein formalism.

Curvature

In affine geometry, as in Riemannian geometry, curvature encodes all local features of affine geometry. Specifically, there is a (1, 3)-tensor called the curvature tensorSimilarly, there is a (0, 2)-tensor obtained by contracting the tensor, called the Ricci curvature tensorthough unlike in Riemannian geometry, the Ricci curvature tensor could be asymmetric. Consequently, it splits to a symmetric and antisymmetric part.

An affine manifold is flat iff its curvature tensor is zero. Similarly for Ricci-flat.

if and only if there exists a volume form invariant under parallel transport.

at either end, then there is no way to extend it further along that end, and thus it is essentially incomplete (rather than incidentally).

Statements:

• is locally symmetric.

• can be interpreted as a pseudo-Riemannian form.

• . Thus, in this case, the affine manifold is canonically just a locally symmetric pseudo-Riemannian manifold.

• the geodesic symmetry is an affine diffeomorphism. Note that the geodesic symmetry generalizes point reflection.

is metrizable.

into a Riemannian manifold. But the Levi-Civita connection for the Blaschke metric may not be the Blaschke connection. Indeed, if the surface is strictly convex but not a quadratic surface, then they are not the same connection.

Affine surfaces

has 2 dimensions, it is an affine surface. Affine surfaces have special properties.

might not be a pseudo-Riemannian manifold.

As for Riemannian surfaces, an affine surface is flat iff it is Ricci-flat.

Problems

Some basic problems in affine differential geometry are:

• whether an affine manifold is metrizable;

• whether an affine manifold can immerse in another affine manifold;

• whether an affine manifold can be geodesically completed;

• classifying affine spheres;

• classifying affine maximal surfaces;

• global structures, especially in its interactions with algebraic geometry;

• etc.

See also

• Affine geometry of curves

• Affine focal set

• Affine sphere

• Affine maximal surface

References