Affine differential geometry
Encyclopedia
Affine differential geometry, is a type of differential geometry in which the differential invariants are invariant under volume-preserving affine transformation
s. 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 form
s over a manifold instead of metric
s.
s of codimension
one. Let be an n-dimensional manifold, and let ξ be a transverse
vector field such that for all 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 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 hypersurface
s for which h is non-degenerate. Interestingly, this is a property of the hypersurface M and 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 inflexion
s. 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, 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.
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.
Since h depends on the choice of transverse vector field ξ it follows that ν does too.
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.
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 transformation
s. 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
.
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 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 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 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 . This has the equation The tangent line at γ(0) has the equation and so the parallel lines are given by for sufficiently small The line intersects the curve at The locus of midpoints is given by 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 In that case the affine normal line to the curve at γ(0) has the equation In fact, direct calculation shows that the affine normal vector at γ(0), namely ξ(0), is given by 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 transformation
In geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...
s. The name affine differential geometry follows from Klein
Felix Klein
Christian Felix Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...
's Erlangen program
Erlangen program
An influential research program and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen...
. The basic difference between affine and Riemannian
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...
differential geometry is that in the affine case we introduce volume form
Volume form
In mathematics, a volume form on a differentiable manifold is a nowhere-vanishing differential form of top degree. Thus on a manifold M of dimension n, a volume form is an n-form, a section of the line bundle Ωn = Λn, that is nowhere equal to zero. A manifold has a volume...
s over a manifold instead of metric
Metric (mathematics)
In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...
s.
Preliminaries
Here we consider the simplest case, i.e. manifoldManifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
s of codimension
Codimension
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, and also to submanifolds in manifolds, and suitable subsets of algebraic varieties.The dual concept is relative dimension.-Definition:...
one. Let be an n-dimensional manifold, and let ξ be a transverse
Transversality
In mathematics, transversality is a notion that describes how spaces can intersect; transversality can be seen as the "opposite" of tangency, and plays a role in general position. It formalizes the idea of a generic intersection in differential topology...
vector field such that for all where ⊕ denotes the direct sum and Span the linear span
Linear span
In the mathematical subfield of linear algebra, the linear span of a set of vectors in a vector space is the intersection of all subspaces containing that set...
.
For a smooth manifold, say N, let Ψ(N) denote the module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
of smooth vector fields over N. Let be the standard covariant derivative
Covariant derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given...
on Rn+1 where
We can decompose DXY into a component tangent
Tangent space
In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....
to M and a transverse component, parallel
Parallel (geometry)
Parallelism is a term in geometry and in everyday life that refers to a property in Euclidean space of two or more lines or planes, or a combination of these. The assumed existence and properties of parallel lines are the basis of Euclid's parallel postulate. Two lines in a plane that do not...
to ξ. This gives the equation of Gauss
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
: where is the induced connexion
Connection (mathematics)
In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner. There are a variety of kinds of connections in modern geometry, depending on what sort of data one wants to transport...
on M and is a bilinear form. Notice that ∇ and h depend upon the choice of transverse vector field ξ. We consider only those hypersurface
Hypersurface
In geometry, a hypersurface is a generalization of the concept of hyperplane. Suppose an enveloping manifold M has n dimensions; then any submanifold of M of n − 1 dimensions is a hypersurface...
s for which h is non-degenerate. Interestingly, this is a property of the hypersurface M and 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 inflexion
Inflection point
In differential calculus, an inflection point, point of inflection, or inflection is a point on a curve at which the curvature or concavity changes sign. The curve changes from being concave upwards to concave downwards , or vice versa...
s. 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, DXξ, can be decomposed into a component tangent to M and a transverse component, parallel to ξ. This gives the Weingarten
Julius Weingarten
Julius Weingarten was a German mathematician. He made some important contributions to the differential geometry of surfaces, such as theWeingarten equations.-External links:...
equation: The type-(1,1)-tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
is called the affine shape operator, the differential one-form
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
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 formVolume form
In mathematics, a volume form on a differentiable manifold is a nowhere-vanishing differential form of top degree. Thus on a manifold M of dimension n, a volume form is an n-form, a section of the line bundle Ωn = Λn, that is nowhere equal to zero. A manifold has a volume...
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
Surface normal
A surface normal, or simply normal, to a flat surface is a vector that is perpendicular to that surface. A normal to a non-flat surface at a point P on the surface is a vector perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a...
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 scaler 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 transportParallel transport
In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection , then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the...
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 BlaschkeWilhelm Blaschke
Wilhelm Johann Eugen Blaschke was an Austro-Hungarian differential and integral geometer.His students included Shiing-Shen Chern, Luis Santaló, and Emanuel Sperner....
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 transformation
Affine transformation
In geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...
s. These transformations are given by where SL(n+1,R) denotes the special linear group
Special linear group
In mathematics, the special linear group of degree n over a field F is the set of n×n matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion....
of matrices with real entries and determinant 1, and ⋉ denotes the semi-direct product. forms a Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
.
The affine normal line
The affine normal line at a point is the line passing through p and parallel to ξ.Plane curves
The affine normal vector field for a curve in the plane has a nice geometrical interpretation. Let be an open interval and let be a smoothSmooth function
In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...
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 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 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 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
Midpoint
The midpoint is the middle point of a line segment. It is equidistant from both endpoints.-Formulas:...
of the line segment
Line segment
In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment...
joining these two intersection points. For each parallel line we get a midpoint, and so the locus
Locus (mathematics)
In geometry, a locus is a collection of points which share a property. For example a circle may be defined as the locus of points in a plane at a fixed distance from a given point....
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
Parabola
In mathematics, the parabola is a conic section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface...
given by the parametrisation . This has the equation The tangent line at γ(0) has the equation and so the parallel lines are given by for sufficiently small The line intersects the curve at The locus of midpoints is given by 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 In that case the affine normal line to the curve at γ(0) has the equation In fact, direct calculation shows that the affine normal vector at γ(0), namely ξ(0), is given by 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 sphereAffine sphereIn mathematics, and especially differential geometry, an affine sphere is a hypersurface for which the affine normals all intersect in a single point...