Affine Structures| Affine Structures |
Affine Space > s.a. vector space.
$ Def: An affine space of dimension
n over \(\mathbb R\) (or a vector space V) is a set E on
which the additive group \(\mathbb R\)n
(or V) acts simply transitively.
* Idea: It can be considered
as a vector space without an origin (therefore without preferred coordinates,
addition and multiplication by a scalar); If v is an element of
\(\mathbb R\)n (or V),
we can write things like P = Q + v, or v
= P − Q, but we cannot give a meaning to P
+ Q.
* Examples: Any vector space
is an affine space over itself, with composition being vector addition.
* Affine subspace: A subset
S ⊂ E of the form S = p + X,
with p ∈ E and X a vector subspace of V;
It is an affine space over X.
* Compatible topology:
A topology on E compatible with translations, in the sense that
all maps p \(\mapsto\) p + v are continuous;
Any such topology on E can be induced (as a "translated
topology") by one on V; If V is a finite-dimensional
real or complex vector space, there is a unique, natural topology on it,
and therefore on E as well, which is first countable.
@ General references: in Greenberg & Harper 81;
Kostrikin & Manin 89.
> Online resources:
see Wikipedia page.
Convex Subset of an Affine Space
$ Def: A subset A of an affine
space X is said to be convex iff the segment connecting x and y
is contained in A,
∀ x, y ∈ A, λ x + (1−λ) y ∈ A, ∀ λ ∈ [0,1] .
* Support function:
For a convex subset K of Euclidean space bounded by γ,
the support function p(φ) gives the distance p
to the origin of the line tangent to the curve γ with angle
φ with respect to the x axis.
* Area of convex 2D set: For a
convex K ⊂ \(\mathbb R\)2,
A(K) = \(1\over2\)\(\displaystyle\oint_{\partial K}\)p(φ) ds = \(1\over2\)\(\displaystyle\oint_0^{2\pi}\)(p2 − p' 2) dφ .
* Star-convex subset:
A subset A ⊂ X is star-convex relative to
x0 ∈ X iff
∀x ∈ A, λx + (1−λ)
x0 ∈ A, ∀λ ∈ [0,1];
Can be seen as the union of segments with one endpoint at x0;
It is typically star-shaped around x0.
* Properties: Any convex set is star-convex relative
to any of its points; The set of convex subsets is stable under intersections, but not under unions.
* Relationships: For any reasonable topology
(e.g., Euclidean), all convex K are sequentially connected, and all sequentially
connected K are connected.
Affine Map / Transformation
* Idea: A map that preserves
the affine structure on a space.
$ Def: In \(\mathbb R\)n,
a composition x' = A x + a of
non-singular linear transformations and translations.
* Properties: It preserves
geometrically independent points.
Affine Structure
> s.a. affine connection; differential
geometry [affine manifold]; projective structure;
Weyl Space.
$ Def: A differentiable
manifold with a preferred set of affinely parametrized geodesics –
an affine geometry at each point.
* Idea: Provides a notion of parallel
propagation of a vector along any curve – expressed by an affine connection.
* Relationships: It implies a projective structure.
Related Concepts
$ Geometrically independent points:
The points a0, ...,
an ∈ \(\mathbb R^n\) are
geometrically independent if, for any given set t0,
..., tn of scalars,
∑i ti
= 0 and ∑i ti
ai = 0 implies
ti = 0, for all i;
In RN: There
can be at most N + 1 geometrically independent points; Using n
of them as vertices, one can make a simplex; An affine transformation can take
these points into the origin and the tips of the first n unit vectors.
@ Affine collineation: Hall & da Costa JMP(88);
Hojman & Núñez JMP(91) [Riemannian manifolds].
> Physics topics:
see affine connections; Affine Gravity;
conservation laws; modified coherent states;
quantum formalism [affine variables].
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– abbreviations
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– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 7 aug 2018