Affine
Structures| Affine
Structures |
Affine Space > s.a. vector
space.
$ Def: An affine space
of dimension n over R (or
a vector space V) is a
set E on which the additive group Rn (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 Rn (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 
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.
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, 
x +
(1–) y A, [0,1]
.
* Support function:
For a convex subset K of Euclidean space bounded
by 
,
the support function p(
)
gives the distance p to
the
origin of the line tangent to the curve with
angle wrt
the x axis.
* Area of convex 2D set: For a convex K R2,
A(K) = 
bdry(K) p()
ds = 02
pi (p2 – p' 2)
d .
* Star-convex subset: A subset A X
is star-convex relative to
x0 X iff x A,
x + (1–)
x0 A, [0,1];
Can be seen as the union of segments with one endpoint at x0; 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 Rn,
a composition x' = A x + a of
nonsingular 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 > s.a. Affine
Gravity.
$ Geometrically independent
points: The points a0, ..., an Rn 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].
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send feedback and suggestions to bombelli at olemiss.edu – modified
2 may 2008