Projective
Structures| Projective
Structures |
Projective Geometry > s.a. [geometry]; statistical
mechanics.
* Idea: The allowed transformations are projections, more general
than the Euclidean group.
$ Def: If V is
a vector space of dimension n + 1 over K,
PG(n, K) is an incidence
structure of subspaces of dimension m,
0 < m < n,
which are (m+1)-dimensional subspaces of V; Incidence is
inclusion.
@ General texts: Veblen & Young 10; Baer 52; Busemann & Kelly
53; Coxeter 87; Samuel 88.
@ Over finite fields: Hirschfeld 79.
@ And physics: Delphenich AdP(06)gq/05
[and special relativity].
Projective Spaces > s.a. topology.
$ Projective plane P2:
The 2-sphere S2 with antipodal points identified: x 
–x;
Or the 2-ball B2 with opposite points on
the bdry identified: x –x on 
B2 =
S1.
$ Projective spaces:
The real one, RPn,
is Sn with
antipodal points identified, the space of lines through the origin of Rn+1;
The complex
one, CPn, is the
space of lines through the origin in Cn+1,
CPn = U(n+1) / [U(n) × U(1)] .
* Properties: The group
of covering transformations of Sn → Pn is
the identity and the antipodal mapping, so, for n 
2, 
1(Pn)
Z/2;
P1 is homeomorphic to S1.
@ References: Boya et al RPMP(03)mp/02 [volumes];
Isidro ht/03,
MPLA(04)ht/03
[quantization].
Projective Structure > s.a. Weyl
Space.
* Idea: A differentiable manifold with a preferred set of geodesics
(non-parametrized); It has a notion of propagation of a direction along itself,
and geodesics are
the lines along which directions are preserved.
@ References: Ehlers & Schild CMP(73) [geometry].
Projective Family / System (a.k.a. inverse system)
$ Def: Given a directed
set I,
a projective family on I is a collection {Xi | i 
I}
of objects in a category, and morphisms in that category {ij: Xi → Xj
| i j I},
such that ii =
idX_i,
and
ij 
jk
= ik.
* Relationships: In many (all?) categories each such family defines
a
projective limit.
Projective Limit (a.k.a. inverse limit) > s.a. tilings [space
of tilings].
$ Def: Given a projective
family (L, {Xs},
{ss'}),
the projective
limit is the set of "coherent" sequences,
X:= {x = {xs}_{s L} ×_{s L} Xs
| if s s'
then ss' xs'
= xs
} .
* Projection: s:
X → Xs is
defined by s x:= xs.
* Topology: If the Xs
are topological spaces, X gets a topology by declaring that
O 
X is
open iff for some s the inverse
image s–1 O is open in Xs,
or O is a union of such sets.
@ General references: Bourbaki Theory of Sets Ch III, #7.
@ Of manifolds: Ashtekar & Lewandowski; Abbati & Manià JGP(99)mp/98.
Related Topics > s.a. FRW spacetime [projective
symmetry]; group
representations [projective]; hilbert
space;
Projective Relativity.
* Projective R-module: If R is a principal ideal domain,
then it is
also free.
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25 may 2008