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 \(\mathbb K\), PG(n,
\(\mathbb K\)) 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];
in Alhamzawi & Alhamzawi a1405 [geometrical interpretation];
Cariglia AP(15)-a1506 [natural Hamiltonian systems];
> s.a. lines [electromagnetism and projective geometry].
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 boundary identified, x ~ −x on
∂B2 = S1.
$ Projective spaces:
The real one, \(\mathbb R\)Pn, is
Sn with antipodal points identified, the space
of lines through the origin of \(\mathbb R\)n+1;
The complex one, \(\mathbb C\)Pn, is the space of
lines through the origin in \(\mathbb C\)n+1,
\(\mathbb C\)Pn = 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,
π1(Pn)
≅ \(\mathbb Z\)/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. conformal structure [compatibility]; lorentzian
geometries [projectively related]; 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; Two manifolds are projectively
related if they have the same set of unparametrized geodesics.
@ References:
Ehlers & Schild CMP(73) [geometry];
Nurowski JGP(12)-a1003 [vs metric structures];
Hall & Lonie CQG(11) [projectively related spacetimes and holonomy].
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 {πij:
Xi
→ Xj
| i ≥ j ∈ I}, such that
πii
= idX_i, and
πij
\(\circ\) πjk
= πik.
* Relationships: In
many (all?) categories each such family defines a projective limit.
Projective Limit (a.k.a. inverse limit)
> s.a. quantum field theory states;
tilings [space of tilings].
$ Def: Given a projective
family (L, {Xs},
{πss'}), the projective
limit is the set of "coherent" sequences,
X:= {x = {xs}s ∈ L ∈ ×s ∈ L Xs | if s ≥ s' then πss' xs' = xs } .
* Projection:
πs:
X → Xs
is defined by πs
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
πs−1 O
is open in Xs,
or O is a union of such sets.
@ General references: Bourbaki L1, Ch III #7.
@ Of manifolds: Ashtekar & Lewandowski JMP(95)gq/94;
Abbati & Manià JGP(99)mp/98.
Related Topics > s.a. FLRW 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.
@ Projective connections: George AIP(08)-a0808 [and algebra of densities].
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