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 P__^{2}:
The 2-sphere S^{2} with antipodal points identified, *x* ~ –*x*;
Or the 2-ball B^{2} with opposite points on
the boundary identified, *x* ~ –*x* on ∂B^{2} =
S^{1}.

$ __Projective spaces__:
The real one, \(\mathbb R\)P^{n},
is S^{n} with
antipodal points identified, the space of lines through the origin of \(\mathbb R\)^{n+1};
The complex one, \(\mathbb C\)P^{n}, is the
space of lines through the origin in \(\mathbb C\)^{n+1},

\(\mathbb C\)P^{n} = U(*n*+1) / [U(*n*) ×
U(1)] .

* __Properties__: The group
of covering transformations of S^{n} → P^{n} is
the identity and the antipodal mapping, so, for *n* ≥ 2, *π*_{1}(P^{n}) ≅ \(\mathbb Z\)/2;
P^{1} is homeomorphic to S^{1}.

@ __References__: Boya et al RPMP(03)mp/02 [volumes];
Isidro ht/03,
MPLA(04)ht/03 [quantization].

**Projective Structure** > s.a. 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 {*X*_{i} | *i* ∈ *I*}
of objects in a category, and morphisms in that category {*π*_{ij}: *X*_{i }→ *X*_{j}
| *i* ≥ *j* ∈ *I*},
such that *π*_{ii} =
id_{X_i},
and *π*_{ij} \(\circ\) *π*_{jk}
= *π*_{ik}.

* __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*, {*X*_{s}},
{*π*_{ss'}}), the projective
limit is the set of "coherent" sequences,

**X**:= {**x** = {*x*_{s}}_{s ∈ L} ∈ ×_{s ∈ L}* X*_{s}
| if *s* ≥ *s*'
then *π*_{ss'}* x*_{s'}
= *x*_{s}
} .

* __Projection__: *π*_{s}:
**X** → *X*_{s} is
defined by *π*_{s}* x*:=

*

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@

**Related Topics** > s.a. FLRW spacetime [projective
symmetry]; group
representations [projective]; hilbert space;
Projective Relativity.

* __Projective R-module__: If

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