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. 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
{*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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