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. 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 | iI} of objects in a category, and morphisms in that category {πij: Xi Xj | ijI}, 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}sL×sL Xs | if ss' then πss' xs' = xs } .

* Projection: πs: XXs is defined by πs x:= xs.
* Topology: If the Xs are topological spaces, X gets a topology by declaring that OX 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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