Noncommutative Distance Theory Note 3 Point Space / December 9, 2007

Noncommutative Distance Theory

Note 3

Point Space

TANAKA Akio

1

Set     X

Points over X     x, y

Distance d over X satisfies next conditions.

d ( x, y ) is nonnegative real number value.

Axiom 1   d ( x, y ) = 0 ⇔ x = y

Axiom 2   d ( x, y ) = d ( y, x )

Axiom 3   d ( x, y ) + d ( y, z) ≥ d ( x, z )

Distance space is ( X, d ).

Distance space is commutative.

2

Manifold    ε, M

Differentiable map   π : ε → M

Manifold     E

Open set of M     U_i

Diffeomorphism     φ_i : π ^-1 (U_i) → U_i ×E

E     Fiber

(ε, π)     Fiber bundle

Ε     Total space

M     Base space

x ∈ U_i ∩ U_j

Linear isomorphism over E    φ_i ○φ_j : {x} ×E → {x} ×E

Diffeomorphism     φ_i : π ^-1 (U_i) → U_i ×E

Vector bundle    π : ε → M

Diffeomorphism     s : M → ε

π ( s ( x ) ) = x

s is cross section of vector bundle π.

Set of infinite differentiable cross section     Γ ( M, ε )

Lie group     G that is structure group

Fiber    G

Fiber bundle     π : P → M

(p ·g) ·h = p·(gh),   p ∈ P,   g, h ∈ G

π(p ·g) = π(p)   p ∈ P, g ∈ G

P     Principle bundle

Manifold    E

Group that is all the diffeomorphic of E     Diff ( E )

ρ : G → Diff ( E )

Direct product    P × E

Equivalence relation     ( p ·g, f ) ~ ( p, ρ(g)f )

Quotient space P ×_G E becomes associated bundle.

Group that is all the endomorphic of E     End ( E )

Representation     ρ : G → End ( E )

Principle bundle     P

Associated bundle     ε = P ×_G E

Dual vector space     E*

Dual representation     ρ* : G → End ( E *)

Associated bundle     ε* = P ×_G E*

Tangent vector bundle of M      TM   

Cross section of TM     X ∈ Γ ( M, TM )

Differential map     φ : M₁ →M₂

φ_* ( v ) ∈ T_φ(x) M₂   v ∈ T_xM₁

φ_* : TM₁ →TM₂

Vector bundle over M that fiber is R^N   ε

Fiber bundle     GL (ε)   Fiber over x ∈ M  is all the linear isomorphism from R^N to fiber ε_x .

GL (ε) is frame bundle of ε.     

Frame bundle of TM     GL ( TM )

Representation space of arbitrary representation ρ over GL ( n )     E

Tensor bundle of M     Associated bundle ε = GL ( TM ) ×_ρ E

Representation E has exterior algebra Γ(M, ΛT*M) that is called differential form of space Ω ( M ).

3

Square matrix     A = ( a_ij )

Diagonal element     a_ii ( i = 1,2, …, n )

Here a_ii is expressed by a_i.

Now there gives i = 1,2, diagonal matrix is  A =(^a1₀ ⁰_a2)

Here A and a_i are seemed to be functions that expressed by f.

f = (^f1₀ ⁰_f2)

f₁ and f₂ is commutative.

Next there gives matrix D = (⁰_μ ^μ₀).

Linear differential form is defined by the next.

Df := [ D, f ] = (⁰_μ(f1-f2) ^μ(f2-f1)₀).

[Reference]

Distance Theory Algebraically Supplemented / 3 Point / Tokyo October 12, 2007

Tokyo December 9, 2007

Sekinan Research Field of Language