Monday, 30 April 2018

Noncommutative Distance Theory Note 3 Point Space


Note 3
Point Space


1
Set     X
Points over X     xy
Distance d over X satisfies next conditions.
xy ) is nonnegative real number value.
Axiom 1   xy ) = 0  y
Axiom 2   xy ) = yx )
Axiom 3   xy ) + yz xz )
Distance space is ( Xd ).
Distance space is commutative.
2
Manifold    ε, M
Differentiable map   π ε → M
Manifold     E
Open set of M     Ui
Diffeomorphism     φi π -1 (Ui Ui ×E
E     Fiber
(επ)     Fiber bundle
Ε     Total space
M     Base space
∈ Ui ∩ Uj
Linear isomorphism over E    φ○φ: {x×E → {x×E
Diffeomorphism     φi π -1 (Ui Ui ×E
Vector bundle    π ε → M
Diffeomorphism     M → ε
π ( s ( ) ) = x
s is cross section of vector bundle π.
Set of infinite differentiable cross section     Γ Mε )
Lie group     that is structure group
Fiber    G
Fiber bundle     π : P → M
(·g) ·h = p·(gh),   p  P,   g G
π(·g) = π(p)   p  P, g  G
P     Principle bundle
Manifold    E
Group that is all the diffeomorphic of     Diff )
ρ G → Diff )
Direct product    P × E
Equivalence relation     ( p ·gf ) ~ ( pρ(g)f )
Quotient space P ×G E becomes associated bundle.
Group that is all the endomorphic of     End )
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 MTM )
Differential map     φ : M1 M2
φ( v )  Tφ(x) M2   v  TxM1
φ* : TM1 TM2
Vector bundle over that fiber is RN   ε
Fiber bundle     GL (ε)   Fiber over x  M  is all the linear isomorphism from RN to fiber εx .
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(MΛT*M) that is called differential form of space Ω ( ).
3
Square matrix     = ( aij )
Diagonal element     aii ( = 1,2, …, n )
Here aii is expressed by ai.
Now there gives = 1,2, diagonal matrix is  A =(a10 0a2)
Here and ai are seemed to be functions that expressed by f.
f = (f10 0f2)
fand fis commutative.
Next there gives matrix D = (0μ μ0).
Linear differential form is defined by the next.
Df := [ D, f ] = (0μ(f1-f2) μ(f2-f1)0).


[Reference]

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