Energy Distance Theory Note 4 Finsler Manifold and Distance

 Energy Distance Theory

Note 4

Finsler Manifold and Distance

TANAKA Akio

1

Banach space     E

C^k manifold       M

Point of M     p

Banach space     T_xM

Norm of T_xM     ||  ||_x

Finsler metric is defined by the next.

(i) Topology by ||  ||_x is equal to topology by norm of Banach space.

(ii) Tangent vector bundle     T (M)

Point     p∈M

Coordinate neighborhood of p     (U_α, α),  α : U_α→E

Ψ_α : U_α×E → π^-1(U_α) ⊂T (M)

||| v |||_x : = ||Ψ_α (x, v)||_x , x∈U_α , v∈E

C > 0

1/C ||| v |||_p ≤ ||| v |||_x ≤C ||| v |||_p ,  x∈U_α , v∈E

2

Banach manifold M that has Finsler metric     Finsler manifold M

Longitude of M     L (σ) : = ∫^b_a ||σ’(t)||dt

p, q∈M

Distance    ρ ( p, q ) : = inf { L (σ) }

Distance space     ( M, ρ )

When ( M, ρ ) is complete distance space, Finsler manifold is called complete.

3

Finsler C^k manifold     M

C^k function over M     f : M → R

Condition (C) is defined by the next.

(i) Subset of M     S

f is boundary over S.

inf_S ||df || = 0

Closure of S     S^-

df = 0 at point p of S^- 

4

Complete Finsler C² manifold     M 

C^k class function     f : M → R satisfies condition ( C ).

Theorem

Connected component of M     M₀

When f is boundary from below, f has minimum value at M₀.

5

1 > m/p , m = dim M

Banach space     L_1,p ( M, R^N )

C^∞ manifold     L_1,p ( M, N )

Distance of L_1,p ( M, R^N )     ρ₀

ρ₀ ( u, v ) = || u – v ||_1,p , u, v ∈ L_1,p ( M, R^N )

Proposition

Finsler manifold (L_1,p ( M, N ) , ||  ||_1,p ) is complete.

[Note]

Word is expressed by closed manifold in Banach space.

Distance is expressed by Finsler metric.

[References]

Distance Theory / Tokyo May 4, 2008

Reversion Theory / Tokyo September 27, 2008

To be continued

Tokyo November 7, 2008

Sekinan Research Field of Language

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