Saturday 17 June 2023

Energy Distance Theory Note 4 Finsler Manifold and Distance

 Energy Distance Theory


Note 4
Finsler Manifold and Distance


1
Banach space     E
Ck manifold       M
Point of M     p
Banach space     TxM
Norm of TxM     ||  ||x
Finsler metric is defined by the next.
(i) Topology by ||  ||is equal to topology by norm of Banach space.
(ii) Tangent vector bundle     T (M)
Point     pM
Coordinate neighborhood of p     (Uαα),  α UαE
Ψα : Uα×→ π-1(UαT (M)
||| v |||x : = ||Ψα (xv)||xUα , vE
> 0
1/C ||| v ||| ||| v |||x C ||| v |||,  xUα , vE
2
Banach manifold M that has Finsler metric     Finsler manifold M
Longitude of M     L (σ) : = ∫ba ||σ’(t)||dt
p, qM
Distance    ρ ( pq ) : = inf { L (σ) }
Distance space     ( M, ρ )
When ( Mρ ) is complete distance space, Finsler manifold is called complete.
3
Finsler Ck manifold     M
Cfunction over M     M  R
Condition (C) is defined by the next.
(i) Subset of M     S
is boundary over S.
infS ||df || = 0
Closure of S     S-
df = 0 at point p of S- 
4
Complete Finsler C2 manifold     M 
Cclass function     M → 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     L1,p MRN )
C manifold     L1,p MN )
Distance of L1,p MRN )     ρ0
ρu, v ) = ||  v ||1,u, v ∈ L1,p MRN )
Proposition
Finsler manifold (L1,p MN ) , ||  ||1,p ) is complete.

[Note]
Word is expressed by closed manifold in Banach space.
Distance is expressed by Finsler metric.
[References]

To be continued
Tokyo November 7, 2008

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