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 || ||x is equal to topology by norm of Banach space.
(ii) Tangent vector bundle T (M)
Point p∈M
Coordinate neighborhood of p (Uα, α), α : Uα→E
Ψα : Uα×E → π-1(Uα) ⊂T (M)
||| v |||x : = ||Ψα (x, v)||x , x∈Uα , v∈E
C > 0
1/C ||| v |||p ≤ ||| v |||x ≤C ||| v |||p , x∈Uα , v∈E
2
Banach manifold M that has Finsler metric Finsler manifold M
Longitude of M L (σ) : = ∫ba ||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 Ck manifold M
Ck function over M f : M → R
Condition (C) is defined by the next.
(i) Subset of M S
f is boundary over S.
infS ||df || = 0
Closure of S S-
df = 0 at point p of S-
4
Complete Finsler C2 manifold M
Ck 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 L1,p ( M, RN )
C∞ manifold L1,p ( M, N )
Distance of L1,p ( M, RN ) ρ0
ρ0 ( u, v ) = || u – v ||1,p , u, v ∈ L1,p ( M, RN )
Proposition
Finsler manifold (L1,p ( M, N ) , || ||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
Postscript
[Reference November 30, 2008]
[Reference November 30, 2008]
