Thursday 2 March 2023

Energy Distance Theory Conjecture 2 Geometry of Word

 
Conjecture 2
Geometry of Word
 
 
 
[Conjecture]
Word is infinite cyclic group.
 
[Explanation]
(Preissmann’s theorem)
When (Mg) is connected Riemann manifold and sectional curvature of M is always KM < 0, non-trivial commutative subset of functional group of Mπ1(M) always becomes infinite cyclic group.
Preparatory proposition for Preissmann’s theorem
(Proposition 1)
When (Mg) and (Nh) are compact Riemann manifold and N is non-positive curvature KN≤0, arbitrary continuous map f ∈C0(MN) is free homotopic with harmonic map uC(M, N).
(Proposition 2)
When M is compact Riemann manifold, Ricci tensor of M is positive semidefinite RicM≥0 , is non-positive curvature KN≤0, and harmonic map is u : MN,  the next is concluded.
When N is negative curvature KN<0, u is constant map or map of u coincides with map of closed geodesic line.
Consideration for the theorem and propositions
1
m-dimensional C class manifold     M
Point of M     x
Tangent space of x     TxM
Inner product of TxM   gx
Coordinate neighborhood of     U
Local coordinate system of U     (x1, …, xm)
Function     gij : gx ( (∂/∂xi)x, (∂/xj)x), 1≤i, jm
gij is C class function over U.
Family of inner product     g = {gx}xM
g is called Riemannian metric.
When M has g, (Mg) is called Riemannian manifold.
2
Riemann manifold      (Mg)
M’s C class vector field    (M)   
Linear connection of M     ∇
XYZ∈X(M)
What ∇ and XYuniquely satisfy the next is called Levi-Civita connection.
(i) Xg(YZ) = g(∇XYZ) + g(Y, ∇XZ)
(ii) ∇XY -YX = [XY]
3
m-dimensional Riemann manifold (Mg)    M
Levi-Civita connection of M     ∇
XYX
R(XY) : = ∇XY - ∇YX - ∇[XY]
Map R : = X(M) ×X(MX(M) → X(M)
R(XYZ) : = R(XY)Z
R is called curvature tensor of M.
4
xM
2-dimensional subspace of tangent space TxM     σ
σ’s normal orthogonal basis on gx     {vw} {v’w’}
K(vw) = R(x)(vwwv) = gx(R(x)(vw)wv)
v’ = cosθv + sinθww’ = ∓sinθv±cosθw  (double sign directly used)
K(σ) : = R(x)(vwwv) = R(x)(v’w’w’v’)
K(σ) is called sectional curvature.
 
[References]
 
To be continued
Tokyo November 23, 2008
 
Postscript
[Reference November 30, 2008]
 
Tokyo
7 January 2018 Reprinted

 



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