Energy Distance Theory / Conjecture 2 / Geometry of Word

Energy Distance Theory

Conjecture 2

Geometry of Word

TANAKA Akio

[Conjecture]

Word is infinite cyclic group.

[Explanation]

Ⅰ

(Preissmann’s theorem)

When (M, g) is connected Riemann manifold and sectional curvature of M is always K_M < 0, non-trivial commutative subset of functional group of M, π₁(M) always becomes infinite cyclic group.

Ⅱ

Preparatory proposition for Preissmann’s theorem

(Proposition 1)

When (M, g) and (N, h) are compact Riemann manifold and N is non-positive curvature K_N≤0, arbitrary continuous map f ∈C⁰(M, N) is free homotopic with harmonic map u_∞∈C^∞(M, N).

(Proposition 2)

When M is compact Riemann manifold, Ricci tensor of M is positive semidefinite Ric^M≥0 , N is non-positive curvature K_N≤0, and harmonic map is u : M→N,  the next is concluded.

When N is negative curvature K_N<0, u is constant map or map of u coincides with map of closed geodesic line.

Ⅲ

Consideration for the theorem and propositions

1

m-dimensional C^∞ class manifold     M

Point of M     x

Tangent space of x     T_xM

Inner product of T_xM   g_x

Coordinate neighborhood of M     U

Local coordinate system of U     (x¹, …, x^m)

Function     g_ij : g_x ( (∂/∂xⁱ)_x, (∂/∂x^j)_x), 1≤i, j≤m

g_ij is C^∞ class function over U.

Family of inner product     g = {g_x}_x∈M

g is called Riemannian metric.

When M has g, (M, g) is called Riemannian manifold.

2

Riemann manifold      (M, g)

M’s C^∞ class vector field    X (M)   

Linear connection of M     ∇

X, Y, Z∈X(M)

What ∇ and X, Y, Z uniquely satisfy the next is called Levi-Civita connection.

(i) Xg(Y, Z) = g(∇_XY, Z) + g(Y, ∇_XZ)

(ii) ∇_XY -∇_YX = [X, Y]

3

m-dimensional Riemann manifold (M, g)    M

Levi-Civita connection of M     ∇

X, Y∈X

R(X, Y) : = ∇_X∇_Y - ∇_Y∇_X - ∇[X, Y]

Map R : = X(M) ×X(M)×X(M) → X(M)

R(X, Y, Z) : = R(X, Y)Z

R is called curvature tensor of M.

4

x∈M

2-dimensional subspace of tangent space T_xM     σ

σ’s normal orthogonal basis on g_x     {v, w} {v’, w’}

K(v, w) = R(x)(v, w, w, v) = g_x(R(x)(v, w)w, v)

v’ = cosθv + sinθw, w’ = ∓sinθv±cosθw  (double sign directly used)

K(σ) : = R(x)(v, w, w, v) = R(x)(v’, w’, w’, v’)

K(σ) is called sectional curvature.

[References]

<Example of word’s infinite cycle is shown by the bellow.>

On Time Property Inherent in Characters / Hakuba March 28, 2003

Prague Theory / Tokyo October 2, 2004

Prague Theory 3 / Tokyo January 28, 2005

TOMONAGA’s Super Multi-Time Theory / Tokyo January 25, 2008

<On minimum unit of meaning, refer to the next.>

Cell Theory / From Cell to Manifold / Tokyo June 2, 2007

Reversion Analysis Theory / Tokyo June 8, 2008

Reversion Analysis Theory 2 /Tokyo June 12, 2008

Holomorphic Meaning Theory / Tokyo June 15, 2008

Holomorphic Meaning Theory 2 / Tokyo June 19, 2008

Energy Distance Theory / Conjecture 1 / Word and Meaning Minimum / Tokyo September 22, 2008

To be continued

Tokyo November 23, 2008

Sekinan Research Field of Language

Postscript
[Reference November 30, 2008]

Distance of Word / November 30. 2008 / Sekinan.wiki.zoho.com