Thursday 18 July 2013

Geometry of Word

Conjecture 2
Geometry of Word

Word is infinite cyclic group.

(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 KN0, 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 KN0, 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
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), 1i, 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.
Riemann manifold      (Mg)
M’s C class vector field    (M)   
Linear connection of M     
What  and XYuniquely satisfy the next is called Levi-Civita connection.
(i) Xg(YZ) = g(XYZ) + g(YXZ)
(ii) XY -YX = [XY]
m-dimensional Riemann manifold (Mg)    M
Levi-Civita connection of M     
R(XY) : = XY - YX - [XY]
Map R : = X(M×X(M)×X(M X(M)
R(XYZ) : = R(XY)Z
R is called curvature tensor of M.
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.

<Example of word’s infinite cycle is shown by the bellow.>
<On minimum unit of meaning, refer to the next.>

To be continued
Tokyo November 23, 2008

[Reference November 30, 2008]

No comments:

Post a Comment