Conjecture 2
Geometry of Word
[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 KM < 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 KN≤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 RicM≥0 , N is non-positive curvature KN≤0, and harmonic map is u : M→N, 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.
Ⅲ
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 M U
Local coordinate system of U (x1, …, xm)
Function gij : gx ( (∂/∂xi)x, (∂/∂xj)x), 1≤i, j≤m
gij is C∞ class function over U.
Family of inner product g = {gx}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 TxM σ
σ’s normal orthogonal basis on gx {v, w} {v’, w’}
K(v, w) = R(x)(v, w, w, v) = gx(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]
To be continued
Tokyo November 23, 2008
Postscript
[Reference November 30, 2008]
[Reference November 30, 2008]
Read more: https://geometrization-language.webnode.page/products/energy-distance-theory-conjecture-2-geometry-of-word/
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