Riemannian Metric, Flow and Entropy
1 In symmetry flow language abbreviated to SFL, Riemannian metric g is presented.
By NASH John’s proof, Riemannian manifold M has isometric embedding i to higher dimensional Euclidean space RN.
i : M → RN
2 From Riemannian metric g, length L of curve C of curved surface is defined.
3 Riemannian manifold ( M, g ) has Levi-Civita connection ∇.
4 M has geodesic.
∇C’ C’ = 0
5 M has geodesic flow.
6 Closed Riemann surface has Anosov dynamical system.
The system has direct sum decomposition T.
T (M) = M1⊕M2
M1 and M2 are linear subspace of manifold M.
7 Containing Anosov dynamical system, Kolmogorov system has positive metric entropy hμ(T).
8 On Bernoulli system in Kolmogorov system, hμ(T) = - ∑p(a) log p(a) that is equal to word entropy and topological entropy in the adequate conditions.
9 Kolmogorov system relates to baker’s transformation and billiard problem.
10 Billiard problem relates to ergodic hypothesis.
11 Now Gromov-Hausdorff distance and Gromov Hausdorff convergence is presented.
12 The convergence leads to diffeomorphic to Riemann manifold.
13 The convergence also leads to Riemann manifold s collapse that is occurred from Ricci flow.
14 Collapse relates to accessibility in entropy.
15 Now Riemann orbit space that makes Alexandrov space is presented.
16 By PERELMAN G., stability theorem is led for Alexandrov space.
From the theorem, a corollary is led as the following. 2 dimensional Alexandrov space is 2 dimensional topological manifold.
17 PERELMAN leads the following theorem on non-compact Alexandrov space X that has non-negative curvature. X’s soul S that is spanning convex closed submanifold becomes compact Alexandrov space that has non-negative curvature. Spanning convex has geodesic that connects arbitrary 2 points.
18 Now half-line is presented.
Alexandrov space X has half-line that has convergence on ¯X’s topology toward + infinity.
19 The convergence relates to finiteness of entropy.
Tokyo May 7, 2007
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