September 2011-October 2011: On Non-Generated Diffeomorphisms

Author Commentary Below: A few notations from this notebook, to indicate topics covered. Author’s page numbering is ignored, any page numbers referenced below are from the scanned PDF documents.

Notebook September 2011 – October 2011:

Non-Generated Diffeomorphisms

(p. 6) What follows from a 1-parameter groupoid of diffeomorphisms?

(p. 07) Some thoughts on the principle of equivalence and Diffeomorphisms. It is argued by Stephen Weinberg in his book Gravitation and Cosmology that the principle of equivalence implies that all theories must be diffeomorphism invariant.

(p.14) Been thinking: Diffeomorphism group and its relation to local translations and Yang-Mills groups…The Diffeomorphism group is more complex in this regard, as there are elements arbitrarily close to the identity that are not on a 1- parameter subgroup. An example is

  • (p. 21) There are diffeomorphisms which:
    • are uniformly close to the identity for N large and ∝ small
    • are therefore contained in the “identity component” of the diffeomorphism
    • do not lie on any 1-parameter subgroup of diffeomorphism
  • Some natural questions present themselves .
  • (p. 32) So, we have:
    • very near the origin
      • circles N wrapped circles
    • unit circles and nearby
      • circles N wrapped ellipses
    • far from origin
      • circles circles (1-wrapped)

(p. 76) So it’s clear induction will play a part.

(p. 135) So all the fun happens when n=0. Generally the differential equations that arise are of the form .

  • (p. 140-145) (review of steps for calculations and conclusions)
    • #1. (p. 140) We want to examine diffeomorphisms that are not generated by a vector field. More specifically, we require that the non-generated diffeomorphism be arbitrarily close to the identity, thereby removing certain easy ‘reasons’ why the diffeomorphisms in question are non-generated. We also only consider small perturbations of a small rotation…
    • #2. We are inspired by a small perturbation on a small rotation
    • #8. (p. 142) …The pattern of the computation is to repeat this calculation until a conjecture can be made and a proof by induction can be done. Even at this point we can conjecture that
    • #11. (p. 145) Now we can calculate the cubic term and compare it with the prediction .

(p. 146) Beginning of the calculation for the cubic (m+n=3).

(p. 151) End of (m+n+3) calculation.