Once in a while we have to learn something new. Once in a while we have to learn again what we have already forgotten. I need to learn again about G-structures. They are as exciting as G-spots, or even more. So, here is the piece from Kobyashi-Nomizu Vol. 1, p. 288
3. Chern [3] defined the notion of a G-structure on a differentiable manifold M, where G is a certain Lie subgroup of GL(n; R) with n = dim M. In our terminologies, a G-structure on M is a reduction of the bundle of linear frames L(M) to the subgroup G. For G = O(n), a G-structure is nothing but a Riemannian metric given on M (see Example 5.7 of Chapter I). For a general theory of G-structures, see Chern [3], Bernard [1] and Fujimoto [1]. We mention some other special cases. Weyl [3] and E. Cartan [3] proved the following. For a closed subgroup G of GL(n; R), n>=3, the following two conditions are equivalent:
(1) G is the group ofall matrices which preserve a certain non-degenerate quadratic form of any signature:
(2) For every n-dimensional manifold M andfor every reduced subbundle
P of L (M) with group G, there is a unique torsion-free connection in P.
The implication (1) -> (2) is clear from Theorem 2.2 of Chapter IV (in which g can be an indefinite Riemannian metric); in fact, if G is such a group, any G-structure on M corresponds to an indefinite Riemannian metric on M in a similar way to Example 5.7 of Chapter I. The implication (2) -> (1) is nontrivial. See also Klingenberg [1].
Let G be the subgroup of GL(n; R) consisting of all matrices which leave the r-dimensional subspace Rr of Rn invariant. A G-structure on an n-dimensional manifold M is nothing but an r-dimensional distribution. Walker [3] proved that an r-dimensional distribution is parallel with respect to a certain torsion-free linear connection if and only if the distribution is integrable. See also Willmore [1, 2].
Let G be GL(n; C) regarded as a subgroup of GL(2n; R) in a natural manner. A G-structure on a 2n-dimensional manifold M is nothing but an almost complex structure on M. This structure will be treated in Volume II.
We will have to talk a little bit about Lie groups ("Lie" for Sophus Lie, Norwegian mathematician, pronounced /liː/ LEE)) and Lie algebras.
This way I will recall for myself, as well as for any interested Reader, the necessary definitions.
Not every smooth manifold can carry a group structure. For instance the two-dimensional sphere. It can be acted upon by a group (for instance the 3D rotation group), but it cannot be a group itself. A good reason for it is that any n-parameter Lie group has n nowhere vanishing, nonsingular, vector fields. Such vector fields exist on the 3D sphere (that is impossible to imagine, because it needs four dimensions), but not on a two-dimensional spherical surface. The fact of life. Or "The hairy ball theorem of algebraic topology (sometimes called the hedgehog theorem in Europe)"
"Hairy ball". Any vector field on the 2D sphere must have at least one singular point. The graphics represents one of the two polarization vector fields of a photon, as seen by an observer moving with the speed of light, drawn on the unit sphere in the momentum space. Thanks are due to JS and AMS for their interest, help, and cooperation with this unfinished project.
Wikipedia has a smoother and prettier illustration of a one-pole vector field on the 2-sphere:
What we will need is a fundamental one-form on a Lie group. But to discuss this form we will need vector fields, the notion of left-invariance, and the construction of the Lie algebra of a Lie group
- Nomizu, Katsumi (2. from left)
- Kobayashi, Shoshichi (2. from right)
- Klingenberg, Wilhelm P.A. (left)
Location: Oberwolfach
Author: Ferus, Dirk (photos provided by Ferus, Dirk)
Source: Dirk Ferus, Berlin
Year: 1974
Copyright: Dirk Ferus, Berlin
6. 11:35: Discipline, discipline, discipline
7. 12:00 Paying attention to reality left and right:
8. 12:30 Freedom and free will. It is true that we can't do everything. But the theory that we do no have free will is only a theory. We are free to accept it or not. Even if we can't do everything, a every moment, there is infinitely many different choices that we are free to make. We can use this freedom or not. It is our free will. Choices that we do not make are being made, randomly or not, by "fate", these are our lost opportunities.
10. 09-02-23 9:30 Fighting with the meaning of U(2) and its double cover I reminded myself about a very good resource that everybody should have: John Baez, "Introduction to Algebraic and Constructive Quantum Field Theory". It has a chapter on Clifford systems and also sections about group representations and renormalization. See also this by the same author (His uncle Albert Baez was a physicist, a co-inventor of the X-ray microscope, and father of singer and progressive activist Joan Baez. Albert interested him in physics as a child.).
