Variational problems in non-Abelian gauge theories. (English) Zbl 0697.58016

Uhlenbeck, Karen K.

Differential geometry and differential equations, Proc. Symp., Shanghai/China 1981, 443-471 (1984).

[For the entire collection see Zbl 0646.00010.]
In this lecture in honor of Su Buchin, the author introduces what she calls the “Symmetric Criticality Principle” - SCP - which combines the use of extremality and symmetry and is useful in Differential Geometry and Mathematical Physics.
The SCP can be formulated as follows: If f is a smooth real-valued function on a smooth G-manifold \({\mathcal M}\), then a critical point of \(f|_{\Sigma}\) is a critical point of f.
Here \(\Sigma =\{p\in {\mathcal M}_{gp}=p\quad \forall g\in G\}\) is the set of symmetric points of \({\mathcal M}\). A point p is called critical for f if the linear functional \(df_ p\in T{\mathcal M}^*_ p\) vanishes.
Section 1 lists a number of examples and counter-examples, such as the case of even functions, the counter-example of a free Hamiltonian flow of a particle in one dimension, etc.
Section 2 proves the SCP in the Riemannian case, with G acting isometrically. In this case the action in geodesic coordinates around the critical point is linear, and \(\Sigma\) is a totally geodesic submanifold.
Section 3 is devoted to the compact case and Section 4 discusses generalizations.
Secton 5 discusses the case of manifolds of sections of G-fibre bundles, in which case \(\Sigma\) becomes the set of equivariant sections. In Section 6 jet bundles and Lagrangians are introduced. Section 7 is devoted to “natural first-order Lagrangians” and Section 8 to the same for the case of Riemannian metrics.
Section 9 contains a discussion of the geometry of the Kaluza-Klein unification of Yang-Mills and gravity. Section 10 deals with geodesic orbits and relative equilibria. Section 11 is entitled ‘Critical orbits of Riemannian manifolds’. Section 12 deals with critical Riemannian metrics. Section 13, “Variational problems with higher cohomogeneity” returns to the general case and ends with a discussion of Kaluza-Klein. The bibliography has 23 entries.

Reviewer: M.E.Mayer

MSC:

58E30	Variational principles in infinite-dimensional spaces
58E10	Variational problems in applications to the theory of geodesics (problems in one independent variable)
58E05	Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
81T08	Constructive quantum field theory

Keywords:

variational problems; critical points; invariance; Symmetric Criticality Principle; symmetry; Kaluza-Klein unification; Yang-Mills; gravity; geodesic orbits; Riemannian metrics; bibliography

Citations:

Zbl 0646.00010