# zbMATH — the first resource for mathematics

Yang-Mills equations in 4-dimensional conformally connected manifolds. (English. Russian original) Zbl 1188.58006
Russ. Math. 53, No. 3, 56-60 (2009); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2009, No. 3, 67-72 (2009).
After a short general description on real quadrics in a 5-dimensional projective space, the author presents the notion of conformal connectivity on 4-dimensional manifolds. The further search of the necessary invariant is related with the Hodge operator. Once the first variation of the Yang-Mills functional and the Yang Mills equation are sketched, the paper ends with three examples of different type of conformal manifolds.
On these examples, it is shown that the only invariant, which is quadratic with respect to the curvature $$\Phi$$ of the connectivity is the Yang-Mills functional $$\int|{\mathrm{tr}} (*\Phi\wedge\Phi)|$$.
Reviewer’s remark: The first sections presented are not new.

##### MSC:
 58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals 53C07 Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) 81T13 Yang-Mills and other gauge theories in quantum field theory
Full Text:
##### References:
 [1] A. L. Besse, Einstein Manifolds (Springer, Berlin, 1987; Mir, Moscow, 1990), Vol. I.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.