Higher order Utiyama-like theorem. (English) Zbl 1126.53016

Janyška, Josef

Rep. Math. Phys. 58, No. 1, 93-118 (2006).

As is well known, Utiyama’s theorem characterizes the Lagrangians for locally gauge invariant gauge fields. Subsequent versions of the theorem were stated and proven globally (within the context of principal bundles and their connections). Further generalizations of the same theorem for operators with values in gauge-natural bundles are known as Utiyama-like theorems. The present paper is dealing with a higher order Utiyama-like theorem based on methods of gauge-natural bundles. It is proved, among other things, that any natural (invariant) operator of order \(r\) (for principal connections on principal \(G\)-bundles and for classical connections on the base manifolds) with values in a \((1,0)\)-order \(G\)-gauge-natural bundle factorizes through the curvature tensors of both kinds of connections and their covariant differentials. Here, the covariant differentials of the curvature tensors of principal connections are taken with respect to both kinds of connections mentioned above. It should be noted that the consideration of auxiliary connections on the base manifold is crucial for the present approach.

Reviewer: Efstathios Vassiliou (Athens)

MSC:

53C05	Connections (general theory)
53C10	\(G\)-structures
53C80	Applications of global differential geometry to the sciences
58A20	Jets in global analysis
58A32	Natural bundles

Keywords:

Utiyama theorem; natural bundles

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