The following article appeared in REPORTS ON MATHEMATICAL PHYSICS, 58 (2006) 93–118. HIGHER ORDER UTIYAMA-LIKE THEOREM JOSEF JANY ˇ SKA Abstract. In this paper we prove higher order version of the Utiyama-like theorem. To prove the Utiyama-like theorem in order r ≥ 2 we have to use auxiliary classical connections on base manifolds. We prove that any natural (invariant) operator of order r for principal connections on principal G-bundles and for classical connections on base manifolds with values in a (1, 0)-order G-gauge-natural bundle factorizes through curvature tensors of both connections and their covariant differentials, where the covariant differential of curvature tensors of principal connections is considered with respect to both connections. Introduction One of the most important assumption of many physical theories is their independence on diffeomorphisms of configuration spaces eventually the independence of gauge transforma- tions. Such independence can be very effectively expressed by using the language of natural and gauge-natural bundles and operators. This is the reason why in last years the theory of natural, [12, 13, 16, 18], and gauge-natural bundles, see Section 2 and [4, 12], became a geometrical background unifying many branches of mathematical physics, see for instance [3, 5, 10]. One of the most famous results in gauge invariant theories is the Utiyama’s theorem that classifies those Lagrangians for gauge fields (principal connections on principal bundles) which are (locally) gauge invariant. In his original paper, [21], Utiyama considered his theorem only locally with specific gauge transformations. Later the Utiyama’s theorem was reproved by many authors also globally, see for instance [3, 4, 15]. The Utiyama’s theorem can be formulated globally as follows: given a principal connection Γ then any gauge invariant first order Lagrangian is given by a gauge invariant Lagrangian of the curvature tensor R[Γ], i.e. Ω(j 1 Γ) =  Ω(R[Γ]). The Utiyama’s theorem can be very simply generalized for operators with values in a gauge-natural bundle of order (1, 0). In this case we shall use the term Utiyama-like theorem instead of the Utiyama’s theorem. The Utiyama-like theorem was proved (in order 1) in [12]. 1991 Mathematics Subject Classification. 53C05, 53C10, 53C80, 58A20, 58A32. Key words and phrases. Gauge-natural bundle, natural operator, principal bundle, principal connection, Utiyama-like theorem, reduction theorem. This research has been supported by the Ministry of Education of the Czech Republic under the Project MSM0021622409 and by the Grant agency of the Czech Republic under the project GA 201/05/0523. 1