All [formula]-gauge-natural operators C sending linear 3-forms [formula] on a smooth [formula] vector bundle E into R-bilinear operators [formula] transforming pairs of linear sections of [formula] into linear sections of [formula] are completely described. The complete descriptions is given of all generalized twisted Dorfman-Courant brackets C (i.e. C as above such that C0 is the Dorfman-Courant bracket) satisfying the Jacobi identity for closed linear 3-forms H . An interesting natural characterization of the (usual) twisted Dorfman-Courant bracket is presented.
There are completely described all [formula]-gauge-natural operators C which, like to the Dorfman-Courant bracket, send closed linear 3-forms [formula]on a smooth (C ∞) vector bundle E into R-bilinear operators [formula] transforming pairs of linear sections of [formula] into linear sections of [formula]. Then all such C which also, like to the twisted Dorfman-Courant bracket, satisfy both some “restricted” condition and the Jacobi identity in Leibniz form are extracted.
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We deduce that all natural operators of the type of the Legendre operator from the variational calculus in fibred manifolds are the constant multiples of the Legendre operator.
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We construct some extension [...] of the flow operator [...] fibred frame bundle functor. Next using operator [...] we present some construction of general connections[...] depending on classical (not necessarily projectable) linear connections V on Y.
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We describe all natural operators A lifting a clasiccal linear connection on an m-dimensional manifold M into a classical linear conection A() on the r-th order frame bundle LrM = invJr/0 (Rm,M).
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Natural bundles admitting natural lifting of linear connections are characterized. Corollaries are presented. Some other similar results are obtained, too.
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Let F : Mfn -> FM. be a natural bundle. We classify all FMm,n-natural operators D transforming projectable vector fields X on (m, n)-dimensional fibered manifolds Y - M to vector fields D{X) on the F-vertical bundle VFY -> M. We apply this classification result to some more known natural bundles F.
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That all natural operators of the type of formal Euler operator from the variational calculus are the constant multiples of the formal Euler operator is deduced.
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For any product-preserving bundle functor F denned on the category F2 M of fibered-fibered manifolds, we determine all natural operators transforming projectable-projectable vector fields on Y 6 Ob(F2M) to vector fields on FY. We also determine all natural affinors on FY and prove a composition property analogous to that concerning Weil bundles.
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We discuss the prolongation of connections to to some non product preserving bundles. We introduce the concept of (r)-connection on a fibered manifold Y and for a given connection F on Y we construct its horizontal extension F(2). We also prove that F(2 ) is the unique (2)-connection on Y canonically dependent on F.
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For integers p ≥ 0, n ≥ p+2 and r ≥ 1 all natural linear operators Λp T*|Mfn → TTr* transforming p-forms from n-manifolds M into vector fields on the r-th order cotangent bundle Tr* M = Jr (M, R)0 of M are completely described.
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Let Q : B - A be an algebra epimorphism of Well algebras and let Q :T M -> T M be the canonical extension of Q over a manifold M. The full classification of natural operators transforming functions TAM -"o R into 2-forms on TBM of finite order with respect to Q is given.
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