Let X, Y, Z, W be manifolds and π : Z → X be a surjective submersion. We characterize π-local regular operators A : C∞(X,Y) → C∞(Z,W) in terms of the corresponding maps à : J∞(X,Y) ×XZ → W satisfying the so-called local finite order factorization property.
Let \(F\) be a bundle functor on the category of all fibred manifolds and fibred maps. Let \(\Gamma\) be a general connection in a fibred manifold \(\mathrm{pr}:Y\to M\) and \(\nabla\) be a classical linear connection on \(M\). We prove that the well-known general connection \(\mathcal{F}(\Gamma,\nabla)\) in \(FY\to M\) is canonical with respect to fibred maps and with respect to natural transformations of bundle functors.
We study how a projectable general connection \(\Gamma\) in a 2-fibred manifold \(Y^2\to Y^1\to Y^0\) and a general vertical connection \(\Theta\) in \(Y^2\to Y^1\to Y^0\) induce a general connection \(A(\Gamma,\Theta)\) in \(Y^2\to Y^1\).
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For natural numbers r,s,q,m,n with s≥r≤q we determine all natural functions g: T *(J (r,s,q)(Y, R 1,1)0)*→R for any fibered manifold Y with m-dimensional base and n-dimensional fibers. For natural numbers r,s,m,n with s≥r we determine all natural functions g: T *(J (r,s)(Y, R)0)*→R for any Y as above.
Let P be a principal fiber bundle with the basis M and with the structural group G. A trivialization of P is a section of P. It is proved that there exists only one gauge natural operator transforming trivializations of P into principal connections in P. All gauge natural operators transforming trivializations of P and torsion free classical linear connections on M into classical linear connections on P are completely described.
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