We reduce the problem of describing all \(\mathcal{M} f_m\)-natural operators transforming general affine connections on \(m\)-manifolds into general affine ones to the known description of all \(GL(\mathbf{R}^m)\)-invariant maps \(\mathbf{R}^{m*}\otimes \mathbf{R}^m\to \otimes^k\mathbf{R}^{m*}\otimes\otimes ^k\mathbf{R}^m\) for \(k=1,3\).
We describe all natural operators \(A\) transforming general connections \(\Gamma\) on fibred manifolds \(Y \rightarrow M\) and torsion-free classical linear connections \(\Lambda\) on \(M\) into general connections \(A(\Gamma,\Lambda)\) on the fibred product \(J^{}Y \rightarrow M\) of \(q\) copies of the first jet prolongation \(J^{1}Y \rightarrow M\).
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 \(Y^2\to Y^1\).
We classify all \(\mathcal{F}^2\mathcal{M}_{m_1,m_2,n_1,n_2}\)-natural operators \(A\) transforming projectable-projectable torsion-free classical linear connections \(\nabla\) on fibered-fibered manifolds \(Y\) of dimension \((m_1,m_2, n_1, n_2)\) into \(r\)th order Lagrangians \(A(r)\) on the fibered-fibered linear frame bundle \(L^{fib-fib}(Y )\) on \(Y\). Moreover, we classify all \(\mathcal{F}^2\mathcal{M}_{m_1,m_2,n_1,n_2}\)-natural operators \(B\) transforming projectable-projectable torsion-free classical linear connections r on fiberedfibered manifolds \(Y\) of dimension \((m_1,m_2, n_1, n_2)\) into Euler morphism \(B(\nabla)\) on \(L^{fib-fib}(Y )\). These classifications can be expanded on the \(k\)th order fibered-fibered frame bundle \(L^{fib-fib,k}(Y )\) instead of \(L^{fib-fib}(Y )\).
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We generalize reduction theorems for classical connections to operators with values in k-th order natural bundles. Using the 2nd order valued reduction theorems we classify all (0,2)-tensor fields on the cotangent bundle of a manifold with a linear (non-symmetric) connection.
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