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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...
We prove that any first order \(\mathcal{F}_2\mathcal{M}_{m_1,m_2,n_1,n_2}\)-natural operator transforming projectable general connections on an \((m_1,m_2, n_1, n_2)\)-dimensional fibred-fibred manifold \(p = (p, p) : (p_Y : Y \to Y ) \to (p_M : M \to M)\) into general connections on the vertical prolongation \(V Y \to M\) of \(p: Y \to M\) is the restriction of the (rather well-known) vertical prolongation...
If (M, g) is a Riemannian manifold then there is the well-known base preserving vector bundle isomorphism TM → T*M given by v → g(v, –) between the tangent TM and the cotangent T*M bundles of M. In the present note first we generalize this isomorphism to the one JrTM → JrT*M between the r-th order prolongation JrTM of tangent TM and the r-th order prolongation JrT*M of cotangent T*M bundles of M....
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 \(Y^2\to Y^1\).
Let \(\mathcal{M} f_m\) be the category of \(m\)-dimensional manifolds and local diffeomorphisms and let \(T\) be the tangent functor on \(\mathcal{M} f_m\). Let \(\mathcal{V}\) be the category of real vector spaces and linear maps and let \(\mathcal{V}_m\) be the category of \(m\)-dimensional real vector spaces and linear isomorphisms. We characterize all regular covariant functors \(F:\mathcal{V}_m\to\mathcal{V}\)...
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 \(J^{<q>}Y \rightarrow M\) of \(q\) copies of the first jet prolongation \(J^{1}Y \rightarrow M\).</q>
Let \(\mathcal{M}f_m\) be the category of \(m\)-dimensional manifolds and local diffeomorphisms and let \(T\) be the tangent functor on \(\mathcal{M}f_m\). Let \(\mathcal{V}\) be the category of real vector spaces and linear maps and let \(\mathcal{V}_m\) be the category of \(m\)-dimensional real vector spaces and linear isomorphisms. Let \(w\) be a polynomial in one variable with real coefficients...
If \(m\geq p+1\geq 2\) (or \(m=p\geq 3\)), all natural bilinear operators \(A\) transforming pairs of couples of vector fields and \(p\)-forms on \(m\)-manifolds \(M\) into couples of vector fields and \(p\)-forms on \(M\) are described. It is observed that any natural skew-symmetric bilinear operator \(A\) as above coincides with the generalized Courant bracket up to three (two, respectively)...
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...
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