Geodesic
On symmetrization of jets
Czechoslovak Mathematical Journal, Tome 61 (2011) no. 1, pp. 157-168
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Let $F=F^{(A,H,t)}$ and $F^1=F^{(A^1,H^1,t^1)}$ be fiber product preserving bundle functors on the category $\mathcal {FM}_m$ of fibred manifolds $Y$ with $m$-dimensional bases and fibred maps covering local diffeomorphisms. We define a quasi-morphism $(A,H,t)\to (A^1,H^1,t^1)$ to be a $GL(m)$-invariant algebra homomorphism $\nu \colon A\to A^1$ with $t^1=\nu \circ t$. The main result is that there exists an $\mathcal {FM}_m$-natural transformation $FY\to F^1Y$ depending on a classical linear connection on the base of $Y$ if and only if there exists a quasi-morphism $(A,H,t)\to (A^1,H^1,t^1)$. As applications, we study existence problems of symmetrization (holonomization) of higher order jets and of holonomic prolongation of general connections.
Let $F=F^{(A,H,t)}$ and $F^1=F^{(A^1,H^1,t^1)}$ be fiber product preserving bundle functors on the category $\mathcal {FM}_m$ of fibred manifolds $Y$ with $m$-dimensional bases and fibred maps covering local diffeomorphisms. We define a quasi-morphism $(A,H,t)\to (A^1,H^1,t^1)$ to be a $GL(m)$-invariant algebra homomorphism $\nu \colon A\to A^1$ with $t^1=\nu \circ t$. The main result is that there exists an $\mathcal {FM}_m$-natural transformation $FY\to F^1Y$ depending on a classical linear connection on the base of $Y$ if and only if there exists a quasi-morphism $(A,H,t)\to (A^1,H^1,t^1)$. As applications, we study existence problems of symmetrization (holonomization) of higher order jets and of holonomic prolongation of general connections.
DOI : 10.1007/s10587-011-0004-3
Classification : 58A05, 58A20, 58A32
Keywords: jets; higher order connections; Ehresmann prolongation; Weil functors; bundle functors; natural operators 
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Mikulski, W. M. On symmetrization of jets. Czechoslovak Mathematical Journal, Tome 61 (2011) no. 1, pp. 157-168. doi: 10.1007/s10587-011-0004-3

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