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Fibered connections and global Poincaré-Cartan forms in higher-order calculus of variations. (English) Zbl 0564.53013
Differential geometry and its applications, Proc. Conf., Nové Město na Moravě/Czech. 1983, Pt. 2, 61-91 (1984).
[For the entire collection see Zbl 0546.00021.]
Given a fibered manifold \(Y\to X\) of fiber dimension \(n\), the frame bundle \(FVY\) of its vertical tangent bundle \(VY\) is a principal fiber bundle over \(Y\) with structure group \(\mathrm{GL}(n,\mathbb R)\). Its first jet prolongation \(J^ 1(FVY)\) over \(X\) can be factorized into an affine bundle \(\hat C(J^ 1Y)=(J^ 1(FVY)/\mathrm{GL}(n,\mathbb R))\to J^ 1Y\) called the bundle of formal connections over \(Y\). A fibered connection over \(Y\) is a pair \((\hat \Gamma,\gamma)\) consisting of a section \(\hat \Gamma: J^ 1Y\to \hat C(J^ 1Y)\) and of a linear connection \(\gamma\) on \(X\). Such a fibered connection transforms naturally every morphism with values in the tensor algebra of \(VY\) and \(TX\) into its covariant derivative with one more covariant component with respect to \(TX\). The author deduces that every fibered connection over \(Y\) determines a unique Poincaré-Cartan form of a \(k\)-th order Lagrangian on \(Y\). For \(k=3\), the explicit coordinate expression of such a form is derived.

MSC:
53C05 Connections (general theory)
58A20 Jets in global analysis
49Q99 Manifolds and measure-geometric topics