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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
##### Keywords:
fibered manifold; fibered connection; Poincaré-Cartan form