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A Riemann-Lagrange geometrization for metrical multi-time Lagrange spaces. (English) Zbl 1106.53019

Summary: We construct a geometrization of the 1-jet fiber bundle \(J^1(T,M)\) for the multi-time quadratic Lagrangian function \[ L= h^{\alpha,\beta}(t) g_{ij}(t,x) x_\alpha^i x_\beta^j+ U_{(i)}^{(\alpha)}(t,x) x_\alpha^i+ F(t,x). \] Our geometrization includes a nonlinear connection \(\Gamma\), a generalized Cartan canonical \(\Gamma\)-linear connection \(C\Gamma\) together with its \(d\)-torsions and \(d\)-curvatures, naturally provided by the given multi-time quadratic Lagrangian function \(L\).

MSC:

53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)
53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
53C80 Applications of global differential geometry to the sciences
58A20 Jets in global analysis
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