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Ricci and Bianchi identities for \(h\)-normal \(\Gamma\)-linear connections on \(J^{1}(T,M)\). (English) Zbl 1040.53009
Summary: The aim of this paper is to describe the local Ricci and Bianchi identities of an \(h\)-normal \(\Gamma\)-linear connection on the first-order jet fibre bundle \(J^{1}(T,M)\). We present the physical and geometrical motives that determined our study and introduce the \(h\)-normal \(\Gamma\)-linear connections on \(J^{1}(T,M)\), emphasizing their particular local features. We describe the expressions of the local components of torsion and curvature \(d\)-tensors produced by an \(h\)-normal \(\Gamma\)-linear connection \(\nabla\Gamma\), and analyze the local Ricci identities induced by \(\nabla\Gamma\), together with their derived local deflection \(d\)-tensors identities. Finally, we expose the local expressions of Bianchi identities which geometrically connect the local torsion and curvature \(d\)-tensors of connection \(\nabla\Gamma\).
MSC:
53B15 Other connections
53B21 Methods of local Riemannian geometry
53B50 Applications of local differential geometry to the sciences
58A20 Jets in global analysis
53B05 Linear and affine connections
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