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 \(\nabla\Gamma\).


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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