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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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