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Semiholonomical, holonomical and trivial spaces of affine connection

Author Shevchenko Yu.
Pages 43-48
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Keywords affine connection, Ricci’s identities, Bianchi’s identities, Laptev’s lemma, holonomicity, semi-holonomicity
Abstract (summary) In n-dimensional space of affine connection An,n with Cartan’s structure equations Ricci’s and Bianchi’s identities were received. Their invariance has been shown. After prolongation of the structure equations using Laptev’s lemma semiholonomical, holonomical and trivial manifolds are defined. The Ricci’s identities allowed us to prove semiholonomicity of the space An,n. This semiholonomicity preserves in the space without torsion A’n,n and in the space without curvature ‘An,n , besides the locally affine space Ά’n,n is trivial. Tensor of non-holonomicity of the space An,n is introduced. Vanishing of this tensor makes the space holonomic, H n n A . Also curvature tensor of associated space of affine connection without torsion A’n,n was introduced. It’s vanishing characterizes trivial space of affine connection, Tn n A .
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