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On the Bianchi identities in a generalized Weyl space. (English) Zbl 1080.53014
Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 2nd international conference on geometry, integrability and quantization, Varna, Bulgaria, June 7–15, 2000. Sofia: Coral Press Scientific Publishing (ISBN 954-90618-2-5/pbk). 151-155 (2001).
An \(n\)-dimensional differential manifold \(W_n^{\star }\), having an asymmetric connection \(\nabla ^{\star }\) and asymmetric conformal metric tensor \(g^{\star }\) preserved by \(\nabla ^{\star }\) is called a generalized Weyl space. The compatibility condition \[ \nabla _k^{\star }g_{ij}^{\star }-2T^{\star }_kg_{ij}^{\star }=0, \] is satisfied, where \(T^{\star }_k\) are the components of a covariant vector field called the complementary vector field of the generalized Weyl space. These spaces V. Murgescu, [Rev. Roum. Math. Pures Appl. 15, 2, 293–301 (1970; Zbl 0193.50503)] are a natural extension of Weyl spaces [I. E. Hiricu and L. Nicolescu, Rend. Circ. Mat. Palermo, II. Ser. 53, 390–400 (2004; Zbl 1173.53305)]. The connection \(\nabla ^{\star }\) is said to be a \(E\)-connection if \(\nabla ^{\star }_k\Omega _i-\nabla ^{\star }_i\Omega _k=0\) holds, where \(\Omega _j=\Omega _{ik}^i\) is the Vrănceanu vector of the connection \(\nabla ^{\star }\) and \(\Omega _{jk}^i\) is the torsion tensor of \(\nabla ^{\star }.\) The connection \(\nabla ^{\star }\) is semi-symmetric if \[ \Omega _{jk}^i={1\over n-1}(\delta _j^i\Omega _k-\delta _k^i\Omega _j). \] In the present paper, generalized Weyl spaces are considered. It is proved that the first Bianchi identity is satisfied if the Vrănceanu vector is a gradient or the space has a semi-symmetric \(E\)-connection. Moreover, it is shown that for a generalized recurrent Weyl space having a semi-symmetric connection the second Bianchi identity is satisfied.
For the entire collection see [Zbl 0957.00038].
MSC:
53B05 Linear and affine connections
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