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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