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Zbl 1080.53014
Çivi, Gülçin
On the Bianchi identities in a generalized Weyl space. (English)
[A] Mladenov, Iva\" ilo 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. 151-155 (2001). ISBN 954-90618-2-5/pbk

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 {\it V. Murgescu}, [Rev. Roum. Math. Pures Appl. 15, 2, 293--301 (1970; Zbl 0193.50503)] are a natural extension of Weyl spaces [{\it I. E. Hiricu} and {\it 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.
[Iulia Hirică (Bucureşti)]

MSC 2000:: *53B05 Linear and affine connections

Keywords: generalized Weyl spaces; semi-symmetric $E$-connections; Bianchi identities; recurrent spaces; Vrănceanu vector

Citations: Zbl 0193.50503; Zbl 1173.53305

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