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The Bianchi identities and curvature tensors of Otsuki spaces. (English) Zbl 0793.53018
Szenthe, J. (ed.) et al., Differential geometry and its applications. Proceedings of a colloquium, held in Eger, Hungary, August 20-25, 1989, organized by the János Bolyai Mathematical Society. Amsterdam: North- Holland Publishing Company. Colloq. Math. Soc. János Bolyai. 56, 547-554 (1992).
An Otsuki space \(O_ n\) means in this paper a manifold equipped with two different affine connections \('\Gamma\) and \(''\Gamma\) and a tensor \(P_ j^ i\), and then the covariant differential of tensors, e.g. of \(X_ j^ i\) is defined as \[ DX_ j^ i:= P_ a^ i P_ j^ b (\partial_ k X_ b^ a+{}'\Gamma_ s{}^ a{}_ k X_ b^ s- {}''\Gamma_ n{}^ s{}_ k X^ a_ s)dx^ k \] [see T. Otsuki, Math. J. Okayama Univ. 9, 99-164 (1960; Zbl 0202.211) and A. Moór, Acta Sci. Math. 40, 129-142 (1978; Zbl 0362.53006)]. A \(W-O_ n\) is an \(O_ n\) endowed with a Weyl metric [A. Moór, Acta Sci. Math. 41, 173-185 (1979; Zbl 0362.53007)]. In this paper several symmetries of the curvature tensors of these spaces are proved, and some Bianchi identities are calculated.
For the entire collection see [Zbl 0764.00002].
MSC:
53B15 Other connections
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