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Ricci type identities for non-basic differentiation in Otsuki spaces. (English) Zbl 1053.53008

In \(N\)-dimensional Otsuki spaces the tensor field \(P^i_j\) and two types of connection coefficients \({}'\Gamma\) and \({}''\Gamma\) have an important role. The basic covariant derivative for the tensor \(V^i_j\) is defined by \[ V^i_{j;k}=V^i_{j,k}+{}'\Gamma_p{}^i{}_kV^p_j-{}''\Gamma_j{}^p{}_kV^i_p, \] and the nonbasic covariant derivative by \(V^i_{j\underset{1}\| k}=P^i_pP^q_jV^p_{q;k}\). The other nonbasic covariant derivative \(\underset{2}\| k\) is obtained from above if the index \(k\) in \({}'\Gamma\) and \({}''\Gamma\) is on the first place; in \(\underset{3}\| k\) and \(\underset{4}\| k\) \(k\) only on one place changes the place.
The author proves several theorems with Ricci-type identities by using different nonbasic covariant derivatives \(\underset\alpha\| k\), \(\alpha=1,2,3,4\).

MSC:

53A40 Other special differential geometries
53B05 Linear and affine connections
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