Transformation of connectednesses (Q2765988)

Let \(X_n\) be a differentiable manifold, \(\nabla\) an affine connection and \((v_1,\dots,v_n)\) be a net of \(n\) independent vector fields which defines the symmetric tensor \(a_{is}=\sum_{\alpha=1}^nv_1^{\alpha}v_s^{\alpha}.\) NEWLINENEWLINENEWLINEIn the present paper it is shown that for any net \((v_1,\dots,v_n)\) there exists only one connection \(\overset{2}{\nabla}\) such that the mixed covariant derivative NEWLINE\[NEWLINE{\overset{3}{\nabla}}_ka_{is}=\partial_ka_{is}-{\overset{1} {\Gamma}}^p_{ki}a_{ps}- {\overset{2}{\Gamma}}^p_{ks}a_{ip}NEWLINE\]NEWLINE determines a Weyl connection NEWLINE\[NEWLINE{\overset{3}\nabla}, {\overset{3}{\nabla}}_ka_{is} = 2\omega_ka_{is},NEWLINE\]NEWLINE where \(\omega_k=-{1\over n}v_i^{\alpha} {\overset{1}{\nabla}}_kv_{\alpha}^i, v_{\alpha}^iv_k^{\alpha} = \delta_k^i.\) NEWLINENEWLINENEWLINEAny transformation between the affine connection \({\overset{1}{\nabla}}\) and \({\overset{2}{\nabla}},\) generated by the net \((v_1,\dots,v_n),\) is called a \((v)\)-transformation. Some properties of \((v)\)-transformations of certain affine connections are obtained. Special cases of nets are also studied and characterized.NEWLINENEWLINEFor the entire collection see [Zbl 0971.00014].