On semi-symmetric connections (Q2640895)

Let \(\nabla\) be a linear connection on M (dim M\(>2)\) and \(\pi\) be a 1- form, R and T the curvature and torsion tensors of \(\nabla\). It is proved that: a) For a \(\pi\)-semisymmetric \(\nabla\) \((T(X,Y)=\pi (Y)X-\pi (X)Y)\) the following statements are equivalent: (i) \(d\pi =0\), \((ii)\quad \sigma_{(X,Y,Z)}\{R(X,Y)Z\}=0,\) \((iii)\quad \sigma_{(X,Y,Z)}\{\nabla_ XT(Y,Z)\}=0.\) b) Let (M,g) be a Riemannian manifold, \(\nabla\) and \({\tilde \nabla}\) two \(\pi\)-, resp. \({\tilde \pi}\)-semi-symmetric metric connections. If d\({\tilde \pi}=0\), the Ricci tensor \(c_ 1\tilde R\) of \({\tilde \nabla}\) is nondegenerate, and \(\sigma_{(X,Y,Z)}\{(\nabla_ X\tilde R)(Y,Z,V)+\tilde R(T(X,Y),Z)V\}=0,\) then \(\nabla ={\tilde \nabla}\). c) If d\({\tilde \pi}=0\), \(\det | c_ 1\tilde R| \neq 0\) and \(R=\tilde R\), then \(\nabla ={\tilde \nabla}\).