On a type of semi-symmetric metric connection on a Riemannian manifold

U.C. De and S.C. Biswas

Abstract: The properties of Riemannian manifolds admitting a semi-symmetric metric connection were studied by many authors ([1], [2], [3], [4], [5], [6]). In [4] an expression of the curvature tensor of a manifold was obtained under assumption that the manifold admits a semi-symmetric metric connection with vanishing curvature tensor and recurrent torsion tensor. Also in [7] Prvanovi\'c and Pusi\'c obtained an expression for curvature tensor of a Riemannian manifold, locally decomposable Riemannian space and the Kähler space which admits a semi-symmetric metric connection $\tilde\nabla $ with vanishing curvature tensor and torsion tensor $T^h_{1m}$ satisfying $\tilde\nabla_k\tilde\nabla_j T^h_{1m}-\tilde\nabla_j\tilde\nabla_k T^h_{1m} =0$. We study a type of semi-symmetric metric connection $\tilde\nabla$ satisfying $\tilde R (X, Y)T=0$ and $\omega(\tilde R(X,Y)Z)=0$, where $T$ is the torsion tensor of the semi-symmetric connection, $\tilde R$ is the curvature tensor corresponding to $\tilde\nabla$ and $\omega$ is the associated 1-form of $T$.

Classification (MSC2000): 53C05

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