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 Riemannian manifolds with a semi${-}$symmetric metric $P$-connection J. Korean Math. Soc.Published online May 17, 2019 Sudhakar Kr Chaubey, Jae Won Lee, and Sunil Kr Yadav Shinas College of Technology, Gyeongsang National University, Poornima College of Engineering Abstract : We define a class of semi-symmetric metric connection on a Riemannian manifold for which the conformal, the projective, the concircular, the quasi conformal and the $m$-projective curvature tensors are invariant. We also study the properties of semi-symmetric, Ricci semi-symmetric and Eisenhart problems for solving second order parallel symmetric and skew-symmetric tensors on the Riemannian manifolds equipped with a semi-symmetric metric $P$-connection. Keywords : Riemannian manifolds, second order parallel tensors, Ricci solitons, symmetric spaces, semi-symmetric metric connection, different curvature tensors MSC numbers : 53D10, 53C25, 53D15 Full-Text :