Study of G −projective curvature tensor on a Riemannian manifold

Abstract

The object of the present paper is to study some properties of the G-projective curvature tensor and the G*-projective curvature tensor in a Riemannian manifold, which have been defined as\nG(Y, Z, U, T) = R(Y, Z, U, T) − (1/2)(n − 1) [ g(Y, U) Ric(Z, T) − g(Y, T) Ric(Z, U) − g(Z, U) Ric(Y, T) + g(Z, T) Ric(Y, U) ],\nG*(Y, Z, U, T) = R(Y, Z, U, T) − (1/2)(n − 1) [ g(Y, U) Z(Z, T) − g(Y, T) Z(Z, U) − g(Z, U) Z(Y, T) + g(Z, T) Z(Y, U) ],\n\nwhere R is the curvature tensor, Ric is the Ricci tensor (i.e., g(QY, Z) = Ric(Y, Z) for the symmetric endomorphism Q of the tangent space at each point), and Z denotes the Z-tensor of type (0,2).\n\nNotes and suggestions:\n- The original text contained garbled or unusual symbols (for example, extraneous characters in the second formula). The above uses conventional notation: G-projective and G*-projective curvature tensors, R for the Riemann curvature tensor, Ric for the Ricci tensor, g for the metric, and Z as the (0,2)-tensor Z.