Journal of the
Korean Mathematical Society JKMS

In this paper we introduce new notions of Ricci-like tensor and many kind of curvature-like tensors such that concircular, projective, or conformal curvature-like tensors defined on semi-Riemannian manifolds. Moreover, we give some geometric conditions which are equivalent to the Codazzi tensor, the Weyl tensor, or the second Bianchi identity concerned with such kind of curvature-like tensors respectively and also give a generalization of Weyl's Theorem given in [18] and [19].

Keywords: semi-Riemannian manifold, projective curvature tensor, conformal curvature tensor, Weyl tensor, harmonic Weyl tensor, curvature-like tensor, curvature-like form, closed, coclosed

MSC numbers: 53C20, 53C50