Some Characterizations for a Quaternion-Valued and Dual Variable Curve

Quaternions, which are found in many fields, have been studied for a long time. The interest in dual quaternions has also increased after real quaternions. Nagaraj and Bharathi developed the basic theories of these studies. The Serret–Frenet Formulae for dual quaternion-valued functions of one real variable are derived. In this paper, by making use of the results of some previous studies, helixes and harmonic curvature concepts in Q D 3 and Q D 4 are considered and a characterization for a dual harmonic curve to be a helix is given.

Pseudo B-symmetric manifolds

In this paper, we introduce a new tensor named [Formula: see text]-tensor which generalizes the [Formula: see text]-tensor introduced by Mantica and Suh [Pseudo [Formula: see text] symmetric Riemannian manifolds with harmonic curvature tensors, Int. J. Geom. Methods Mod. Phys. 9(1) (2012) 1250004]. Then, we study pseudo-[Formula: see text]-symmetric manifolds [Formula: see text] which generalize some known structures on pseudo-Riemannian manifolds. We provide several interesting results which generalize the results of Mantica and Suh [Pseudo [Formula: see text] symmetric Riemannian manifolds with harmonic curvature tensors, Int. J. Geom. Methods Mod. Phys. 9(1) (2012) 1250004]. At first, we prove the existence of a [Formula: see text]. Next, we prove that a pseudo-Riemannian manifold is [Formula: see text]-semisymmetric if and only if it is Ricci-semisymmetric. After this, we obtain a sufficient condition for a [Formula: see text] to be pseudo-Ricci symmetric in the sense of Deszcz. Also, we obtain the explicit form of the Ricci tensor in a [Formula: see text] if the [Formula: see text]-tensor is of Codazzi type. Finally, we consider conformally flat pseudo-[Formula: see text]-symmetric manifolds and prove that a [Formula: see text] spacetime is a [Formula: see text]-wave under certain conditions.