The theory of Lie derivatives and its applications

In this paper we introduce a new notion of Z-tensor and a new kind of Riemannian manifold that generalize the concept of both pseudo Ricci symmetric manifold and pseudo projective Ricci symmetric manifold. Here the Z-tensor is a general notion of the Einstein gravitational tensor in General Relativity. Such a new class of manifolds with Z-tensor is named pseudoZ symmetric manifold and denoted by (PZS)n. Various properties of such an n-dimensional manifold are studied, especially focusing the cases with harmonic curvature tensors giving the conditions of closeness of the associated one-form. We study (PZS)n manifolds with harmonic conformal and quasi-conformal curvature tensor. We also show the closeness of the associated 1-form when the (PZS)n manifold becomes pseudo Ricci symmetric in the sense of Deszcz (see [A. Derdzinsky and C. L. Shen, Codazzi tensor fields, curvature and Pontryagin forms, Proc. London Math. Soc.47(3) (1983) 15–26; R. Deszcz, On pseudo symmetric spaces, Bull. Soc. Math. Belg. Ser. A44 (1992) 1–34]). Finally, we study some properties of (PZS)4 spacetime manifolds.

/pdf/pseudo-z-symmetric-riemannian-manifolds-with-harmonic-2n194fjyf3.pdf

Pseudo z symmetric riemannian manifolds with harmonic curvature tensors

In this paper we study the closedness properties of generalized cur- vature 2-forms, which are said to be Riemannian, Conformal, Projective, Concircular and Conharmonic curvature 2-forms, associated to each generalized curvature tensors on a Riemannian manifold. Corresponding to each curvature tensors, such generalized curvature 2-forms are the associated curvature 2-forms. In particular, we focus on the closedness of difierential 2-forms associated to the divergence of generalized curvature tensors, which is weaker than the notion of harmonic curvature. In this case, we give an algebraic condition involving the Riemann curvature tensor and the Ricci tensor arising from an old identity due to Lovelock.

/pdf/the-closedness-of-some-generalized-curvature-2-forms-on-a-2rbzmaxpms.pdf

The closedness of some generalized curvature 2-forms on a Riemannian manifold I

In this paper, we introduce the notion of recurrent conformal 2-forms on a pseudo-Riemannian manifold of arbitrary signature. Some theorems already proved for the same differential structure on a Riemannian manifold are proven to hold in this more general contest. Moreover other interesting results are pointed out; it is proven that if the associated covector is closed, then the Ricci tensor is Riemann compatible or equivalently, Weyl compatible: these notions were recently introduced and investigated by one of the present authors. Further some new results about the vanishing of some Weyl scalars on a pseudo-Riemannian manifold are given: it turns out that they are consequence of the generalized Derdzinski–Shen theorem. Topological properties involving the vanishing of Pontryagin forms and recurrent conformal 2-forms are then stated. Finally, we study the properties of recurrent conformal 2-forms on Lorentzian manifolds (space-times). Previous theorems stated on a pseudo-Riemannian manifold of arbitrary signature are then interpreted in the light of the classification of space-times in four or in higher dimensions.

Recurrent conformal 2-forms on pseudo-Riemannian manifolds

http://webbuild.knu.ac.kr/~yjsuh/Publications/2004/[2004]4%20Baek,%20Cheng,%20Suh.pdf

Complete space-like hypersurfaces in locally symmetric Lorentz spaces

In this paper we introduce new notions of Ricci-like tensor and many kind of curvature-like tensors such that concir- cular, 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).

On semi-riemannian manifolds satisfying the second bianchi identity

https://projecteuclid.org/download/pdf_1/euclid.rmjm/1181069782

Conformally Recurrent Semi-Riemannian Manifolds

The purpose of this paper is to give a theorem of Liouvilletype for complete Riemannian manifolds as an extension of the Theorem of Nishikawa [6].

A liouville-type theorem for complete riemannian manifolds

http://webbuild.knu.ac.kr/~yjsuh/Publications/2006/%5B2006%5D3%20Suh,Kwon,Yang.pdf

Conformally symmetric semi-Riemannian manifolds

O'Neill introduced a notion of Riemannian submersion [7]. In this paper we give a new notion of semi-Riemannian submersion and want to investigate some geometric properties concerned with sectional and Ricci curvatures of this submersion.

https://projecteuclid.org/download/pdf_1/euclid.kmj/1138043720

Jung-Hwan Kwon

Papers

On semi-riemannian manifolds satisfying the second bianchi identity

Conformally Recurrent Semi-Riemannian Manifolds

A liouville-type theorem for complete riemannian manifolds

Conformally symmetric semi-Riemannian manifolds

On sectional and Ricci curvatures of semi-Riemannian submersions