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RECURRENT Z FORMS ON RIEMANNIAN AND KAEHLER MANIFOLDS

CARLO ALBERTO MANTICA

Physics Department, Università Degli Studi di Milano, Via Celoria 16, 20133, Milano, Italy

I.I.S. Lagrange, Via L. Modignani 65, 20161, Milano, Italy

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and

YOUNG JIN SUH

Department of Mathematics, Kyungpook National University, Taegu 702-701, Korea

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https://doi.org/10.1142/S0219887812500594Cited by:19 (Source: Crossref)

Abstract

In this paper, we introduce a new kind of Riemannian manifold that generalize the concept of weakly Z-symmetric and pseudo-Z-symmetric manifolds. First a Z form associated to the Z tensor is defined. Then the notion of Z recurrent form is introduced. We take into consideration Riemannian manifolds in which the Z form is recurrent. This kind of manifold is named (ZRF)_n. The main result of the paper is that the closedness property of the associated covector is achieved also for rank(Z_kl) > 2. Thus the existence of a proper concircular vector in the conformally harmonic case and the form of the Ricci tensor are confirmed for(ZRF)_n manifolds with rank(Z_kl) > 2. This includes and enlarges the corresponding results already proven for pseudo-Z-symmetric (PZS)_n and weakly Z-symmetric manifolds (WZS)_n in the case of non-singular Z tensor. In the last sections we study special conformally flat (ZRF)_n and give a brief account of Z recurrent forms on Kaehler manifolds.

Keywords:

AMSC: Primary 53C15, Secondary 53C25

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