Czechoslovak Mathematical Journal, Vol. 67, No. 1, pp. 197-206, 2017


A characterization of the Riemann extension in terms of harmonicity

Cornelia-Livia Bejan, Şemsi Eken

Received August 25, 2015.  First published February 24, 2017.

Abstract:  If $(M,\nabla)$ is a manifold with a symmetric linear connection, then $T^*M$ can be endowed with the natural Riemann extension $\bar{g}$ (O. Kowalski and M. Sekizawa (2011), M. Sekizawa (1987)). Here we continue to study the harmonicity with respect to $\bar{g}$ initiated by C. L. Bejan and O. Kowalski (2015). More precisely, we first construct a canonical almost para-complex structure $\mathcal{P}$ on $(T^*M,\bar{g})$ and prove that $\mathcal{P}$ is harmonic (in the sense of E. Garciá-Río, L. Vanhecke and M. E. Vázquez-Abal (1997)) if and only if $\bar{g}$ reduces to the classical Riemann extension introduced by E. M. Patterson and A. G. Walker (1952).
Keywords:  semi-Riemannian manifold; cotangent bundle; natural Riemann extension; harmonic tensor field
Classification MSC:  53C07, 53C50, 53B05, 53C43, 58E20
DOI:  10.21136/CMJ.2017.0459-15


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Affiliations:   Cornelia-Livia Bejan, Department of Mathematics, \Gh. Asachi\ Technical University of Iaşi, Postal address: Seminar Matematic, Universitatea \Al. I. Cuza\ Iaşi, Bd. Carol I no. 11, Iaşi, 700506, Romania, e-mail: bejanliv@yahoo.com; Şemsi Eken, Department of Mathematics, Mersin University, Çiftlikköy Merkez Street, 33343 Yenişehir, Mersin, Turkey, e-mail: semsieken@hotmail.com

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