A characterization of the Riemann extension in terms of harmonicity

Cornelia-Livia Bejan; Şemsi Eken

Czechoslovak Mathematical Journal (2017)

  • Volume: 67, Issue: 1, page 197-206
  • ISSN: 0011-4642

Abstract

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If ( M , ) is a manifold with a symmetric linear connection, then T * M can be endowed with the natural Riemann extension g ¯ (O. Kowalski and M. Sekizawa (2011), M. Sekizawa (1987)). Here we continue to study the harmonicity with respect to g ¯ initiated by C. L. Bejan and O. Kowalski (2015). More precisely, we first construct a canonical almost para-complex structure 𝒫 on ( T * M , g ¯ ) and prove that 𝒫 is harmonic (in the sense of E. García-Río, L. Vanhecke and M. E. Vázquez-Abal (1997)) if and only if g ¯ reduces to the classical Riemann extension introduced by E. M. Patterson and A. G. Walker (1952).

How to cite

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Bejan, Cornelia-Livia, and Eken, Şemsi. "A characterization of the Riemann extension in terms of harmonicity." Czechoslovak Mathematical Journal 67.1 (2017): 197-206. <http://eudml.org/doc/287883>.

@article{Bejan2017,
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. García-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).},
author = {Bejan, Cornelia-Livia, Eken, Şemsi},
journal = {Czechoslovak Mathematical Journal},
keywords = {semi-Riemannian manifold; cotangent bundle; natural Riemann extension; harmonic tensor field},
language = {eng},
number = {1},
pages = {197-206},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A characterization of the Riemann extension in terms of harmonicity},
url = {http://eudml.org/doc/287883},
volume = {67},
year = {2017},
}

TY - JOUR
AU - Bejan, Cornelia-Livia
AU - Eken, Şemsi
TI - A characterization of the Riemann extension in terms of harmonicity
JO - Czechoslovak Mathematical Journal
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 1
SP - 197
EP - 206
AB - 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. García-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).
LA - eng
KW - semi-Riemannian manifold; cotangent bundle; natural Riemann extension; harmonic tensor field
UR - http://eudml.org/doc/287883
ER -

References

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