Distinguished connections on ( J 2 = ± 1 ) -metric manifolds

Fernando Etayo; Rafael Santamaría

Archivum Mathematicum (2016)

  • Volume: 052, Issue: 3, page 159-203
  • ISSN: 0044-8753

Abstract

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We study several linear connections (the first canonical, the Chern, the well adapted, the Levi Civita, the Kobayashi-Nomizu, the Yano, the Bismut and those with totally skew-symmetric torsion) which can be defined on the four geometric types of ( J 2 = ± 1 ) -metric manifolds. We characterize when such a connection is adapted to the structure, and obtain a lot of results about coincidence among connections. We prove that the first canonical and the well adapted connections define a one-parameter family of adapted connections, named canonical connections, thus extending to almost Norden and almost product Riemannian manifolds the families introduced in almost Hermitian and almost para-Hermitian manifolds in [13] and [18]. We also prove that every connection studied in this paper is a canonical connection, when it exists and it is an adapted connection.

How to cite

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Etayo, Fernando, and Santamaría, Rafael. "Distinguished connections on $(J^{2}=\pm 1)$-metric manifolds." Archivum Mathematicum 052.3 (2016): 159-203. <http://eudml.org/doc/286697>.

@article{Etayo2016,
abstract = {We study several linear connections (the first canonical, the Chern, the well adapted, the Levi Civita, the Kobayashi-Nomizu, the Yano, the Bismut and those with totally skew-symmetric torsion) which can be defined on the four geometric types of $(J^2=\pm 1)$-metric manifolds. We characterize when such a connection is adapted to the structure, and obtain a lot of results about coincidence among connections. We prove that the first canonical and the well adapted connections define a one-parameter family of adapted connections, named canonical connections, thus extending to almost Norden and almost product Riemannian manifolds the families introduced in almost Hermitian and almost para-Hermitian manifolds in [13] and [18]. We also prove that every connection studied in this paper is a canonical connection, when it exists and it is an adapted connection.},
author = {Etayo, Fernando, Santamaría, Rafael},
journal = {Archivum Mathematicum},
keywords = {$(J^2=\pm 1)$-metric manifold; $\alpha $-structure; natural connection; Nijenhuis tensor; second Nijenhuis tensor; Kobayashi-Nomizu connection; first canonical connection; well adapted connection; connection with totally skew-symmetric torsion; canonical connection; Riemannian almost product structure; para-Hermitian structure; Hermitian structure; Norden structure; biparacomplex; 3-web},
language = {eng},
number = {3},
pages = {159-203},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Distinguished connections on $(J^\{2\}=\pm 1)$-metric manifolds},
url = {http://eudml.org/doc/286697},
volume = {052},
year = {2016},
}

TY - JOUR
AU - Etayo, Fernando
AU - Santamaría, Rafael
TI - Distinguished connections on $(J^{2}=\pm 1)$-metric manifolds
JO - Archivum Mathematicum
PY - 2016
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 052
IS - 3
SP - 159
EP - 203
AB - We study several linear connections (the first canonical, the Chern, the well adapted, the Levi Civita, the Kobayashi-Nomizu, the Yano, the Bismut and those with totally skew-symmetric torsion) which can be defined on the four geometric types of $(J^2=\pm 1)$-metric manifolds. We characterize when such a connection is adapted to the structure, and obtain a lot of results about coincidence among connections. We prove that the first canonical and the well adapted connections define a one-parameter family of adapted connections, named canonical connections, thus extending to almost Norden and almost product Riemannian manifolds the families introduced in almost Hermitian and almost para-Hermitian manifolds in [13] and [18]. We also prove that every connection studied in this paper is a canonical connection, when it exists and it is an adapted connection.
LA - eng
KW - $(J^2=\pm 1)$-metric manifold; $\alpha $-structure; natural connection; Nijenhuis tensor; second Nijenhuis tensor; Kobayashi-Nomizu connection; first canonical connection; well adapted connection; connection with totally skew-symmetric torsion; canonical connection; Riemannian almost product structure; para-Hermitian structure; Hermitian structure; Norden structure; biparacomplex; 3-web
UR - http://eudml.org/doc/286697
ER -

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