Distinguished connections on $(J^{2}=\pm 1)$-metric manifolds

Etayo, Fernando ; Santamaría, Rafael

Archivum mathematicum, Tome 52 (2016) no. 3, pp. 159-203.

Résumé

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.

Détail

DOI : 10.5817/AM2016-3-159

Classification : 53C05, 53C07, 53C15, 53C50
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