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

Fernando Etayo and Rafael Santamaría

Corresponding author: Fernando Etayo, Departamento de Matemáticas, Estadística y Computación, Facultad de Ciencias, Universidad de Cantabria, Avda, de los Castros, s/n, 39071 Santander, Spain
Rafael Santamaría, Departamento de Matemáticas, Escuela de Ingenierías Industrial e Informática, Universidad de León, Campus de Vegazana, 24071 León, Spain


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.

AMSclassification: primary 53C15; secondary 53C05, 53C50, 53C07.

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.

DOI: 10.5817/AM2016-3-159