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# Article

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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
Summary:
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.
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