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

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 $(J^2=\pm1)$-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. We also prove that every connection studied in this paper is a canonical connection, when it exists and it is an adapted connection.Comment: Corrected typos and updated reference

On the Geometry of Almost Golden Riemannian Manifolds

“…The first canonical connection have been studied in a particular case of almost product Riemannian manifolds in [5]. More precisely, Lemmas 3.10 and 3.12 of that paper prove that ∇ 0 is adapted to the almost product Riemannian structure and show how to build the set of adapted covariant derivatives starting from ∇ 0 .…”

“…; [8] and the references therein). An almost product structure is a particular case of an α-structure, which is a way of generalizing simultaneously almost product and almost complex structures on a manifold (see [5,Sec. 2]).…”

“…Proof. Bearing in mind that one can describe the set of natural derivation laws of J ϕ by means of ∇ 0 as above (see [5,Prop. 2.4]), Lemma 3.1 allows to claim that the set of adapted derivation laws to an almost Golden structure ϕ is the set presented above.…”

“…This set coincides with the previous one. An easy calculation shows that the set L of the previous proposition is the image of T 1 2 (M ) by the associated Obata operator (see [5,Prop. 2.3]).…”

“…The most known manifolds in these conditions are almost Riemannian product with null trace manifolds. This kind of manifolds are a particular class of (J 2 = ±1)-metric ones (see [5]). If the metric g satisfies the condition (1.2) then g is called a pure metric respect to the polynomial structure J (see, e.g., [6] and [10]).…”