The well adapted connection of a $$(J^{2}=\pm 1)$$ ( J 2 = ± 1 ) -metric manifold

Abstract

In this paper, we study the well adapted connection attached to a (J 2 = ±1)-metric manifold, proving it exists for any of the four geometries and obtaining a explicit formula as a derivation law. Besides we characterize the coincidence of the well adapted connection with the Levi Civita and the Chern connections.

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

“…The G-structure defined by an (α, ε)-structure will be denoted as a G (α,ε) -structure. The corresponding structure groups and Lie algebras have been studied in [7]. In particular one has: As in the case of an α-structure, we introduce the following:…”

“…In [6] we have studied in a unified way the geometric properties of (J 2 = ±1)-metric manifolds. In the recent paper [7] we introduce the well adapted connection of any (J 2 = ±1)-metric manifold, thus being our first approach to this unified vision of connections in the four geometries.…”

“…2). We follow recalling the unified presentation of the Chern connection in the case αε = −1 obtained in [7] (Theorem 5. 3).…”

“…Integrabilty means M is a holomorphic manifold in cases i) and ii), and M has two complementary foliations in cases iii) and iv).A linear connection is said to be reducible, natural or adapted to a manifold M having an (α, ε)-structure (J, g) if its covariant derivative ∇ a parallelizes both structures, i.e., ∇ a J = 0, ∇ a g = 0. The most significative natural connection is the well adapted connection ∇ w , which has been intensively studied in [7]. We say that it is the most significative connection because it measures the integrability of the G (α,ε) -structure defined by (J, g): it is integrable if and only if the torsion and the curvature tensors of the well adapted connection vanish.…”

On the Geometry of Almost Golden Riemannian Manifolds

“…The first canonical connection is not the unique distinguished adapted connection to the almost Golden Riemannian structure which can be introduced with the help of the induced almost product Riemannian structure. There exists another adapted connection previously introduced in [4], attached to an almost product Riemannian structure in the paracomplex case; i.e., r = s. However, the most part of the results and proofs showed there are also still valid in the case r = s.…”