On operators associated with tensor fields

Abstract

The aim of this paper is to introduce some operators which are applied to pure tensor fields. In this context Tachibana, Vishnevskii, Yano-Ako operators and their generalizations can be found. (2000). 53A45, 53C56, 47B47, 15A72. Mathematics Subject Classification

On anti-Hermitian metric connections

“…∇ g = 0) and consequently the connection ∇ = ∇ + 1 2 J (∇ J ) is anti-Hermitian metric connection of type I. From (4) and (9), we see that the torsion tensor of the connection ∇ = ∇ + S is given by:…”

“…In [2,4,5], we have given the anti-Hermitian metric g and considered exclusively the Levi-Civita connection ∇ of g. This is the unique connection that satisfies ∇ g = 0, and has no torsion. But there are many other connections ∇ with torsion parallelizing the metric g. We call these connections anti-Hermitian metric connections.…”

Musical isomorphisms and problems of lifts

“…Structures of this kind have also been studied under the name: Almost complex structures with pure (or B-)metric. An anti-Kähler (Kähler-Norden) manifold can be defined as a triple (M, g, ϕ) which consists of a smooth manifold M endowed with an almost complex structure ϕ and an anti-Hermitian metric g such that ∇ϕ = 0, where ∇ is the Levi-Civita connection of g. It is well known that the condition ∇ϕ = 0 is equivalent to C-holomorphicity (analyticity) of the anti-Hermitian metric g (see [6]), i.e., Z) is the twin anti-Hermitian metric. It is a remarkable fact that (M, g, ϕ) is anti-Kähler if and only if the twin anti-Hermitian structure (M, G, ϕ) is anti-Kähler.…”

Remarks on natural differential operators with tensor fields

“…then ψ is said to be pure with respect to ϕ. Natural R-bilinear differential operators Φ(ϕ, ψ) on pure tensor fields were studied in [4,7]. We recall the main result.…”