Notes on Para-Norden–walker 4-Manifolds

Abstract

A Walker 4-manifold is a pseudo-Riemannian manifold, (M4, g) of neutral signature, which admits a field of parallel null 2-plane. The main purpose of the present paper is to study almost paracomplex structures on 4-dimensional Walker manifolds. We discuss sequently the problem of integrability, para-Kähler (paraholomorphic), quasi-para-Kähler and isotropic para-Kähler conditions for these structures. The curvature properties for para-Norden–Walker metrics with respect to the almost paracomplex structure and some properties of para-Norden–Walker metrics in context of almost product Riemannian manifolds are also investigated. Also, we discuss the Einstein conditions for these structures.

On operators associated with tensor fields

“…Many authors, among them Ganchev [2], Kruchkovich [7,8], Norden [16], Salimov [17][18][19][20][21][22][23][24][25][26][27][28][29], Sato [30], Sekizawa [31], Shirokov [33], Tachibana [34,35], Vishnevskii [37][38][39][40][41], Yano [43,44] have considered pure tensors with respect to an algebraic structure.…”

Almost analytic forms with respect to a quadratic endomorphism and their cohomology

“…) is an usual almost Hermitian manifold, b) for ε = +1 the triple (M, F, g) is an almost para-Norden manifold; see, for instance, [11].…”

On the (1, 1)-tensor bundle with Cheeger–Gromoll type metric

“…A Riemannian almost product manifold (M, ϕ, g) is a Riemannian almost product W 3 -manifold if σ X,Y,Z g((∇ X ϕ)Y, Z) = 0, where σ is the cyclic sum by X, Y, Z [15]. In [12], the authors proved that σ for any X ∈ 1 0 (M) and A ∈ 1 1 (M). The Cheeger-Gromoll type metric CG g is pure with respect to J, i.e.…”