Some remarks concerning hyperholomorphic B-manifolds

Abstract

The authors consider a differentiable manifold with Π-structure which is an isomorphic representation of an associative, commutative and unitial algebra. For Riemannian metric tensor fields, the Φ-operators associated with r-regular Π-structure are introduced. With the help of Φ-operators, the hyperholomorphity condition of B-manifolds is established.

On operators associated with tensor fields

“…We note that, when ϕ is an almost complex structure on M , a tensor field t of type (r, s) which satisfies t ∈ Kerφ ϕ is said to be almost analytic(holomorphic)(see, for example, [5,7,25,30,[34][35][36]44]). Analytic(holomorphic) function with respect to the different algebra is studied in [1,7,24,40].…”

“…Tachibana operators of the type considered and their generalizations by using an algebraic structures were studied in [7,17,18,25,33]. The applications of the Tachibana operators to the theory of complete lifts in the tensor bundle are studed in [14,19,20,23].…”

“…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.…”

Applications of $${\Phi}$$ -Operators to Hypercomplex Geometry

“…Condition (1) will be called the Scheffers condition [9] . In particular, in case of the algebra of complex numbers A 2 = R(i), i 2 = −1, the Scheffers condition coincides with Cauchy-Riemann conditions.…”

“…Explicit formulae of Φ ϕ -operator for pure tensor fields of types (1, q) and (o, q) are given in [9]. Also in [9] derives relations between the geometry of hyperholomorphic B-manifolds (Norden manifolds) and Φ ϕ -operator. We note that, Φ ϕ -operator is extension of the operator of Lie derivation L X , X ∈ 1 0 (M n ) to affinor fields.…”