3-quasi-Sasakian manifolds

In the present paper we carry on a systematic study of 3-quasi-Sasakian manifolds. In particular we prove that the three Reeb vector fields generate an involutive distribution determining a canonical totally geodesic and Riemannian foliation. Locally, the leaves of this foliation turn out to be Lie groups: either the orthogonal group or an abelian one. We show that 3-quasi-Sasakian manifolds have a well-defined rank, obtaining a rank-based classification. Furthermore, we prove a splitting theorem for these manifolds assuming the integrability of one of the almost product structures. Finally, we show that the vertical distribution is a minimum of the corrected energy.Comment: 17 pages, minor modifications, references update

On a two-dimensional analogue of the Lebesgue function for Fourier-Chebyshov sums

This article considers the problem of approximating a function of two variables f(x,y) by Fourier sums over Chebyshev polynomials orthogonal on a discrete grid. The paper shows that if Tα,βn(x, N) (α,β > -1, n = 0,1,2,...) are classical Chebyshev polynomials, orthogonal on a discrete grid, then the system of polynomials of two variables {Zα,βm,n(x, y)}km,n=0 = {Tα,βm(x, N)}km,n=0, where k = m + n ≤ N −1 is orthogonal on the set ΩN × N = {(xi, yj)}i,j=1N - 1. For an arbitrary function f(x,y) continuous on [−1,1]2, partial Fourier Chebyshev sums Sα,βm,n,N(f, x,y) are constructed over the system of polynomials τα,βm,n,N(x,y) orthonormal on the grid ΩN × N = {(xi, yj)}i,j=1N - 1. The task is to estimate the partial sum Sα,βm,n,N(f, x,y) of the Fourier series of the function f(x,y) from the system of polynomials τα,βm,n,N(x,y) from the function itself f ∈ C[-1, 1]2, in the case when(x,y) ∈ [−1,1]×[−1,1], which in turn reduces to the problem of estimating the Lebesgue function Wα,βm,n,N(x,y). The result of the work is a theorem in which it is proved that the order of the Lebesgue constants ‖Sα,βm,n,N‖ of the indicated discrete sums under certain conditions is O((mn)q+1/2), where q = max{α,β}. As a consequence of the obtained result, some approximation properties of discrete sums Sα,βm,n,N(f, x,y) are considered

On Some Aspects of Geometry of Almost C(λ)-manifolds

In this paper almost C(λ)-manifolds are considered. The local structure of Ricci-flat almost C(λ)-manifolds is obtained. On the space of the adjoint G-structure, necessary and sufficient conditions are obtained under which the al-most C(λ)-manifolds are manifolds of constant curvature and the structure of the Riemannian curvature tensor of an almost C(λ)-manifold of constant curvature is obtained. Relations are obtained that characterize the Einstein almost C(λ)-manifolds. It is proved that a complete almost C(λ)-Einstein manifold is either holomorphically isometrically covered by the product of a real line by a Ricciflat Kähler manifold, or is compact and has a finite fundamental group. For almost C(λ)-manifolds that are -Einstein, analytic expressions for the functions  and  characterizing these manifolds are obtained. It is shown that an almost C(λ)-manifold has an Ф-invariant Ricci tensor. We study also almost C(λ)-manifolds of pointwise constant Ф-holomorphic sectional curvature.</jats:p

AXIOM OF Φ-HOLOMORPHIC (2r+1)-PLANES FOR GENERALIZED KENMOTSU MANIFOLDS

In this paper we study generalized Kenmotsu manifolds (shortly, a GK-manifold) that satisfy the axiom of Φ-holomorphic (2r+1)-planes. After the preliminaries we give the definition of generalized Kenmotsu manifolds and the full structural equation group. Next, we define Φ- holomorphic generalized Kenmotsu manifolds and Φ-paracontact generalized Kenmotsu manifold give a local characteristic of this subclasses. The Φ-holomorphic generalized Kenmotsu manifold coincides with the class of almost contact metric manifolds obtained from closely cosymplectic manifolds by a canonical concircular transformation of nearly cosymplectic structure. A Φ- paracontact generalized Kenmotsu manifold is a special generalized Kenmotsu manifold of the second kind. An analytical expression is obtained for the tensor of Ф-holomorphic sectional curvature of generalized Kenmotsu manifolds of the pointwise constant Φ-holomorphic sectional curvature. Then we study the axiom of Φ-holomorphic (2r+1)-planes for generalized Kenmotsu manifolds and propose a complete classification of simply connected generalized Kenmotsu manifolds satisfying the axiom of Φ-holomorphic (2r+1)-planes. The main results are as follows. A simply connected GK-manifold of pointwise constant Φ-holomorphic sectional curvature satisfying the axiom of Φ-holomorphic (2r+1)-planes is a Kenmotsu manifold. A GK-manifold satisfies the axiom of Φ-holomorphic (2r+1)-planes if and only if it is canonically concircular to one of the following manifolds: (1) CPn×R; (2) Cn×R; and (3) CHn×R having the canonical cosymplectic structure.</jats:p