On almost cosymplectic manifolds
Zbigniew Olszak1981, Kodai Mathematical Journal
The purpose of the present paper is to study the structure of almost cosymplectic manifolds. §2 presents the basic definitions and some preliminary properties of an almost cosymplectic structure. Examples of such structures are given in § 3. In § 4 we state many important curvature identities for almost cosymplectic manifolds. In § 5 we give certain sufficient conditions for an almost contact metncstructure to be almost cosymplectic. Basing on the identities from §4 we prove in §6 that almost cosymplectic manifolds of non-zero constant sectional curvature do not exist in dimensions greater than three. However it is known (cf. [2]) that such manifolds of zero sectional curvature (i. e. locally flat) exist and they are cosymplectic. Moreover we give certain restrictions on the scalar curvature of almost cosymplectic manifolds which are conformally flat or of constant ^-sectional curvature. All manifolds considered in this paper are assumed to be connected and of class C°°. All tensor fields, including differential forms, are of class C°°. The notation and terminology will be the same as that employed in [2]. § 2. Preliminaries. Let (M, φ, ξ, 77, g) be a (2n+l)-dimensional almost contact metric manifold, that is, Mis a differentiate manifold and {φ, ξ, η, g) an almost contact metric structure on M, formed by tensor fields φ, ξ, η of type (1, 1), (1, 0), (0, 1), respectively, and a Riemannian metric g such that φξ=O, η°φ=0, η(X)=g(X, ξ), g(φX, φY)=g(X, Y)-η(X)η{Y). On such manifold we may always define a 2-form Φ by Φ(X, Y)~g(φX, Y). (Λf, φ, ξ, 7], g) is said to be an almost cosymplectic manifold (cf. [2]) if the forms Φ and Ύ] are closed, i.e. dΦ=0 and dη^O, where d is the operator of exterior differentiation. In particular, if the almost contact structure of an almost cosymplectic manifold is normal, then it is said to be a cosymplectic manifold (cf. [1]).
