On nearly Sasakian and nearly cosymplectic manifolds

Abstract

We prove that every nearly Sasakian manifold of dimension greater than five is Sasakian. This provides a new criterion for an almost contact metric manifold to be Sasakian. Moreover, we classify nearly cosymplectic manifolds of dimension greater than five

Nearly Sasakian manifolds revisited

“…Definition 2.3. A nearly Sasakian manifold is an almost contact metric manifold (M, g, φ, ξ, η) such that (4) (∇ X φ)X = g(X, X)ξ − η(X)X.…”

“…Concerning other dimensions, there have been many attempts of finding explicit examples of proper nearly Sasakian non-Sasakian manifolds until the recent result obtained in [4] showing that every nearly Sasakian structure of dimension greater than five is always Sasakian. Such result depends on the early work [3] by the first and third authors, which in turn draws many properties proved in [6].…”

Some Results on Nearly Cosymplectic Manifolds

“…is skew symmetric and anticommutes with ϕ. It satisfies 5) and the following formulas hold [3], [4] g…”

A Geometric Obstruction for CR-Slant Warped Products in a Nearly Cosymplectic Manifold

“…In the subsequent literature on this topic, quite important were the papers of H. Endo [3,4]. The best known example of a non-cosymplectic nearly cosymplectic manifold is the 5-sphere S 5 as a totally geodesic hypersurface in S 6 .…”

“…Furthermore, They proved that every 5-dimensional nearly cosymplectic manifold is an Einstein manifold with positive scalar curvature. In [6], the authors proved that a non-cosymplectic nearly cosymplectic manifoldM of dimension 2n + 1 > 5 is locally isometric to one of the Riemannian products: R ×Ñ 2n ,M 5 ×Ñ 2n−4 , whereÑ 2n is a non-Kaehler nearly Kaehler manifold, N 2n−4 is a nearly Kaehler manifold, andM 5 is a non-cosymplectic nearly cosymplectic manifold.…”