Sewing cells in almost cosymplectic and almost Kenmotsu geometry

TL;DR

This paper introduces a method to construct almost contact metric manifolds by sewing together 3-dimensional building blocks called cells, and studies how nullity conditions are preserved or altered in the resulting manifolds.

Contribution

It provides a new construction technique for almost contact metric manifolds from cells and analyzes the nullity conditions in the sewn manifolds, including generalized cases.

Findings

Sewn manifolds inherit nullity conditions from cells with possible constant modifications.

Construction applies to almost cosymplectic and almost α-Kenmotsu manifolds.

Generalized nullity conditions are preserved in sewn manifolds if satisfied by cells.

Abstract

For a finite family of 3-dimensional almost contact metric manifolds with closed the structure form $\eta$ is described a construction of an almost contact metric manifold, where the members of the family are building blocks - cells. Obtained manifold share many properties of cells. One of the more important are nullity conditions. If cells satisfy nullity conditions - then - in the case of almost cosymplectic or almost $\alpha$-Kenmotsu manifolds - "sewed cells" also satisfies nullity condition - but generally with different constants. It is important that even in the case of the generalized nullity conditions - "sewed cells" are the manifolds which satisfy such conditions provided the cells satisfy the generalized nullity conditions.