A dimensional restriction for a class of contact manifolds

TL;DR

This paper proves that a specific class of contact manifolds with certain geometric structures must be 5-dimensional if their horizontal sectional curvature is constant at one point, revealing a dimensional restriction.

Contribution

The work establishes a new dimensional restriction for a class of contact manifolds with almost contact metric structures under curvature conditions.

Findings

Manifolds in the considered class are necessarily 5-dimensional under the given curvature assumption.

The class includes nearly cosymplectic manifolds and certain classes defined by Chinea and Gonzalez.

The result links geometric curvature conditions to the global dimension of the manifold.

Abstract

In this work we consider a class of contact manifolds $(M,\eta)$ with an associated almost contact metric structure $(\phi, \xi, \eta,g)$. This class contains, for example, nearly cosymplectic manifolds and the manifolds in the class $C_9\oplus C_{10}$ defined by Chinea and Gonzalez. All manifolds in the class considered turn out to have dimension $4n+1$. Under the assumption that the sectional curvature of the horizontal $2$-planes is constant at one point, we obtain that these manifolds must have dimension $5$.