[Paper Review] Almost {\alpha}-Cosymplectic ({\kappa},{\mu},{ u})-Spaces

Hakan Öztürk, Nesip Aktan+1 more|arXiv (Cornell University)|Jul 4, 2010

Geometric Analysis and Curvature Flows10 references18 citations

TL;DR

This paper investigates almost $\alpha$-cosymplectic manifolds satisfying a generalized $(\kappa,\mu,\nu)$-nullity condition, proving that for dimensions greater than three, $\kappa$, $\mu$, and $\nu$ need not be constant functions. It establishes invariance under $D$-homothetic deformations and constructs a 3-dimensional example with non-constant $\kappa$ and $\mu$, demonstrating the existence of such structures beyond the constant case.

ABSTRACT

Main interest of the present paper is to investigate the almost {\\alpha}-cosymplectic manifolds for which the characteristic vector field of the almost {\\alpha}-cosymplectic structure satisfies a specific ({\\kappa},{\\mu},{\ u})-nullity condition. This condition is invariant under D-homothetic deformation of the almost cosymplectic ({\\kappa},{\\mu},{\ u})-spaces in all dimensions. Also, we prove that for dimensions greater than three, {\\kappa},{\\mu},{\ u} are not necessary constant smooth functions such that df^{\\eta}=0. Then the existence of the three-dimensional case of almost cosymplectic ({\\kappa},{\\mu},{\ u})-spaces are studied. Finally, we construct an appropriate example of such manifolds.

Motivation & Objective

• To investigate almost $\alpha$-cosymplectic manifolds satisfying a generalized $(\kappa,\mu,\nu)$-nullity condition.
• To determine whether $\kappa$, $\mu$, and $\nu$ can be non-constant smooth functions in dimensions greater than three.
• To prove invariance of the $(\kappa,\mu,\nu)$-nullity condition under $D$-homothetic deformations.
• To establish the existence of 3-dimensional almost $\alpha$-cosymplectic $(\kappa,\mu,\nu)$-spaces with non-constant $\kappa$ and $\mu$.
• To construct an explicit example of a 3-dimensional almost $\alpha$-cosymplectic manifold satisfying the $(\kappa,\mu,\nu)$-nullity condition with non-constant functions.

Proposed method

• Introduces the concept of almost $\alpha$-cosymplectic manifolds and derives curvature formulas involving the tensor field $h$ and the Reeb vector field $\xi$.
• Uses the $D$-homothetic deformation to show invariance of the $(\kappa,\mu,\nu)$-nullity condition across dimensions.
• Applies the $\eta$-parallel condition on $h$ to derive geometric properties of integral submanifolds of the $\mathcal{D}$-distribution.
• Derives a curvature identity for $R(X,Y)\xi$ in terms of $\kappa$, $\mu$, and $\nu$, showing their dependence on $\eta$-closed functions.
• Constructs a 3-dimensional manifold $M^3 = \{(x,y,z) \in \mathbb{R}^3 \mid z \neq 0\}$ with explicitly defined tensor fields $\phi$, $\xi$, $g$, and $\eta$.
• Verifies the almost $\alpha$-cosymplectic structure via $d\eta = 0$ and $d\Phi = 2\alpha \eta \wedge \Phi$, and computes $R(X,Y)\xi$ to identify $\kappa = -(e^{-4\alpha z} + \alpha^2)$, $\mu = 2z$, $\nu = 0$.

Experimental results

Research questions

• RQ1 Can almost $\alpha$-cosymplectic manifolds satisfy the $(\kappa,\mu,\nu)$-nullity condition with non-constant smooth functions $\kappa$, $\mu$, and $\nu$ in dimensions greater than three?
• RQ2 Is the $(\kappa,\mu,\nu)$-nullity condition invariant under $D$-homothetic deformations in all dimensions?
• RQ3 What are the geometric implications of $\eta$-parallelism of the tensor field $h$ in almost $\alpha$-cosymplectic manifolds?
• RQ4 Do 3-dimensional almost $\alpha$-cosymplectic $(\kappa,\mu,\nu)$-spaces exist with non-constant $\kappa$ and $\mu$?
• RQ5 Can an explicit example of a 3-dimensional almost $\alpha$-cosymplectic manifold be constructed that satisfies the $(\kappa,\mu,\nu)$-nullity condition with non-constant $\kappa$ and $\mu$?

Key findings

• For dimensions greater than three, the functions $\kappa$, $\mu$, and $\nu$ in the $(\kappa,\mu,\nu)$-nullity condition are not necessarily constant, but must lie in the subring $\mathcal{R}_\eta(M^{2n+1})$ where $df \wedge \eta = 0$.
• The $(\kappa,\mu,\nu)$-nullity condition is invariant under $D$-homothetic deformations in all dimensions, preserving the structure's geometric properties.
• When the tensor field $h$ is $\eta$-parallel, the integral submanifolds of the $\mathcal{D}$-distribution inherit a K\