[Paper Review] Almost Kenmotsu metric as quasi Yamabe soliton

Dibakar Dey, Pradip Majhi|arXiv (Cornell University)|May 4, 2020

Geometric Analysis and Curvature Flows17 references4 citations

TL;DR

This paper investigates quasi Yamabe solitons on $(k,\mu)'$-almost Kenmotsu manifolds where the potential vector field $V$ is pointwise collinear with the characteristic vector field $\xi$. It proves that under this condition, $V$ must be a constant multiple of $\xi$, is a strict infinitesimal contact transformation, and satisfies $\pounds_V h' = 0$. The result is illustrated via a 5-dimensional example with $k = -2$, $\mu = -2$, confirming a shrinking quasi Yamabe soliton with $\lambda = -41$.

ABSTRACT

In the present paper, we characterize a class of almost Kenmotsu manifolds admitting quasi Yamabe soliton. It is shown that if a $(k,μ)'$-almost Kenmotsu manifold admits a quasi Yamabe soliton $(g,V,λ,α)$ with $V$ pointwise collinear with $ξ$, then (1) $V$ is a constant multiple of $ξ$, (2) $V$ is a strict infinitesimal contact transformation and (3) $(£_V h')X = 0$ for any vector field $X$. Finally an illustrative example is presented to support the result.

Motivation & Objective

• To characterize $(k,\mu)'$-almost Kenmotsu manifolds admitting a quasi Yamabe soliton with the potential vector field $V$ pointwise collinear to the characteristic vector field $\xi$.
• To investigate the geometric and dynamical properties of such solitons, particularly focusing on the behavior of $V$ and the tensor field $h'$.
• To establish conditions under which the soliton structure implies conservation of the $h'$ tensor under Lie derivative along $V$.
• To provide a concrete example of a shrinking quasi Yamabe soliton in a 5-dimensional $(k,\mu)'$-almost Kenmotsu manifold with $k = -2$, $\mu = -2$.
• To verify that the soliton is shrinking, steady, or expanding based on the value of the constant $b$ in $V = b\xi$.

Proposed method

• Utilizes the definition of a quasi Yamabe soliton: $\frac{1}{2}\pounds_V g = (r - \lambda)g + \alpha V^\# \otimes V^\#$, where $r$ is scalar curvature, $\lambda$ is constant, and $\alpha$ is a smooth function.
• Assumes $V = b\xi$ with $b$ constant, reducing the problem to analyzing the Lie derivative of the metric and connection along $\xi$.
• Employs the structure equations of $(k,\mu)'$-almost Kenmotsu manifolds, including $R(X,Y)\xi = k[\eta(Y)X - \eta(X)Y] + \mu[\eta(Y)h'X - \eta(X)h'Y]$, to compute curvature and Lie derivative actions.
• Derives the expression for $\pounds_V \nabla$ using the identity $g((\pounds_V \nabla)(X,Y),Z) = \frac{1}{2}[(\nabla_X \pounds_V g)(Y,Z) + \cdots]$, and substitutes known expressions for $\pounds_V g$ and $\nabla \eta$.
• Computes $\pounds_V R$ via the formula $(\pounds_V R)(X,Y)Z = (\nabla_X \pounds_V \nabla)(Y,Z) - (\nabla_Y \pounds_V \nabla)(X,Z)$, leading to $\pounds_V R(X,\xi)\xi = 0$.
• Uses the identity $\pounds_V R(X,\xi)\xi = -2\pounds_V h' X$ and equates it to the earlier result to conclude $\pounds_V h' = 0$.

Experimental results

Research questions

• RQ1 What geometric constraints arise when a $(k,\mu)'$-almost Kenmotsu manifold admits a quasi Yamabe soliton with $V$ pointwise collinear to $\xi$?
• RQ2 Under what conditions is $V$ a constant multiple of $\xi$ in such a soliton structure?
• RQ3 Is $V$ a strict infinitesimal contact transformation when $V = b\xi$ with $b$ constant?
• RQ4 Does the tensor field $h'$ remain invariant under the Lie derivative along $V$ in this setting?
• RQ5 Can a concrete example of a shrinking quasi Yamabe soliton be constructed on a 5-dimensional $(k,\mu)'$-almost Kenmotsu manifold?

Key findings

• If a $(k,\mu)'$-almost Kenmotsu manifold admits a quasi Yamabe soliton $(g,V,\lambda,\alpha)$ with $V$ pointwise collinear to $\xi$, then $V$ must be a constant multiple of $\xi$.
• $V$ is a strict infinitesimal contact transformation, meaning $\pounds_V \eta = 0$.
• The Lie derivative of the tensor field $h'$ along $V$ vanishes: $\pounds_V h' = 0$.
• The scalar curvature $r = 2n(k - 2n)$ is constant, and $\lambda = r - b$, with $b$ the constant such that $V = b\xi$.
• For the 5-dimensional example with $k = -2$, $\mu = -2$, $r = -80$, and $V = \xi$, the soliton is shrinking with $\lambda = -41 < 0$.
• The example confirms $\pounds_\xi \eta = 0$ and $\pounds_\xi h' = 0$, verifying the theoretical results.