[Paper Review] Ricci solitons on Ricci pseudosymmetric $(LCS)_n$-manifolds

Shyamal Kumar Hui, Richard S. Lemence+1 more|arXiv (Cornell University)|Jul 12, 2017

Geometric Analysis and Curvature Flows21 references3 citations

TL;DR

This paper investigates Ricci solitons on $(LCS)_n$-manifolds that are Ricci pseudosymmetric with respect to various curvature tensors—concircular, projective, $W_3$, conharmonic, and conformal. It derives explicit expressions for the function $L_S$ in each case and identifies critical values of $L_S$ that determine whether the Ricci soliton is shrinking, steady, or expanding, with a concrete example constructed for the concircular case in dimension 3.

ABSTRACT

The object of the present paper is to study some types of Ricci pseudosymmetric $(LCS)_n$-manifolds whose metric is Ricci soliton. We found the conditions when Ricci soliton on concircular Ricci pseudosymmetric, projective Ricci pseudosymmetric, $W_{3}$-Ricci pseudosymmetric, conharmonic Ricci pseudosymmetric, conformal Ricci pseudosymmetric $(LCS)_n$-manifolds to be shrinking, steady and expanding. We also construct an example of concircular Ricci pseudosymmetric $(LCS)_3$-manifold whose metric is Ricci soliton.

Motivation & Objective

• To investigate Ricci solitons on $(LCS)_n$-manifolds that are Ricci pseudosymmetric with respect to different curvature tensors.
• To determine the conditions under which such Ricci solitons are shrinking, steady, or expanding.
• To derive explicit expressions for the function $L_S$ associated with each type of Ricci pseudosymmetry.
• To construct a concrete example of a concircular Ricci pseudosymmetric $(LCS)_3$-manifold supporting a Ricci soliton.
• To identify the critical value of $L_S$ that separates the three soliton types (shrinking, steady, expanding) for each curvature type.

Proposed method

• Define Ricci pseudosymmetry via the condition $(R(X,Y)\cdot S)(Z,U) = L_S \cdot Q(g,S)(Z,U;X,Y)$ on the set where $S \neq \frac{r}{n}g$.
• Apply the Ricci soliton condition $\mathcal{L}_V g + 2S + 2\lambda g = 0$ to the $(LCS)_n$-manifold structure.
• Utilize the geometric properties of $(LCS)_n$-manifolds, including the concircular vector field $P$, the 1-form $\eta$, and the associated functions $\alpha$, $\rho$, and $\lambda$.
• Compute the action of each curvature tensor (concircular, projective, $W_3$, conharmonic, conformal) on the Ricci tensor and derive $L_S$ expressions via tensorial identities.
• Use the derived $L_S$ expressions to analyze the sign of $\lambda$, which determines the soliton type (shrinking, steady, expanding).
• Construct a 3-dimensional example of a concircular Ricci pseudosymmetric $(LCS)_3$-manifold with a Ricci soliton to verify Theorem 3.1.

Experimental results

Research questions

• RQ1 Under what conditions is a Ricci soliton on a concircular Ricci pseudosymmetric $(LCS)_n$-manifold shrinking, steady, or expanding?
• RQ2 How does the function $L_S$ vary across different types of Ricci pseudosymmetry (projective, $W_3$, conharmonic, conformal) in $(LCS)_n$-manifolds?
• RQ3 What is the critical value of $L_S$ that determines the soliton type in each curvature-based Ricci pseudosymmetric $(LCS)_n$-manifold?
• RQ4 Can a concrete example of a concircular Ricci pseudosymmetric $(LCS)_3$-manifold with a Ricci soliton metric be explicitly constructed?
• RQ5 How do the geometric invariants $\alpha$, $\rho$, and $n$ influence the classification of Ricci solitons in these manifolds?

Key findings

• For concircular Ricci pseudosymmetric $(LCS)_n$-manifolds, $L_S = (\alpha^2 - \rho) + \frac{\lambda}{n-1} + \frac{\alpha}{n}$, and the soliton is shrinking, steady, or expanding depending on whether $L_S < \frac{\alpha}{n} + (\alpha^2 - \rho)$, $= \frac{\alpha}{n} + (\alpha^2 - \rho)$, or $> \frac{\alpha}{n} + (\alpha^2 - \rho)$.
• In projective Ricci pseudosymmetric $(LCS)_n$-manifolds, $L_S = \left(1 + \frac{2\lambda}{\alpha}\right)(\alpha^2 - \rho)$, and the critical value for $L_S$ is $\alpha^2 - \rho$, provided $\alpha > \alpha^2 - \rho$.
• For $W_3$-Ricci pseudosymmetric $(LCS)_n$-manifolds, $L_S = \left(1 + \frac{2\lambda}{\alpha}\right)(\alpha^2 - \rho) - \frac{\alpha + \lambda}{n-1}$, with critical value $\alpha^2 - \rho - \frac{\alpha}{n-1}$, valid when $\frac{(n-1)\alpha}{2(n-1)(\alpha^2) - \alpha} > 0$.
• In conharmonic Ricci pseudosymmetric $(LCS)_n$-manifolds, $L_S = \frac{\alpha + 2\lambda}{n-2} - (\alpha^2 - \rho)$, and the critical value is $\frac{\alpha}{n-2} - (\alpha^2 - \rho)$.
• For conformal Ricci pseudosymmetric $(LCS)_n$-manifolds, $L_S = \frac{\lambda}{n-1} - (\alpha^2 - \rho)$, and the critical value is $-(\alpha^2 - \rho)$, leading to $\lambda < 0$, $= 0$, or $> 0$ when $L_S < -(\alpha^2 - \rho)$, $= -(\alpha^2 - \rho)$, or $> -(\alpha^2 - \rho)$ respectively.
• An explicit example of a concircular Ricci pseudosymmetric $(LCS)_3$-manifold with a Ricci soliton metric is constructed, verifying Theorem 3.1 and confirming the derived $L_S$ expression and soliton type classification.