[Paper Review] Almost $η$-Ricci solitons in $(LCS)_n$-manifolds

Adara M. Blaga|arXiv (Cornell University)|Jul 28, 2017

Geometric Analysis and Curvature Flows15 references27 citations

TL;DR

This paper investigates almost η-Ricci solitons on (LCS)n-manifolds under curvature conditions (ξ,·)R·S = 0 and (ξ,·)S·R = 0, deriving bounds for the Ricci curvature norm and a Bochner-type formula in the gradient case. The key contribution is a sharp double inequality for |S|² in terms of ∇ξ, μ, |ξ|², Δf, and scal, along with a Bochner-type identity linking the Laplacian of |ξ|² to curvature and soliton functions, revealing geometric constraints on the manifold's scalar curvature and potential vector field behavior.

ABSTRACT

We consider almost $η$-Ricci solitons in $(LCS)_n$-manifolds satisfying certain curvature conditions. We provide a lower and an upper bound for the norm of the Ricci curvature in the gradient case, derive a Bochner-type formula for an almost $η$-Ricci soliton and state some consequences of it on an $(LCS)_n$-manifold.

Motivation & Objective

• To analyze almost η-Ricci solitons in (LCS)n-manifolds under specific curvature conditions (ξ,·)R·S = 0 and (ξ,·)S·R = 0.
• To derive lower and upper bounds for the norm of the Ricci curvature tensor in the gradient case.
• To establish a Bochner-type formula for gradient almost η-Ricci solitons on (LCS)n-manifolds.
• To characterize the scalar curvature and geometric constraints of the manifold under these soliton and curvature conditions.

Proposed method

• Utilizes the definition of almost η-Ricci solitons via the equation Lξg + 2S + 2λg + 2μη⊗η = 0, with λ, μ smooth functions.
• Applies the gradient condition ξ = grad(f), transforming the soliton equation into Hess(f) + S + λg + μη⊗η = 0.
• Derives a Bochner-type formula by taking divergences and traces of the soliton equation and using curvature identities from (LCS)n-structure properties.
• Employs the (LCS)n-structure relations, including ∇ξ = α(I + η⊗ξ), ϕ = I + η⊗ξ, and curvature identities like R(X,Y)ξ = (α²−ρ)(η(Y)X − η(X)Y).
• Uses the Ricci operator Q and its ϕ-invariance to analyze curvature symmetries and derive the norm bounds.
• Applies the maximum principle and differential identities to analyze the behavior of |ξ|² and scalar curvature under soliton conditions.

Experimental results

Research questions

• RQ1 What are the bounds for the norm of the Ricci curvature tensor in a gradient almost η-Ricci soliton on an (LCS)n-manifold?
• RQ2 How does the scalar curvature of an (LCS)n-manifold behave under an almost η-Ricci soliton with curvature conditions (ξ,·)R·S = 0 and (ξ,·)S·R = 0?
• RQ3 Can a Bochner-type formula be derived for gradient almost η-Ricci solitons in (LCS)n-geometry?
• RQ4 What geometric constraints arise when the Ricci tensor is Ricci symmetric or η-recurrent in the context of almost η-Ricci solitons on (LCS)n-manifolds?
• RQ5 Is it possible to classify gradient almost η-Ricci solitons on (LCS)n-manifolds via the derived Bochner-type identity?

Key findings

• A sharp double inequality bounds |S|²: |∇ξ|² + μ²|ξ|⁴ + μ∇ξ(|ξ|²) −(Δf + μ|ξ|²)²/n ≤ |S|² ≤ |∇ξ|² + μ²|ξ|⁴ + μ∇ξ(|ξ|²) + (scal)²/n.
• For a steady gradient almost η-Ricci soliton with scal = 0 and Δf = −μ|ξ|², equality holds in the norm bound if |ξ|² is constant.
• The scalar curvature scal = (1−n)[α −n(α² + ξ(α)) + μ] is constant if and only if dμ = (1−2nα)ξ(α)η + nd(ξ(α)).
• The Bochner-type formula ½(Δ−∇ξ)(|ξ|²) = |∇ξ|² + λ|ξ|² + μ|ξ|²(|ξ|²−2Δf) + (n−2)ξ(λ) −|ξ|²ξ(μ) holds for gradient almost η-Ricci solitons.
• On an (LCS)n-manifold, |∇ξ|² = α²(n−1), and the soliton parameters satisfy μ−λ = (n−1)(α²−ρ).
• No gradient Ricci soliton exists on an (LCS)n-manifold, as shown by the derived condition 2α² −[2(n−2)α−1]ξ(α) −(n−2)ξ(ξ(α)) = 0 having no solution for constant α.