[Paper Review] A note on Almost Riemann Soliton and gradient almost Riemann soliton

Krishnendu De, Uday Chand De|arXiv (Cornell University)|Aug 24, 2020

Geometric Analysis and Curvature Flows14 references6 citations

TL;DR

This paper investigates almost Riemann solitons (ARS) and gradient almost Riemann solitons (GARS) in 3-dimensional non-cosymplectic normal almost contact metric (acm) manifolds. It establishes that under constant α, β, a GARS implies the manifold is either quasi-Sasakian or of constant sectional curvature −(α²−β²). The paper further proves that if the potential vector field is divergence-free and pointwise collinear with ξ, the ARS reduces to a Riemann soliton. An explicit example of a 3D normal acm manifold admitting a Riemann soliton is constructed.

ABSTRACT

The quest of the offering article is to investigate \emph{almost Riemann soliton} and \emph{gradient almost Riemann soliton} in a non-cosymplectic normal almost contact metric manifold $M^3$. Before all else, it is proved that if the metric of $M^3$ is Riemann soliton with divergence-free potential vector field $Z$, then the manifold is quasi-Sasakian and is of constant sectional curvature -$λ$, provided $α,β=$ constant. Other than this, it is shown that if the metric of $M^3$ is \emph{ARS} and $Z$ is pointwise collinear with $ξ$ and has constant divergence, then $Z$ is a constant multiple of $ξ$ and the \emph{ARS} reduces to a Riemann soliton, provided $α,\;β=$constant. Additionally, it is established that if $M^3$ with $α,\; β=$ constant admits a gradient \emph{ARS} $(γ,ξ,λ)$, then the manifold is either quasi-Sasakian or is of constant sectional curvature $-(α^2-β^2)$. At long last, we develop an example of $M^3$ conceding a Riemann soliton.

Motivation & Objective

• To investigate almost Riemann solitons (ARS) and gradient ARS in 3-dimensional non-cosymplectic normal almost contact metric (acm) manifolds.
• To determine geometric conditions under which ARS or GARS imply constant curvature or quasi-Sasakian structure.
• To establish the relationship between divergence-free vector fields and the reduction of ARS to Riemann solitons.
• To construct a concrete example of a 3D normal acm manifold admitting a Riemann soliton.
• To generalize results from Ricci solitons and Riemann solitons to the almost Riemann soliton framework in contact geometry.

Proposed method

• Utilizes the definition of ARS via the equation $ 2R + ilde{ abla}Z fimes g + ilde{ abla}Z imes g = 0 $, where $ R $ is the Riemann curvature tensor and $ ilde{ abla} $ denotes the Kulkarni-Nomizu product.
• Applies the gradient ARS condition $ 2R + abla^2 ilde{ abla} ilde{ abla} ilde{ abla} = 0 $, with $ abla^2 ilde{ abla} $ as the Hessian of the potential function $ ilde{ abla} $.
• Employs the structure equations of normal acm manifolds, including $ abla_{u_i}u_j $ computations via Koszul’s formula.
• Uses the Nijenhuis tensor condition $ [ abla, abla] + 2d ilde{ abla} imes ilde{ abla} = 0 $ to verify normality.
• Derives curvature relations using the Riemann tensor components computed from the Levi-Civita connection.
• Applies inner product and contraction techniques to derive scalar constraints, such as $ \alpha(\frac{r}{2} + 3(\alpha^2 - \beta^2)) = 0 $, to classify the manifold.

Experimental results

Research questions

• RQ1 Under what conditions does a gradient almost Riemann soliton on a 3D non-cosymplectic normal acm manifold imply constant sectional curvature?
• RQ2 When does a divergence-free potential vector field in an ARS imply that the ARS reduces to a Riemann soliton?
• RQ3 What geometric structure arises when the potential vector field is pointwise collinear with the characteristic vector field $ \xi $ and has constant divergence?
• RQ4 Can a 3D normal acm manifold admit a Riemann soliton, and if so, under what conditions?
• RQ5 How do the constants $ \alpha $ and $ \beta $, defining the curvature of the acm structure, influence the classification of the manifold under ARS or GARS?

Key findings

• If the metric of a 3D non-cosymplectic normal acm manifold is a Riemann soliton with a divergence-free potential vector field $ Z $, and $ \alpha, \beta $ are constant, then the manifold is quasi-Sasakian and has constant sectional curvature $ -\lambda $.
• If the ARS has a potential vector field $ Z $ that is pointwise collinear with $ \xi $ and has constant divergence, then $ Z $ is a constant multiple of $ \xi $, and the ARS reduces to a Riemann soliton.
• For a 3D normal acm manifold with constant $ \alpha, \beta $, a gradient ARS $ (\gamma, \xi, \lambda) $ implies the manifold is either quasi-Sasakian or has constant sectional curvature $ -(α^2 - β^2) $.
• The scalar curvature $ r $ satisfies $ r = -6(\alpha^2 - \beta^2) $ when the manifold is Einstein under the GARS condition.
• The example manifold $ M = \{(x,y,z) \in \mathbb{R}^3, z \neq 0\} $ with $ u_1 = z\partial_x, u_2 = z\partial_y, u_3 = z\partial_z $ admits a Riemann soliton with $ \lambda = 1 $, $ \xi = u_3 $, and $ \pounds_\xi g = 0 $.
• The Riemann curvature tensor components in the example satisfy $ R(u_1,u_2)u_3 = 0 $, $ R(u_2,u_3)u_3 = -u_2 $, $ R(u_1,u_3)u_3 = -u_1 $, confirming non-trivial curvature and soliton structure.