Classification of base of warped product almost Ricci solitons

TL;DR

This paper classifies certain Ricci-Hessian type manifolds related to almost Ricci solitons, revealing their structure and conditions under which they are locally isometric to warped products, advancing understanding of geometric flows.

Contribution

It provides a classification of Ricci-Hessian manifolds and characterizes when they are locally warped products, extending the theory of almost Ricci solitons.

Findings

The vector field ∇λ lies in the module generated by ∇f and ∇φ.

Under certain conditions, the manifold is locally isometric to a warped product.

The classification applies to cases where ∇f and ∇φ are either linearly independent or proportional.

Abstract

In this paper we study a Ricci-Hessian type manifold $(\Bbb{M},g,\varphi,f,\lambda)$ which is closely related to the construction of almost Ricci soliton realized as a warped product. We classify certain classes of the Ricci-Hessian type manifolds and derive some implications for almost Ricci solitons and generalized $m$--quasi-Einstein manifolds. We consider two complementary cases: $\nabla f$ and $\nabla\varphi$ are linearly independent in $C^\infty(\Bbb{M})$--module $\mathfrak{X}(\Bbb{M})$; and $\nabla f=h\nabla\varphi$ for a smooth function $h$ on $\Bbb{M}$. In the first case we show that the vector field $\nabla\lambda$ belongs to the $C^\infty(\Bbb{M})$--module generated by $\nabla f$ and $\nabla\varphi$, while in the second case, under additional hypothesis, the manifold is, around any regular point of $f$, locally isometric to a warped product.