Classification of base of warped product almost Ricci solitons
In this paper we study a Ricci-Hessian type manifold $(\Bbb{M},g,φ,f,λ)$ 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 $\nablaφ$ are linearly independent in $C^\infty(\Bbb{M})$--module $\mathfrak{X}(\Bbb{M})$; and $\nabla f=h\nablaφ$ for a smooth function $h$ on $\Bbb{M}$. In the first case we show that the vector field $\nablaλ$ belongs to the $C^\infty(\Bbb{M})$--module generated by $\nabla f$ and $\nablaφ$, while in the second case, under additional hypothesis, the manifold is, around any regular point of $f$, locally isometric to a warped product.