Classification of base of warped product almost Ricci solitons

Abstract

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

18 pages

References in corpus (2)

• Gradient Ricci almost soliton warped product
• The Ricci-Bourguignon Flow

Cited by in corpus (1)

• Gradient Ricci almost soliton warped product