Gradient Einstein-type Sasakian manifolds with $α=0$ are trivial

TL;DR

The paper establishes a rigidity result for gradient Einstein-type manifolds in the Sasakian setting: if the defining constants satisfy $α=0$, then any gradient Einstein-type Sasakian manifold must be trivial, i.e., the potential function is constant. The proof leverages Sasakian structure equations and curvature identities, proceeding through two cases ($β≠0$ and $β=0$) to force the gradient of the potential to vanish. Consequently, many Sasakian solitons that fall under the Einstein-type umbrella become trivial, reinforcing the unifying perspective of Einstein-type manifolds. The result also implies a broader statement: for $α=β=0$ and dimension $n≥2$, gradient Einstein-type manifolds are necessarily trivial, highlighting a strong rigidity in this regime.

Abstract

Catino, Mastrolia, Monticelli, and Rigoli have launched an ambitious program to study known geometric solitons from a unified perspective, termed Einstein-type manifolds. This framework allows one to treat Ricci solitons, Yamabe solitons, and all of their generalizations simultaneously. Einstein-type manifolds are characterized by four constants $α, β, μ$ and $ρ$. In this paper, we show that when $α= 0$ gradient Einstein-type Sasakian manifolds are trivial. As a consequence, many geometric solitons on Sasakian manifolds turn out to be trivial.

Theorems & Definitions (14)

• Theorem 1.1
• Definition 2.1: Almost contact manifold
• Remark 2.2
• Definition 2.3: Almost contact metric manifolds
• Definition 2.4: Contact metric manifold
• Definition 2.5: Nijenhuis tensor
• Definition 2.6: Sasakian manifold
• Theorem 2.7
• Lemma 2.8
• Definition 2.9: $K$-contact