Integral formulas in $h$-Almost Ricci-Bourguignon solitons

TL;DR

This work studies compact gradient $h$-almost Ricci-Bourguignon solitons by deriving integral formulas that connect Hessians of the potential, Ricci and scalar curvature, and the soliton data. Using the $Hodge$–de $Rham$ decomposition and standard identities, the authors obtain three principal integral relations, including a bound involving $\int_M h \lvert \dot{\nabla}^2 h \rvert^2$, $\int_M h \lvert \nabla^2 h - (\Delta h/n)g \rvert^2$, and $\int_M (1/h) \lvert Ric - (R/n)g \rvert^2$. These formulas lead to a rigidity result: for $n>3$, a compact nontrivial soliton is isometric to a Euclidean sphere provided either the potential is conformal or the scalar curvature is constant. The paper also extends integral formulas to general compact $h$-almost Ricci-Bourguignon solitons and presents an explicit expression for the curvature-variation tensor $\mathcal{F}(X,Y)$. These results generalize known sphere-characterization results for Ricci solitons to the broader $h$-almost Ricci-Bourguignon setting, offering tools for classification and further study of soliton structures.

Abstract

The aim of this paper is to investigate some integral formulas for compact gradient $h$-almost Ricci-Bourguignon solitons. Consequently, we generalize the results previously ob tained for Ricci almost solitons. Moreover, we prove that a compact, non-trivial $h$-almost Ricci-Bourguignon soliton with dimension greater than or equal to 3 is isometric to a Euclidean sphere, provided either the potential vector field is conformal or its scalar curvature is constant. Finally, we generalize the integral formula for compact h-almost Ricci-Bourguignon.

Theorems & Definitions (18)

• Definition 2.1
• Definition 2.2
• Lemma 2.3
• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Corollary 3.3
• proof
• Proposition 3.4