Harmonic forms and generalized solitons

TL;DR

This paper investigates conditions under which certain 1-forms associated with generalized solitons satisfy harmonic and Schrödinger-Ricci equations, extending known results for Ricci and Yamabe solitons with applications and examples.

Contribution

It provides necessary and sufficient conditions for dual 1-forms of potential vectors to solve harmonic and Schrödinger-Ricci equations in generalized solitons, including gradient cases.

Findings

Characterization of harmonic and Schrödinger-Ricci harmonic forms for generalized solitons.

Conditions for 1-forms orthogonal to the potential vector's dual form.

Applications and examples illustrating the theoretical results.

Abstract

For a generalized soliton $(g,\xi,\eta,\beta,\gamma,\delta)$, we provide necessary and sufficient conditions for the dual $1$-form $\xi^{\flat}$ of the potential vector field $\xi$ to be a solution of the Schr\"{o}dinger-Ricci equation, a harmonic or a Schr\"{o}dinger-Ricci harmonic form. We also characterize the $1$-forms orthogonal to $\xi^{\flat}$, underlying the results obtained for the Ricci and Yamabe solitons. Further, we formulate the results for the case of gradient generalized solitons. Several applications and examples are also presented.