Triviality of k-Yamabe gradient solitons immersed in certain warped product spaces

Abstract

We deal with complete noncompact and stochastically complete $k$-Yamabe gradient solitons immersed in a warped product space obeying a suitable curvature constraint. In this context, we establish a necessary and sufficient condition for a Riemannian manifold immersed in a warped product to be a $k$-Yamabe gradient soliton, under the hypothesis that the potential function agrees with the height function. Proceeding with this setting, we use a suitable Bochner type formula jointly with integrability conditions and some maximum principles dealing, in particular, with the notions of convergence to zero at infinity and polynomial volume growth, to obtain new triviality results concerning $k$-Yamabe gradient solitons. Moreover, we present some applications of our main results to a class of pseudo-hyperbolic spaces.