Sobre estruturas gradiente Einstein-type produto torcido

Abstract

In this work, we study gradient Einstein-type structures immersed in a warped product space of an interval I and a Riemannian manifold M, with a potential function h given by the height function. First, we obtained the necessary conditions for an Einstein-type gradient structure immersed in a warped product to be minimal, totally umbilic or totally geodesic. In addition, we provide triviality results for the potential function h. Next, we characterize the rotational hypersurfaces having a gradient Einstein-type structure in warped product, where the fiber is a space form. We also study particular cases of gradient Einstein-type structures in warped product spaces, namely, gradient Ricci-harmonic solitons (GRHS). In this case, we prove triviality results for the potential and warped functions when they reach a maximum or minimum. Finally, we provide a family non-enumerable of non-trivial geodesically complete examples of GRHS considering the base and fiber of a warped product conformal to a semi-Euclidean space invariant under the action of a translation group of codimension one.