On the geometry of Einstein-type structures

Gradient Einstein-Type Structures immersed into a Riemannian Warped Product

“…Note that du(∇h) is tangent to Σ, while nH is normal to Σ, therefore, by (1) it follows that Σ n is harmonic Einstein minimally immersed in I × f M n . In addition, by [4] we get that λ is constant. Replacing R u = nλ in (6), we obtain…”

“…In [4] the authors proved that any compact Einstein-type manifold with bounded Ricci tensor satisfying some inequalities involving the functions µ and λ is Einstein harmonic. In this way, our first result provide a necessary condition for a compact immersion with a gradient Einstein-type structure to be trivial.…”

“…Next, we present to the following formula quoted from [4], in which it is an adaptation of the Bochner's formula to Einstein-type structure, and will be useful in our next result.…”

“…Regarding the geometry of an Einstein-type structure, it is possible to study it from two different points of view, namely, the intrinsic geometry and the submanifolds geometry. From the intrinsic point of view, Anselli et al in [4] obtained several results for the Einstein-type structure introducing the concept of the u−curvatures. In this direction, performing with a conformal deformation of a harmonic Einstein metric, the authors obtain a solution of (1), for n ≥ 3 and µ = −1/(n − 2).…”

Tachibana-type theorems on complete manifolds

“…Harmonic Einstein structures are fixed points of the coupled Ricci-harmonic map flow introduced by R. Buzano, [25]. In [3], in a more general context it has been introduced an algebraic curvature tensor, the ϕ-Weyl tensor W ϕ , which reflects the part of the Riemann tensor of M that is not prescribed by the algebraic structure of Ric ϕ = Ric M −αϕ * g N . For a harmonic Einstein structure the tensor W ϕ satisfies the second Bianchi identity, see [23,Proposition 3.2], but in general it is not divergence free (unless the pull-back metric ϕ * g N is a Codazzi tensor on M ).…”