Fernando Dobarro

Curvature of multiply warped products

Abstract In this paper, we study Ricci-flat and Einstein–Lorentzian multiply warped products. We also consider the case of having constant scalar curvatures for this class of warped products. Finally, after we introduce a new class of space–times called as generalized Kasner space–times, we apply our results to this kind of space–times as well as other relativistic space–times, i.e., Reissner–Nordstrom, Kasner space–times, Banados–Teitelboim–Zanelli and de Sitter black hole solutions.

Singular Casimir Elements of the Euler Equation and Equilibrium Points

The problem of the nonequivalence of the sets of equilibrium points and energy-Casimir extremal points, which occurs in the noncanonical Hamiltonian formulation of equations describing ideal fluid and plasma dynamics, is addressed in the context of the Euler equation for an incompressible inviscid fluid. The problem is traced to a Casimir deficit, where Casimir elements constitute the center of the Poisson algebra underlying the Hamiltonian formulation, and this leads to a study of singularities of the Poisson operator defining the Poisson bracket. The kernel of the Poisson operator, for this typical example of an infinite-dimensional Hamiltonian system for media in terms of Eulerian variables, is analyzed. For two-dimensional flows, a rigorously solvable system is formulated. The nonlinearity of the Euler equation makes the Poisson operator inhomogeneous on phase space (the function space of the state variable), and it is seen that this creates a singularity where the nullity of the Poisson operator (the “dimension” of the center) changes. The problem is an infinite-dimension generalization of the theory of singular differential equations. Singular Casimir elements stemming from this singularity are unearthed using a generalization of the functional derivative that occurs in the Poisson bracket.

About curvature, conformal metrics and warped products

We consider the curvature of a family of warped products of two pseduo-Riemannian manifolds (B, gB) and (F, gF) furnished with metrics of the form c2gB ⊕ w2gF and, in particular, of the type w2μgB ⊕ w2gF, where c, w:B → (0, ∞) are smooth functions and μ is a real parameter. We obtain suitable expressions for the Ricci tensor and scalar curvature of such products that allow us to establish results about the existence of Einstein or constant scalar curvature structures in these categories. If (B, gB) is Riemannian, the latter question involves nonlinear elliptic partial differential equations with concave–convex nonlinearities and singular partial differential equations of the Lichnerowicz–York-type among others.