Abstract. We prove that shear-free perfect fluid solutions of Einstein's field equations must be either expansion-free or non-rotating (as conjectured by Ellis and Treciokas) for all linear equations of state p = wρ except for w ∈ − IntroductionFor the Einstein field equations with perfect fluid sources, an important insight into the structure of solutions can be obtained under two fairly general and observationally defensible assumptions: the equation of state for the pressure p and energy density ρ is barotropic (i.e. p = p(ρ)) and the flow is shear-free (i.e. fluid's velocity is a transversally conformal vector). While the former restriction is specific to isoentropic fluids, the latter is essentially a kinematic condition of isotropy. In cosmology, the shear-free condition expresses the isotropy of the relative recessional motion of the galaxies (but allows the red shift and the cosmic microwave background radiation to be anisotropic) and is a common feature of standard spacetimes, such as the FRW and Gödel models. In kinetic theory this characterizes [35] the velocity of collision-dominated gases with isotropic distribution function, under the Einstein-Boltzmann equations.The first result indicating the nature of solutions in this context was stated by Gödel ([12]): a shear-free dust fluid (i.e. p = 0, in particular a congruence of timelike geodesics) in a spatially homogeneous spacetime of type IX cannot both expand and rotate. This contrasts Newtonian cosmology where expanding and rotating (shear-free) solutions exist and can avoid the singularity formation [24]. Gödel's result was later proved [15] to remain true without symmetry assumptions and it can be seen as a timelike analogue of the Goldberg-Sachs theorem [18]. After proving that the same conclusion holds for radiation fluids (i.e. p = If the velocity vector field of a self-gravitating barotropic perfect fluid (p + ρ = 0 and p = p(ρ)) is shear-free, then either the expansion or the rotation of the fluid vanishes.2010 Mathematics Subject Classification. 53B30, 53C43, 53Z05, 83C55.

Abstract. The present status of the shear-free fluid conjecture in general relativity is discussed: a review is given of recent partial proofs, a new and complete proof is given for the case of a linear equation of state, including a non-zero cosmological constant, and a number of useful results are presented which might help in proving the conjecture for a general equation of state.

Abstract. The higher-power derivative terms involved in both Faddeev and Skyrme energy functionals correspond to σ 2 -energy, introduced by Eells and Sampson in [13]. The paper provides a detailed study of the first and second variation formulae associated to this energy. Some classes of (stable) critical points are outlined. IntroductionCommon tools in field theory, non-linear σ-models are known in differential geometry mainly through the problem of harmonic maps between Riemannian manifolds. Namely a (smooth) mapping ϕ : (M, g) → (N, h) is harmonic if it is critical point for the Dirichlet energy functional [13],a generalization of the kinetic energy of classical mechanics. Less discussed from differential geometric point of view are Skyrme and Faddeev-Hopf models, which are σ-models with additional fourth-power derivative terms (for an overview including recent progress concerning both models, see [25]).The first one was proposed in the sixties by Tony Skyrme [37], to model baryons as topological solitons (see [31]) of pion fields. Meanwhile it has been shown [47] to be a low energy effective theory of quantum chromodynamics that becomes exact as the number of quark colours becomes large. Thus baryons are represented by energy minimising, topologically nontrivial maps ϕ :with the boundary condition ϕ({|x| → ∞}) = I 2 , called skyrmions. Their topological degree is identified with the baryon number. The static (conveniently renormalized) Skyrme energy functional isThis energy has a topological lower bound [14]: E Skyrme (ϕ) ≥ 6π 2 |degϕ|. In the second one, stated in 1975 by Ludvig Faddeev and Antti J. Niemi [15], the configuration fields are unitary vector fields ϕ : R 3 → S 2 ⊂ R 3 with the boundary condition ϕ({|x| → ∞}) = (0, 0, 1). The static energy in this case is given bywhere c 2 , c 4 are coupling constants. Again the field configurations are indexed by an integer, their Hopf invariant : Q(ϕ) ∈ π 3 (S 2 ) ∼ = Z and the energy has a topological lower bound: cf. [45]. Although this model can be viewed as a constrained variant of the Skyrme model, it exhibits important specific properties, e.g. it allows knotted solitons. Moreover, in [16] it has been proposed that it arises as a dual description of strongly coupled SU (2) Yang-Mills theory, with the solitonic strings (possibly) representing glueballs. See also [17] for an alternative approach to these issues.Both models rise the same kind of topologically constrained minimization problem: find out static energy minimizers in each topological class (i.e. of prescribed baryon number or Hopf invariant). We can give an unitary treatment for both if we take into account that they are particular cases of the following energy-type functional:

Certain critical points of a sextic sigma-model Lagrangian reminiscent of Skyrme model correspond to perfect fluids with stiff matter equation of state. We analyze from a differential geometric perspective this correspondence extended to general barotropic fluids.

The Walczak formula is a very nice tool for understanding the geometry of a Riemannian manifold equipped with two orthogonal complementary distributions. M. Svensson [Holomorphic foliations, harmonic morphisms and the Walczak formula, J. London Math. Soc. (2) 68 (3) (2003) 781-794] has shown that this formula simplifies to a Bochner-type formula when we are dealing with Kähler manifolds and holomorphic (integrable) distributions. We show in this paper that such results have a counterpart in Sasakian geometry. To this end, we build on a theory of (contact) holomorphicity on almost contact metric manifolds. Some other applications for (pseudo-)harmonic morphisms on Sasaki manifolds are outlined.