Eigenvalues of harmonic almost submersions

Abstract

Abstract. Maps between Riemannian manifolds which are submersions on a dense subset, are studied by means of the eigenvalues of the pull-back of the target metrics, the first fundamental form. Expressions for the derivatives of these eigenvalues yield characterizations of harmonicity, totally geodesic maps and biconformal changes of metric preserving harmonicity. A Schwarz lemma for pseudo harmonic morphisms is proved, using the dilatation of the eigenvalues and, in dimension five, a Bochner technique method, involving the Laplacian of the difference of the eigenvalues, gives conditions forcing pseudo harmonic morphisms to be harmonic morphisms.

Perfect Fluids From High Power Sigma-Models

“…[divS(ϕ)](V ) = 0, ∀ V ∈ Ker dϕ) for any map ϕ, not necessarily for critical maps. In particular, for σ 3 -energy we have the following general result, whose independent proof can be find in [27,Proposition 3]. …”

“…Then (27) tells us simultaneously that (U, p = −ρ/3, ρ = λ 2 ) satisfies the equations of a (shear-free) perfect fluid and that the associated submersion ϕ is a harmonic morphism.…”

A steady Euler flow on the 3-sphere and its associated Faddeev-Skyrme solution

“…It is known [18,Prop.3] that, given a smooth map ϕ : (M 3 , g) → (N 2 , h) between Riemannian manifolds (exponents indicate the dimension), the vector field V = λ 1 λ 2 U (locally defined around a regular point of ϕ) is divergence-free, where λ 2 i are the eigenvalues of ϕ * h w.r.t. g and U is a unit vector spanning ker(dϕ).…”

Biharmonic holomorphic maps and conformally Kähler geometry

“…dϕ(U ) = 0 and ϕ * h(E i , X) = λ 2 i g(E i , X) for any vector X. Since, for any i, we have ( [17])…”