[Paper Review] Biconformal changes of metric and pseudo-harmonic morphisms

Radu Slobodeanu|ArXiv.org|Aug 27, 2004

Geometric Analysis and Curvature Flows5 references3 citations

TL;DR

This paper investigates biconformal changes of metric on Riemannian manifolds and proves that pseudo-harmonic morphisms—maps preserving harmonicity and compatibility of an almost complex structure on the horizontal distribution—are invariant under such metric transformations. The key result shows that PHWC (Pseudo-Horizontally Weakly Conformal) and PHH (Pseudo-Horizontally Homothetic) conditions, along with harmonicity, are preserved under biconformal changes, with PHH harmonic morphisms remaining invariant only if the conformal factor is constant.

ABSTRACT

Pseudo-harmonic morphisms give rise on the domain space to a distribution which admits an almost complex structure compatible with the given Riemannian metric. We shall show that this property, together with the harmonicity, are preserved by a biconformal change of the domain metric. The special case of the pseudo-horizontally homothetic harmonic morphisms is also treated.

Motivation & Objective

• To investigate how biconformal changes of metric affect the geometric properties of pseudo-harmonic morphisms.
• To determine under which metric transformations the PHWC (Pseudo-Horizontally Weakly Conformal) and PHH (Pseudo-Horizontally Homothetic) conditions are preserved.
• To analyze the invariance of harmonicity under biconformal metric changes in the context of submersions into Kähler manifolds.
• To establish conditions under which PHH harmonic morphisms remain harmonic and pseudo-harmonic under metric deformations.
• To extend known results on harmonic morphisms to the broader class of pseudo-harmonic morphisms via biconformal transformations.

Proposed method

• Introduces biconformal metric changes defined by independent conformal factors σ and ρ on horizontal and vertical distributions: g̅ = σ⁻²gᴴ + ρ⁻²gⱽ.
• Derives the transformed tension field τ̅(φ) using Koszul formula and adapted orthonormal frames, showing τ̅(φ) = σ²[τ(φ) + dφ(grad(ln(ρ²ⁿ⁻ᵐσ²⁻²ⁿ)))].
• Applies the formula to compute the change in the divergence of the f-structure F^φ on the horizontal bundle, yielding F^φ div̅ᴴ F^φ = σ²[F^φ divᴴ F^φ + (2n−2)grad(ln σ)].
• Uses the relation τ(φ) = −(m−2n)dφ(μᵛ) for PHH harmonic morphisms to analyze invariance under metric changes.
• Applies the transformation of the horizontal connection ∇̅ᴴ to derive the condition for PHH invariance: ∇̅ᴴ(X)F^φ(Y) = 0 for all X,Y ∈ Γ(ℋ).
• Establishes that PHH harmonic morphisms are preserved under gσ only if σ is constant, via analysis of the transformed connection and divergence terms.

Experimental results

Research questions

• RQ1 Under what biconformal metric changes is the PHWC condition preserved for pseudo-harmonic morphisms?
• RQ2 How does the tension field transform under biconformal changes, and when does harmonicity remain invariant?
• RQ3 What conditions ensure that the PHH (Pseudo-Horizontally Homothetic) property is preserved under biconformal metric deformations?
• RQ4 Is there a class of biconformal changes that preserve both harmonicity and the PHH condition simultaneously?
• RQ5 For which conformal factors σ is a PHH harmonic morphism preserved under the metric change gσ = σ⁻²gᴴ + σ^(4n−4)/(m−2n) gⱽ?

Key findings

• The tension field of a PHWC submersion transforms under biconformal changes as τ̅(φ) = σ²[τ(φ) + dφ(grad(ln(ρ²ⁿ⁻ᵐσ²⁻²ⁿ)))], showing explicit dependence on the conformal factors.
• For any smooth σ > 0, setting gσ = σ⁻²gᴴ + σ^(4n−4)/(m−2n) gⱽ preserves the pseudo-harmonic morphism property if and only if φ is already a pseudo-harmonic morphism with respect to g.
• PHH harmonic morphisms are preserved under biconformal changes gσ only when the conformal factor σ is constant.
• The condition for PHH invariance reduces to requiring that the horizontal divergence of F^φ vanishes under the new metric, which holds only if σ is constant.
• The mean curvature of the fibers transforms as μ̅ᵛ = σ²[μᵛ + ℋ(grad(ln ρ))], showing explicit dependence on the vertical conformal factor.
• The result generalizes known invariance properties of harmonic morphisms to the broader class of pseudo-harmonic morphisms under biconformal changes.