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Stability and integrability of horizontally conformal maps and harmonic morphisms
Author(s) -
Kim J.J.,
Yun G.
Publication year - 2013
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201200183
Subject(s) - mathematics , morphism , conformal map , harmonic map , integrable system , manifold (fluid mechanics) , distribution (mathematics) , harmonic , riemannian manifold , pure mathematics , geodesic , mathematical analysis , corollary , homothetic transformation , totally geodesic , geometry , physics , mechanical engineering , quantum mechanics , engineering
In this article, we study stability of minimal fibers and integrability of horizontal distribution for horizontally conformal maps and harmonic morphisms. Let φ : ( M n , g ) → ( N m , h )be a horizontally conformal submersion. We prove that if the horizontal distribution is integrable, then any minimal fiber of φ is volume‐stable. This result is an improved version of the main theorem in [15]. As a corollary, we obtain if φ is a submersive harmonic morphism whose fibers are totally geodesic, and the horizontal distribution is integrable, then any fiber of φ is volume‐stable and so such a map φ is energy‐stable if M is compact. We also show that if φ : ( M n , g ) → ( N m , h )is a horizontally conformal map from a compact Riemannian manifold M into an orientable Riemannian manifold N which is horizontally homothetic, and if the pull‐back of the volume form of N is harmonic, then the horizontal distribution is integrable and φ is a harmonic morphism.