Harmonic morphisms between riemannian manifolds

Abstract

A harmonic morphism <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mi>M</mml:mi> <mml:mo>→</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> between Riemannian manifolds <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>M</mml:mi> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> is by definition a continuous mappings which pulls back harmonic functions. It is assumed that dim <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>≥</mml:mo> </mml:mrow> </mml:math> dim <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> , since otherwise every harmonic morphism is constant. It is shown that a harmonic morphism is the same as a harmonic mapping in the sense of Eells and Sampson with the further property of being semiconformal, that is, a conformal submersion of the points where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mi>f</mml:mi> </mml:mrow> </mml:math> vanishes. Every non-constant harmonic morphism is shown to be an open mapping.