Harmonic Morphisms and Minimal Conformal Foliations on Lie Groups

Abstract Let G be a Lie group equipped with a left-invariant Riemannian metric. Let K be a semisimple and normal subgroup of G generating a left-invariant conformal foliation $$\mathcal {F}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> on G . We then show that the foliation $$\mathcal {F}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> is Riemannian and minimal. This means that locally the leaves of $$\mathcal {F}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> are fibres of a harmonic morphism. We also prove that if the metric restricted to K is biinvariant then $$\mathcal {F}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> is totally geodesic.