Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Harmonic Morphisms on Lie groups and Minimal Conformal Foliations of Codimension two

Sigmundur Gudmundsson, Thomas Jack Munn

Abstract

Let G be a Lie group equipped with a left-invariant semi-Riemannian metric. Let K be a semisimple subgroup of G generating a left-invariant conformal foliation F of codimension two on G. We then show that the foliation F is minimal. This means that locally the leaves of F are fibres of a complex-valued harmonic morphism. In the Riemannian case, we prove that if the metric restricted to K is biinvariant then F is totally geodesic.

Harmonic Morphisms on Lie groups and Minimal Conformal Foliations of Codimension two

Abstract

Let G be a Lie group equipped with a left-invariant semi-Riemannian metric. Let K be a semisimple subgroup of G generating a left-invariant conformal foliation F of codimension two on G. We then show that the foliation F is minimal. This means that locally the leaves of F are fibres of a complex-valued harmonic morphism. In the Riemannian case, we prove that if the metric restricted to K is biinvariant then F is totally geodesic.
Paper Structure (8 sections, 10 theorems, 54 equations)

This paper contains 8 sections, 10 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Conformal Foliations of Codimension Two
  3. Lie Foliations of Codimension Two
  4. Minimality and Total Geodecity
  5. The Proofs of our Main Results
  6. The Berger metrics on the subgroup $\text{\bf SU}(2)$
  7. The Riemannian Leaf Space $L^2=G/K$
  8. Sectional curvatures in Riemannian Lie groups

Key Result

Theorem 1.1

Bai-Eel Let $\phi:(M,g) \to (N^2,h)$ be a horizontally conformal submersion from a Riemannian manifold to a surface. Then $\phi$ is harmonic if and only if $\phi$ has minimal fibres.

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.1
  • Proposition 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof : Proof of Theorem \ref{['theorem-totally-geodesic']}
  • ...and 6 more