A loop group method for affine harmonic maps into Lie groups

Abstract

Abstract. We generalize the Uhlenbeck-Segal theory for harmonic maps into compact semi-simple Lie groups to general Lie groups equipped with torsion free bi-invariant connection.

Harmonic Maps into Homogeneous Spaces According to a Darboux Homogeneous Derivative

“…In [5], Dorfmeister-Inoguchi-Kobayashi develop a similar work in the context of Lie groups. In fact, they assume that M is a Riemannian surface and adopt a Lie group G with a left invariant connection instead G/H with a G-invariant connection.…”

“…This method has been used to obtain important results about harmonic maps on Lie groups and homogeneous space, see for example Dai-Shoji-Urakawa [4], Higaki [10], Khemar [11], Dorfmeister-Inoguchi-Kobayashi [5], Sharper [14], Uhlenbeck [16]. Finally, interesting theorems and nice examples are founded in two papers of Urakawa (see [17] and [18]).…”

Harmonic and biharmonic homomorphisms between Riemannian Lie groups

“…This generalizes (ii) of Proposition 2.5. One can see [8] where harmonicity of maps into affine Lie groups is investigated.…”

“…Let study now the converse of our study above. Let (n, h) be two Lie algebras such that n carries an Euclidean product and h two Euclidean products , 1 and , 2 , ρ : h −→ Der(n) and ω ∈ ∧ 2 h * ⊗ n satisfying (8). Define on g = n ⊕ h the bracket [ , ] g (10) […”

“…Proof. Since Der(n) = ad(n) then ρ(u) = ad F (u) and from (8) we deduce that ω must have the following form…”