Harmonic Maps into Homogeneous Spaces According to a Darboux Homogeneous Derivative

Our purpose is to use a Darboux homogenous derivative to understand the harmonic maps with values in homogeneous space. We present a characterization of these harmonic maps from the geometry of homogeneous space. Furthermore, our work covers all type of invariant geometry in homogeneous space.


Introduction
Harmonic maps into Lie groups and homogeneous space has received much attention due to importance of Lie groups and homogeneous spaces in mathematics and other Sciences. In this context, a useful tool is the Darboux derivative, which provides a kind of linearization in the sense to transfer the problem of Lie group to Lie algebra. Considering a smooth manifold M , a Lie group G and its Lie algebra g then the Darboux derivative of a smooth map F : M → G is a g-valued 1-form given by α F = F * ω, where ω is the Maurer-Cartan form on G. It is well-known that from α F we can establish conditions to F become harmonic.
Our purpose is to study the harmonic maps using the direct relation between the reductive homogeneous space G/H and m, where m is the horizontal component of the Lie algebra g of G. Currently, the main technique is based on the lift of the smooth map F : M → G/H to a smooth map of M into G.
About our main result, let G be a Lie group, H ⊂ G a closed Lie subgroup of G, g and h the Lie algebras of G and H, respectively. Consider G/H with a G-invariant connection ∇ G/H . Our main tools in this context are the Maurer-Cartan form for homogeneous space (see ), denoted by µ, and the Darboux homogeneous derivative. The Darboux homogeneous derivative of a smooth map F : M → G/H is defined as the m-valued 1-form given by µ F = F * µ. Hence it follows our main result: Theorem Let ∇ G/H and ∇ G be connections on G/H and G, respectively. Suppose that π : G → G/H is an affine submersion with horizontal distribution. Let (M, g) be a Riemannian manifold and F : M → G/H a smooth map. Then F is harmonic map if and only if It is direct from equation (1) that the harmonicity of smooth maps with values in G/H depends only on structure of G/H and of the invariant geometry given by ∇ G/H . Furthermore, this is a analogous linearization to the Lie group case.
It is well known that the G-invariant connection ∇ G/H is associated to a Ad(H)invariant bilinear form β : m × m → m. Thus assuming that β is skew-symmetric we conclude that a smooth map F is harmonic if and only if d * µ F = 0. When β has a symmetric part, and this is context of Riemmanian homogeneous space, the study becomes more interesting. But here we study just the special case of Sl(n, R)/SO(n, R). In fact, we give applications of the above theorem to the homogeneous space Sl(n, R)/SO(n, R). First, we consider Sl(n, R)/SO(n, R) as symmetric space. Second, we present an application with a Fisher α-connection in Sl(n, R)/SO(n, R) (see also Fernandes-San Martin [10], [11]). We obtain an explicit solution of geodesics whit values in the nilpontent group of the Iwasawa decomposition of Sl(n, R)/SO(n, R) with respect to a special case of the α-connections.
About the structure of this paper, in section 2 we give a brief summary of some concepts of connection on homogeneous spaces. Then in section 3 we prove our main result (see the above theorem). Finally, in section 4 we present some applications of the main results in case of homogeneous spaces Sl(n, R)/SO(n, R).

Connections on homogeneous spaces
In this section we introduce the notations and results about homogeneous spaces that will be necessary in the sequel. We begin by introducing the kind of homogeneous space that we will work. For more details see Helgason [7].
Let G be a Lie group and H ⊂ G a closed Lie subgroup of G. Denote by g and h the Lie algebras of G and H, respectively. We assume that the homogeneous space G/H is reductive, that is, there is a subspace m of g such that g = h ⊕ m and Ad(H)(m) ⊂ m. Take π : G → G/H the natural projection. It is well-known that there exists a neighborhood U ⊂ m of 0 such that the smooth map φ : For each a ∈ G we denote the left translation as τ a : G/H → G/H where τ a (gH) = agH. Furthermore, if L a is the left translation on G, then π • L a = τ a • π.
As the left translation L g is a diffeomorphism we have Denoting T G h := {(L g ) * e h; ∀ g ∈ G} and T G m := {(L g ) * e m; ∀ g ∈ G} it follows that The horizontal projection of T G into T G m is written as h.
From above equality we have the following facts about theory of connection. First, we can view π : G → G/H as a H-principal fiber bundle. Furthermore, Theorem 11.1 in [12] shows that the principal fiber bundle G(G/H, H) has the vertical part of the Maurer-Cartan form on G, which is denoted by ω and is a connection form with respect to decomposition (2). It is clear that T G m is a connection in G(G/H, H). Thus we denote the horizontal lift from G/H to G by H.
Take A ∈ m. The left invariant vector fieldÃ on G is denoted byÃ(g) = L g * A and the G-invariant vector field A * on G/H is defined by A * = τ g * A. It is clear thatÃ is a horizontal lift vector field of A * and π * (Ã) = A * . In other words,Ã and A * are π-related. Now we introduce some geometric aspects of G and G/H. With intention of using the theory of principal fiber bundles, our idea is choose a good connection ∇ G such that it is horizontally projected over ∇ G/H . In other words, π : G → G/H will be an affine submersion with horizontal distribution (see Abe-Hasegawa [1] for detail of these connections or Proposition 2.1 below).
We choose a G-invariant connection ∇ G/H on G/H. Theorem 8.1 in Nomizu [13] assures the existence of a unique Ad(H)-invariant bilinear form β : In the following we construct a connection ∇ G from ∇ G/H such that the projection π is an affine submersion with horizontal distribution. We extend β to a bilinear form α from g × g into g satisfying Proof. Take A, B ∈ m and the left invariant vectors fieldsÃ,B on exp(U ) and the G-invariant vector fields A * , B * on N . It is clear thatÃ,B are horizontal, π * (Ã) = A * and π * (B) = B * . By construction of ∇ G , for g ∈ exp(U ), we have This gives h(∇ G AB ) = H(∇ G/H A * B * ). Using properties of connection it is easy to show that this result holds for any vector fields X, Y on N . Now taking X, Y on G/K we translate they to N ( by invariance of ∇ G/H ) and the result follows.
In the following sections we regard π as an affine submersion with horizontal distribution following the choice given by (3) for connections ∇ G and ∇ G/H .

Harmonic maps on homogeneous spaces
In this section we proof our main result and present two examples that will be important to the applications (see next section). We begin introducing the Maurer-Cartan form in the homogeneous space G/H.
Denoting by ω the Maurer-Cartan form on G we define the 1-form µ : where H is the horizontal lift from G/H to G given by decomposition g = h ⊕ m. We note that µ depends on the direct sum g = h ⊕ m. It is clear that if A ∈ m, then µ(π(g))(A * (π(g))) = A.

Remark 1.
Here, we note that µ can be seen as Maurer-Cartan form (see Burstall- . The difference is the direction of construction, while Burstall-Ranwnsley go to m for T (G/H) we go in the inverse direction.
It is easy to show that the Maurer-Cartan form and homogeneous Darboux derivative satisfy analogous properties of the Maurer-Cartan form and Darboux derivative in Lie groups.
Before the main result we need the following technical Lemmas.
Proof. From definition of co-differential d * , for any orthonormal frame field {e 1 , . . . , e n } on M in a neighborhood of p ∈ M , we have where ∇ g is the Levi-Civita connection associated to the metric g. By definition of dual connection, Since θ : m → R is a linear map and using the definition of co-differential in the last equality, we have Similarly, we show the second equality of the Lemma.
∈ N , since µ is invariant by τ , we consider the neighborhood τ (N ). Let A 1 , . . . , A m be a basis on m and A 1 * , . . . , A m * a frame field associated to basis on N .
where we use the fact that ∇ G/H µ is a tensor. From definitions of dual connection and µ we see that ). Note that we use the left invariant property of ∇ G in the above second equality. Since the connection map β is a bilinear form, it follows Now we have the main theorem. In the proof we use freely the concepts and notations of Catuogno [3] and Emery [5].   (2) in [15]) we deduce that .
From the last term in above right side we have Suppose that F is an harmonic map. Thus F (B t ) is a martingale in G/H. On the other side, t 0 (F * µ)dB s is a real local martingale. Then from equation (7) and Doob-Meyer decomposition it follows that Hence, since B is arbitrary, Finally, Lemma 3.3 yields (9) is true, and let θ be a 1-form on G/H. Here, it is sufficient to consider the 1-forms that are invariant by µ, namely, 1-forms that

Analogous computations show that
Applying Lemma 3.2 in the above equality we obtain By hypothesis, Since B t is a Brownian motion, t 0 (F * θ)dB s is a real martingale. Consequently, t 0 θd G/H F (B s ) is a real martingale and, being θ arbitrary, F (B t ) is a ∇ G/Hmartingale. Thus, F is a harmonic map.
Example 3.1. Let <, > be a scalar product in m × m which is invariant by Ad(H), that is, < Ad(h)X, Ad(h)Y >=< X, Y > for h ∈ H and X, Y ∈ m. This scalar product gives a G -invariant metric ≪, ≫ on G/H. The invariant connection associated to ≪, ≫ is given by the connection function where U : m × m → m is a symmetric bilinear form defined by for all X, Y, Z ∈ m (see Nomizu [13]). Now Theorem 3.4 shows that the necessary and sufficient condition for F : M → G/H to be a harmonic map is that where {e 1 , . . . , e n } is an orthonormal frame field in M .

Remark 2.
In [4], Day-Shojy-Urakawa presents a study about reductive Riemannian homogeneous spaces, however the authors work with lift of smooth maps to Lie group to characterize the harmonic maps with values in homogeneous space (see Theorem 2.1 in [4]).
Assuming that M is a Riemannian surface and that ∇ G/H is a canonical connection, that is, β(X, Y ) = 0 or β(X, Y ) = 1 2 [X, Y ], Higaki in [8] obtains an equivalent result of above theorem. Furthermore, Khemar in [9, Th.7.1.1] also presents an equivalent result for a Riemannian surface M and a family of invariant connections given by β t (X, Y ) = t[X, Y ] for 0 ≤ t ≤ 1.
In [6], 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.
In the sequel, we classify the harmonic maps for a large class of G-invariant connections. We will work with G-invariant connections ∇ G/H given by Ad(H)invariant bilinear forms β which satisfy (10) β(X, X) = 0, ∀ X ∈ m.
It is clear that every bilinear form β can be written as a sum of a symmetric part and a skew-symmetric part. The condition (10) is not so restrictive, in fact we can find these bilinear forms in the reductive homogeneous spaces G/H that admit a G-invariant metric with U ≡ 0 and in the Riemannian symmetric spaces. Other examples, following Nomizu [13], are the canonical invariant connection of the first kind, which has β(X, Y ) = 1 2 [X, Y ] m , for all X, Y ∈ m, and the canonical invariant connection of the second kind, which has β(X, Y ) = 0, for all X, Y ∈ m. Furthermore, every skew-symmetric bilinear form satisfies the condition (10). Summarizing, the next result characterizes every harmonic map under this conditions. Corollary 3.5. Consider as above the connections ∇ G/H , ∇ G and the submersion π. Let M be a Riemannian manifold and take G/H a reductive homogeneous space. If β, associated to ∇ G/H , satisfies β(X, X) = 0 for all X ∈ m, then a map F : M → G/H is harmonic if and only if A direct consequence of the above corollary is that the harmonic maps depend only on µ, the geometry of β has no influence. Now for the next corollary, let {A 1 , . . . , A r } be a basis of m. In this basis we have that µ = r k=1 µ k A k . Then In the basis of m we have that µ = r k=1 µ k A k , so µ F (e i ) = r k=1 µ k F (e i )A k . Then In the sequel we establish two examples that are necessary in our applications.
Example 3.2. Let G be connected non-compact semisimple Lie group with finite center and denote by g its Lie algebra. Take a Cartan decomposition g = h ⊕ s. Choose a maximal abelian subspace a ⊂ s and denote by Π + the corresponding set of positive roots and Σ the set of simple roots. Put where g α stands for the α-root space. Then we have the Iwasawa decomposition on the Lie algebra, g = h ⊕ a ⊕ n + . We denote by H = exph, N + = expn + and A = expa the connected subgroups with corresponding Lie algebras. Therefore, it follows the Iwasawa decomposition of G, G = H × A × N + . Thus for a smooth map Taking a vector v ∈ T x M we have that µF x * (v) is given by Under condition that β(X, X) = 0, from Corollary 3.5 we have that F is an harmonic map if and only if Hence, from Corollary 3.5 we have that F is an harmonic map if and only if

Applications
In this section we present a study of harmonic maps with values in the homogeneous spaces Sl(n, R)/SO(n, R). First we treat the symmetric space Sl(n, R)/SO(n, R) with canonical Riemannian connection β(X, Y ) = 1 2 [X, Y ] m . In the second part we consider the connection given by β(X, Y ) = XY + Y X 2 − tr(XY ) n Id , that is a special case of Fisher α-connections studied, for example, in [10] and [11].
Using the Iwasawa decomposition of Sl(n, R) we have an explicit description of its homogeneous spaces. Hence, as a particular case of the Example 3.2, we consider the following construction for the Iwasawa decomposition of sl(n, R), the Lie algebra of Sl(n, R). Take the Cartan decomposition given by sl(n, R) = so(n, R) ⊕ s, where so(n, R) = {X ∈ sl(n, R) : X t = −X} and s = {X ∈ sl(n, R) : X t = X}. Now consider a a maximal abelian subalgebra of s given by all diagonal matrices with vanishing trace. Also with canonical choices we have the following Iwasawa decomposition sl(n, R) = so(n, R) ⊕ a ⊕ n + , where n + is the subalgebra of all upper triangular matrices with null entries on the diagonal.
Consider a smooth map F : M → Sl(n, R)/SO(n, R). Then

4.1.
Harmonic maps on canonical symmetric space. In case of canonical Riemannian connection, β : m × m → m is given by Then from Example 3.2 we have that F is a harmonic map if and only if d * µ F 1 (x) = 0 and d * µ F 2 (x) = 0.
As an immediate consequence we obtain the geodesics.

4.2.
Harmonic maps with respect to α-connections. In this case, the bilinear form β : m × m → m is given by First we need to compute tr(µ * F ∇ G /H) in Theorem 3.4. In fact, for an orthonormal frame {e 1 , . . . , e n } on M we have that because the trace of any matrix in a ⊕ n + is null. Considering µ F 1 (e i ) ∈ a and µ F 2 (e i ) ∈ n + , Theorem 3.4 assures that a smooth map F : M → Sl(n, R)/SO(n, R) is harmonic if and only if 1 n tr(µ 2 F 1 (e i ))Id] = 0 and (14) As in dimension 2 we have a better description we consider now the simply case of Sl(2, R)/SO(2, R).
4.2.1. Geodesic in N + . Now our purpose is to study geodesics in N + . In the other hand, Fernandes-San Martin [11] studied the geodesics with respect to α-connections in A.
where the function f : N → N is defined by recurrence as Proof. First, it is clear from equation (15) For n ≥ 3 we will proof by induction. Suppose that we know the solution of all nilpotent matrix of dimension n. Taking a nilpotent matrix of dimension n + 1 we see that there are two nilpotent matrix of dimension n. Namely, the matrix (b ij ) 1≤i<j≤n and (b ij ) 2≤i<j≤n+1 , which we know the solutions. Then we need to find the solution of function b 1(n+1) . In fact, from equation (15) we see that this function satisfies the differential equation 1i (t)ḃ i(n+1) (t) = 0.
We now know thatḃ