Constructions of harmonic maps between Hadamard manifolds

3.1. DAMEK-RICCI SPACES 61
where [ , ] Ò
is the Lie bracket on n defined by (3.1.1). In this way, s becomes a Lie algebra
with an inner product. The simply connected Lie group S associated with s, equipped with
the left invariant metric g S , is called a Damek-Ricci space.
Example. Let v = R 2k , u = R 2 , and µ(z,v) =(−y, x),µ(w, v) =(x, y), where {z,w} is
an orthonormal basis of R 2 and v =(x, y) ∈ R k ⊕ R k . Then n is a (classical) Heisenberg
algebra and (S, g S ) is isometric to the complex hyperbolic space whose sectional curvature
K satisfies −4 ≤ K ≤−1. The quaternion hyperbolic space and the Cayley hyperbolic
plane are also Damek-Ricci spaces. These three spaces are called classical (cf. [8]). Thus
Damek-Ricci spaces are generalizations of rank one symmetric spaces of noncompact type.
Let ∇ be the Levi-Civita connection on (S, g S ). Using the formula
2〈∇ X Y,Z〉 = 〈[X, Y ],Z〉 + 〈[Z, X],Y〉 + 〈[Z, Y ],X〉
for X, Y, Z ∈ s, one obtains
⎧
∇ h X =0, ∇ X h = −[h, X],
⎪⎨ ∇ v v ′ = 〈v, v ′ 〉h + 1 2 [v, v′ ],
∇ v z = ∇ z v = −µ(z,v),
⎪⎩
∇ z z ′ =2〈z,z ′ 〉h,
where X ∈ s,v,v ′ ∈ v,z,z ′ ∈ z and µ is the composition of quadratic forms in the definition
of n. Then the following results have been proved.
Theorem 3.1.2.
([7], [8]).
(1) A Damek-Ricci space is a symmetric space if and only if it is classical
(2) A Damek-Ricci space has strictly negative sectional curvature if and only if it is classical
([13]).
In fact, Damek constructed a simple example of Damek-Ricci space whose sectional
curvature attains zero for some two-dimensional plane. Other examples have been given