Rend. Sem. Mat. Univ. Pol. Torino
Vol. 60, 2 (2002)
M. J. Druetta ∗
P-SPACES OF IWASAWA TYPE AND ALGEBRAIC RANK ONE
Abstract. A Riemannian manifold is a P-space if the Jacobi operators
along the geodesics are diagonalizable by a parallel orthonormal basis.
We show that a solvable Lie group of Iwasawa type and algebraic rank
one which is a P-space is a symmetric space of noncompact type and rank
one. In particular, irreducible, non-flat homogeneous Einstein P-spaces
with nonpositive curvature and algebraic rank one, are rank one symmetric
spaces of noncompact type.
Let M be a Riemannian manifold, R its curvature tensor and R X the Jacobi operator
defined by R X Y = R(Y, X)X, where X is a unit tangent vector. Following [3], we say
that M is a P-space if for every geodesic γ in M the associated Jacobi operators Rγ 0 (t )
are diagonalizable by a parallel orthonormal basis along γ , condition that is satisfied
for symmetric spaces.
In this paper we study the homogeneous P-spaces of Iwasawa type and algebraic
rank one and in particular, those with nonpositive curvature which are Einstein, since
irreducible, non-flat homogeneous Einstein spaces with nonpositive curvature are represented as Lie groups of Iwasawa type (see [6]). The geometry of Lie groups of
Iwasawa type and algebraic rank one which at a first glance seems to be complicated,
becomes very simple when they are P-spaces: they are Damek–Ricci spaces whose
geometric structure is well known (Damek-Ricci spaces are defined in Section 1, following [2], Chapter 4).
An outline of the paper is as follows. In Sections 1 and 2, we give the basic results
concerning the Lie algebras of Iwasawa type and algebraic rank one, its geodesics in
various directions and the expression of the Jacobi operator along them. Properties of
its eigenvalues in the special case of parallel eigenvectors are obtained in Section 3. The
geometric hypothesis involving condition (P) about the eigenvalues is strongly used in
Section 4 to obtain algebraic properties of the solvable Lie algebra. This information
allows us to show that an Iwasawa type Lie group of algebraic rank one satisfying
condition (P) is a Damek-Ricci space. By applying a result from [2], we obtain the
following:
T HEOREM . If S is a solvable Lie group of Iwasawa type and algebraic rank one
which is an P-space, then S is a rank one symmetric space of noncompact type.
The class of Riemannian manifolds obtained by considering Lie groups of Iwasawa
type contains as a subclass the Damek-Ricci spaces, and more generally the irreducible,
∗ Partially supported by Conicet, Conicor and Secyt (UNC). Part of this work was done during a stay of
the author at ICTP, Trieste-Italy, by funds from IMPA (CNPq, Brazil).
55
56
M. J. Druetta
non-flat homogeneous Einstein spaces with nonpositive curvature. As a consequence,
we have
C OROLLARY. An irreducible, non-flat homogeneous Einstein P-space with nonpositive curvature and algebraic rank one is a symmetric space of noncompact type
and rank one.
1. Lie algebras of Iwasawa type and algebraic rank one
A solvable Lie algebra s with inner product h, i is a metric Lie algebra of Iwasawa type
if it satisfies the conditions
(i) s = n ⊕ a where n = [s, s] and a, the orthogonal complement of n, is abelian.
(ii) The operators ad H are symmetric for all H ∈ a.
(iii) For some H0 ∈ a, ad H0 |n has positive eigenvalues.
The simply connected Lie group S with Lie algebra s and left invariant metric
induced by the inner product h, i will be called of Iwasawa type. The Levi Civita
connection and the curvature tensor associated to the metric can be computed by
2 h∇ X Y, Z i
R(X, Y )
=
=
h[X, Y ], Z i − h[Y, Z ], Xi + h[Z , X], Y i ,
[∇ X , ∇Y ] − ∇[X,Y ]
for any X, Y, Z in s.
For each unit vector X in s, R X , the Jacobi operator associated to X, is the symmetric endomorphism of s defined by R X Y = R(Y, X)X. We will say that either the
metric Lie algebra s satisfies condition (P) or S is a P-space if for every geodesic γ in
S the associated Jacobi operator Rγ 0 (t ) can be diagonalized by a parallel orthonormal
basis of Tγ (t ) S. We note that condition (P) is equivalent to the fact R X ◦ R 0X = R 0X ◦ R X
for all X ∈ s (see Corollary 5 of [3] and note that S is a real analytic C ∞ -manifold).
We recall that S is an Einstein space if
Ric(X) = trR X = c |X|2 , c constant, for all X ∈ s.
If s = n⊕a is a metric Lie algebra of Iwasawa type, let z denote the center of n = [s, s]
and let v be the orthogonal complement of z with respect to the metric h, i restricted
to n. Thus n decomposes as n = z ⊕ v, and for all H ∈ a ad H : z → z and hence,
ad H : v → v since ad H is symmetric. We recall that n is said to be 2-step nilpotent if
[n, n] = [v, v] ⊂ z.
For any Z ∈ z the skew-symmetric linear operator j Z : v → v is defined by
h j Z X, Y i = h[X, Y ], Z i for all X, Y ∈ v and Z ∈ z.
Equivalently, j Z X = (ad X )∗ Z for all X ∈ v, where (ad X )∗ denotes the adjoint of ad X .
The operators j Z coincide with the usual one in the case of a 2-step nilpotent n (see
[5]) and their properties determine the geometry of n and s, as we will see.
We now assume that s = n⊕a is a metric Lie algebra of Iwasawa type and algebraic
rank one; that is, a = RH where H, |H | = 1, is chosen such that all eigenvalues of
P-spaces of Iwasawa type
57
ad H |n are positive. S is called a Damek-Ricci space in the special case that ad H |z = Id,
ad H |v = 12 Id and j Z2 = − |Z |2 Id for all Z ∈ z (see [2], p. 78).
We recall that since ad H |n is a symmetric operator, n has an orthogonal direct
sum decomposition into eigenspaces nµ , for all eigenvalues µ of ad H |n , which are invariant by ad H with the property [nµ , nµ0 ] ⊂ nµ+µ0 (by the Jacobi identity), whenever
µ+µ0 is an eigenvalue of ad H |n (see [7]). Moreover, since z and v are ad H -invariant,
P by
the same P
argument they also have decompositions into their eigenspaces as z = λ zλ ,
and v = µ vµ .
1.1. Algebraic structure of the Lie algebra s
The definition of the Lie algebra structure on s implies that, as a Lie algebra, s is
the semidirect sum s = n +σ a of n and a = RH, by considering the R-algebra
homomorphism σ = ad: a →der n, H → (ad H : n → n). Carrying this over to
the group level means that S = N ×τ A is a semidirect product of N and A = R
(considered in the canonical way), where
τ : A → AutN, τa : x → axa −1, (dτa )e = Ad(a),
is given by a exp Xa −1 = expn(Exp(tad H )X) for all X ∈ n, a = t, and Exp denotes the exponential mapof matrices. Note that S is diffeomorphic to s under the
map (X, r ) → expn X, r since expn : n → N , the exponential map of N, is a
diffeomorphism.
We assume that n is 2-step nilpotent. In this case we have that for any Z ∈ z and
X ∈ v, if Z ∗ and Y are eigenvectors of ad H restricted to z and v, with associated
eigenvalues λ and µ, respectively, then the product in S yields
=
(expn(Z + X), r ) · (expn(Z ∗ + Y ), s)
1
(expn(Z + erλ Z ∗ + erµ [X, Y ] + X + erµ Y ), r + s).
2
In fact, note that by the definition of the product in S we have
(expn(Z + X), r ) · (expn(Z ∗ + Y ), s)
=
=
=
expn(Z + X)τr (expn(Z ∗ + Y )), r + s
expn(Z + X) expn Exp(r ad H )Z ∗ + Exp(r ad H )Y , r + s
expn(Z + X) expn(erλ Z ∗ + erµ Y ), r + s ,
since expn X expn Y = expn(X + Y + 21 [X, Y ]) gives the multiplication law in N (see
the Campbell-Hausdorff formula in [7]).
1.2. Global coordinates in S
We introduce global coordinates in S given by ϕ = (x 1 , ..., x k , y1 , ..., ym , r ), defined
as follows. If {Z 1 , ..., Z k } and {X 1 , ..., X m } (k = dim z, m = dim v) are orthonormal
58
M. J. Druetta
bases of eigenvectors of ad H in z and v, with associated eigenvalues {λ1 , ..., λk } and
{µ1 , ..., µm } respectively, then
ϕ(x 1 , ..., x k , y1 , ..., ym , r ) = expn(x 1 Z 1 + ... + x k Z k + y1 X 1 + ... + ym X m ), r .
Following the same argument as the given in [2], p. 82 for Damek-Ricci spaces, we
see in the case of 2-step nilpotent n that
∂ = e−rλi Z i (ϕ(x 1 , ..., x k , y1 , ..., ym , r )),
∂ x i ϕ(x1 ,...,xk ,y1 ,...,ym ,r)
∂ = e−rµi X i (ϕ(x 1 , ..., x k , y1 , ..., ym , r ))
∂y i ϕ(x 1 ,...,x k ,y1 ,...,ym ,r)
+
1 X −rλs
e
y j h j Z s X i , X j iZ s (ϕ(x 1 , ..., x k , y1 , ..., ym , r )),
2
j,s
∂ ∂r = H (ϕ(x 1, ..., x k , y1 , ..., ym , r )),
ϕ(x 1 ,...,x k ,y1 ,...,ym ,r)
where Z i , X i and H on the right-hand side denote the left invariant vector fields on S
associated to the corresponding vectors in s.
1.3. Curvature formulas
By applying the connection formula given at the beginning of this section, one obtains
∇ H = 0 and if Z , Z ∗ ∈ z, X, Y ∈ v then ∇ Z Z ∗ = ∇ Z ∗ Z = h[H, Z ], Z ∗ iH, ∇ X Z =
∇ Z X = − 21 j Z X and
∇ X Y = 21 [X, Y ] + h[H, X], Y iH, in case of 2-step nilpotent n. Consequently, by a
direct computation, we obtain the following formulas (see [4], Section 2):
(i) R H = −ad2H .
(ii) If either Z ∈ zλ , |Z | = 1, or X ∈ vµ , |X| = 1,
R Z H = −λ2 H and R X H = −µ2 H.
(iii) If Z ∈ zλ , |Z | = 1, for any Z ∗ ∈ z and X ∈ v, we have
1
R Z Z ∗ = λ hZ , ad H Z ∗ iZ − ad H Z ∗ and R Z X = − j Z2 (X) − λad H X.
4
In the case that n is 2-step nilpotent, we obtain
(iv) If X ∈ vµ , |X| = 1, for any Z ∈ z , Y ∈ v
RX Z =
3
1
[X, j Z X] − µad H Z and R X Y = − j[X,Y ] X − µad H Y.
4
4
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P-spaces of Iwasawa type
2. Geodesics and associated Jacobi operators
Throughout this section and the following ones, s = n ⊕ a will denote a metric Lie
algebra of Iwasawa type and algebraic rank one, where a = RH, |H | = 1, is chosen
such that all eigenvalues of ad H |n are positive, and n = z ⊕ v is expressed as in the
previous section. Let γY denote the geodesic in S satisfying γY (0) = e (the identity of
S) and γY0 (0) = Y. For any X ∈ s, the associated left invariant field along the geodesic
γY will be denoted by X (t) = X (γY (t)) = (dLγY (t ) )e X.
Next, we compute the geodesic γY with Y ∈ n, an eigenvector of ad H |n .
L EMMA 1. If Y ∈ n is an eigenvector of ad H |n with eigenvalue α, then
1
tanh αt
Y, − ln(cosh αt)
γY (t) = expn
α
α
with associated tangent vector field
γY0 (t) =
1
Y (t) − tanh αt H (t).
cosh αt
Proof. Let S0 be the simply connected Lie group associated to s0 , the Lie algebra
spanned by {Y, H }. Note that S0 has global coordinates ϕ(x, r ) = (expn xY, r ) and it
is a totally geodesic subgroup of S with connection ∇ S0 = ∇| S0 satisfying
∇Y Y = α H, ∇Y H = −[H, Y ] = −αY, ∇ H = 0.
Since the coordinate fields associated to ϕ are given by
∂ ∂ −rα
= e Y (ϕ(x, r )),
= H (ϕ(x, r )),
∂ x ϕ(x,r)
∂r ϕ(x,r)
the Christoffel symbols are easily computed by the formulas
∂ ∂ ∂ −2rα
∇∂
=
αe
H
(ϕ(x,
r
))
,
∇
=
−α
,
∂
∂x ∂ x ∂ x ∂r ∂ x ϕ(x,r)
ϕ(x,r)
ϕ(x,r)
∂ ∂ ∇∂
= 0= ∇∂
.
∂r ∂r ∂r ∂ x ϕ(x,r)
ϕ(x,r)
Hence, we obtain the geodesic γY (t) = ϕ(x(t), r (t)), where x(t) and r (t) are solutions
of the differential equations
x 00 − 2αx 0r 0
=
0,
r + αe−2rα (x 0 )2 = 0.
∂ , γY0 (t) = 1, with
Using that γY0 (t) = x 0 (t) ∂∂x γ (t ) + r 0 (t) ∂r
γ (t )
00
Y
Y
∂ 0
2αr(t )
x (t) = e
,
∂x =e
γY (t )
−αr(t )
∂ Y (t) and
= H (t),
∂r γY (t )
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M. J. Druetta
we have the equivalent equations
x 0 (t)
=
e2αr(t ),
r 00 (t) + α(1 − (r 0 (t))2
=
0
whose solutions satisfy
x(t) =
Z
t
e2αr(u)du and r 0 (t) = − tanh αt.
0
Therefore, we get
r (t) = −
1
1
ln(cosh αt), x(t) = tanh αt,
α
α
and the expression of γY and γY0 follows as claimed.
P ROPOSITION 1. If Z ∈ z and X ∈ v are eigenvectors of ad H with associated
eigenvalues λ and µ, respectively, then for any Y ∈ v, we have
1
(dL γ Z (t ))e
(i ) Rγ Z0 (t ) Y (t) =
cosh2 λt
1
· R Z Y − sinh2 λt ad2H Y − sinh λt j Z
λ Id − ad H Y
2
1
(ii ) Rγ X0 (t ) Z (t) =
(dL γ X (t ) )e
cosh2 µt
1
· R X Z − λ2 sinh2 µt Z + (λ − µ) sinh µt j Z X
2
1
(iii ) Rγ X0 (t ) Y (t) =
(dL γ X (t ))e
cosh2 µt
1
µId − ad H [X, Y ]
· R X Y − sinh2 µt ad2H Y − sinh µt
2
in the case of 2-step nilpotent n, with Y ⊥X in v.
Proof. (i) Let Z ∈ z and γ Z (t) be the associated geodesic. Since
γ Z0 (t) =
1
Z (t) − tanh λt H (t), we have that
cosh λt
1
(dL γ Z (t ) )e
R(Y (t), γ Z0 (t))γ Z0 (t) =
2
cosh
λt
· R Z Y + sinh2 λt R H Y − sinh λt (R(Y, Z )H + R(Y, H )Z ) .
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P-spaces of Iwasawa type
Using the Bianchi identity and the connection formulas we compute
R(Y, Z )H + R(Y, H )Z
=
2R(Y, H )Z − R(Z , H )Y
=
2∇[H,Y ] Z − ∇[H,Z ] Y
1
− j Z (ad H Y ) + λj Z Y,
2
=
that substituted in the above expression, gives (i) as stated since R H = −ad2H .
(ii)-(iii) Assume that X ∈ v is an eigenvector of ad H with eigenvalue µ, and let
Y ⊥ X in v. Using the expression of γ X0 (t), in the same way as (i) we get
1
(dL γ X (t ) )e
R(Z (t), γ X0 (t))γ X0 (t) =
cosh2 µt
· R X Z + sinh2 µt R H Z − sinh µt (R(Z , X)H + R(Z , H )X) .
Hence, the expression of Rγ X0 (t ) Z (t) follows as claimed since
R(Z , X)H + R(Z , H )X
= 2∇[H,Z ] X − ∇[H,X ] Z = 2λ∇ Z X − µ∇ X Z
1
= −(λ − µ) j Z X.
2
Finally, we have
1
(dL γ X (t ))e
R(Y (t), γ X0 (t))γ X0 (t) =
cosh2 µt
· R X Y + sinh2 µt R H Y − sinh µt (R(Y, X)H + R(Y, H )X) .
In the same way as above, in the case of 2-step nilpotent n, we compute
R(Y, X)H + R(Y, H )X
=
=
2∇[H,Y ] X − ∇[H,X ] Y
1
−[H, [X, Y ]] + µ[X, Y ] (hX, Y i = 0),
2
which completes the proof of the proposition.
3. Eigenvectors and eigenvalues along Jacobi operators
In this section we assume that n = z ⊕ v is non-abelian and j Z is non-singular on v for
all Z ∈ z. Note that if λ is an eigenvalue of ad H |z and Z ∈ zλ then j Z |vµ : vµ → vλ−µ
for any eigenvalue µ of ad H |v . In fact, for X ∈ vµ and Y ∈ vµ0
h j Z X, Y i = h[X, Y ], Z i 6= 0 ⇒ [X, Y ] 6= 0,
and thus µ + µ0 = λ since [X, Y ] ∈ nµ+µ0 and has non-zero component in zλ .
62
M. J. Druetta
Hence, for any eigenvalue µ of ad H |v, λ − µ is also an eigenvalue of ad H |v, λ > µ
and the symmetric operator j Z2 preserves vµ . Moreover, the map Z → [X, j Z X] defines a symmetric operator on z (h[X, j Z X], Z ∗ i = h j Z ∗ X, j Z Xi = h[X, j Z ∗ X], Z i)
such that [X, j Z X] ∈ zλ for all X ∈ vµ since h[X, j Z X], Z i = | j Z X|2 6= 0,
[X, j Z X] ∈ nλ (the Jacobi identity) and λ > µ.
Next, we describe the eigenvalues of the operators Rγ 0 (t ), Rγ 0 (t ) and the parallel
Z
X
vector fields along the geodesics γ Z (t) and γ X (t) for some Z ∈ zλ and X ∈ vµ .
L EMMA 2. Let Z ∈ zλ and X ∈ vµ be unit vectors. We set Y =
jZ X
| jZ X | .
(i) If X is an eigenvector of j Z2 , then Rγ Z0 (t ) has an eigenvector U (t) = x(t)X (t) +
y(t)Y (t) with x(t)2 + y(t)2 = 1 and associated eigenvalue a Z (t) satisfying
1
x(t)
a Z (t) cosh2 λt = hR Z Y, Y i − (λ − µ)2 sinh2 λt − ( λ − µ) | j Z X|
sinh λt
2
y(t)
whenever y(t) 6= 0.
(ii) Assume that n is 2-step nilpotent. If X satisfies [X, j Z X] = | j Z X|2 Z , then
Rγ X0 (t ) has an eigenvector U (t) = x(t)Z (t) + y(t)Y (t), x(t)2 + y(t)2 = 1, whose
associated eigenvalue a X (t) is given by
=
=
a X (t) cosh2 µt
1
y(t)
1
| j Z X|2 − λµ − λ2 sinh2 µt + (λ − µ) | j Z X|
sinh µt or
4
2
x(t)
a X (t) cosh2 µt
1
3
x(t)
− | j Z X|2 − (λ − µ)(µ + (λ − µ) sinh2 µt ) + (λ − µ) | j Z X|
sinh µt
4
2
y(t)
according to x(t) 6= 0 or y(t) 6= 0, respectively.
Proof. Note that by the properties of Z and X, the spaces span{X (t), Y (t)} and
span{Z (t), Y (t)} are invariant under the symmetric operators Rγ Z0 (t ) and Rγ X0 (t ) , respectively. The assertion of the lemma follows from the expression of Rγ 0 (t ) and Rγ 0 (t )
Z
X
given in Proposition 1 and using in each case the equalities
Rγ 0 (t ) (U (t)) = a Z (t)(U (t)) and Rγ 0 (t )(U (t)) = a X (t)U (t).
Z
X
Note that in the last case,
RX Z =
1
3
| j Z X|2 Z − λµZ and R X Y = − | j Z X|2 Y − µ(λ − µ)Y
4
4
since n is 2-step nilpotent.
P ROPOSITION 2. Let Z ∈ zλ be a unit vector. If X ∈ vµ , |X| = 1, is an eigenvector of j Z2 , then the parallel vector field U along the geodesic γ Z with U (0) = x 0 X +y0 Y
63
P-spaces of Iwasawa type
(Y =
jZ X
| j Z X | ),
x 02 + y02 = 1, is given by U (t) = x(t)X (t) + y(t)Y (t) where
x(t) = x 0 cos s(t) − y0 sin s(t) , y(t) = x 0 sin s(t) + y0 cos s(t), and
Z t
du
1
.
s(t) = | j Z X|
2
0 cosh λu
Proof. We first note that (dL γ Z (t ))e (span{X, j Z X}) is invariant under ∇γ Z0 (t ) since
γ Z0 (t) = cosh1 λt Z (t) − tanh λt H (t), ∇ Z X = − 21 j Z X, ∇ Z j Z X = 12 | j Z X|2 X and
∇ H = 0.
Hence the parallel vector field U along γ Z with U (0) = x 0 X + y0 Y is given by U (t) =
x(t)X (t) + y(t)Y (t) satisfying the equation ∇γ Z0 (t )U (t) = 0, which gives
x 0 (t)X + x(t)
1
1
∇ Z X + y 0 (t)Y + y(t)
∇ Z Y = 0 for all real t
cosh λt
cosh λt
since X and Y are left invariant. Thus,
x 0 (t)X − x(t)
1
1
j Z X + y 0 (t)Y − y(t)
j Z Y = 0,
2 cosh λt
2 cosh λt
and x(t), y(t) are solutions of the differential equations
x 0 (t) +
| j Z X|
| j Z X|
y(t) = 0, y 0 (t) −
x(t) = 0
2 cosh λt
2 cosh λt
since j Z Y = − | j Z X| X. By expressing these equations in the matrix form as
0
1 | j Z X|
0 −1
x(t)
x (t)
,
=
y(t)
y 0 (t)
2 cosh λt 1 0
the solutions are given by
0 −1
x(t)
x0
, where J =
,
= Exps(t)J
1 0
y0
y(t)
R t du
s(t) = 21 | j Z X| 0 cosh
exponential map of matrices). The assertion
λu (Exp denotes the cos s − sin s
of the proposition follows since Exps J =
.
sin s
cos s
P ROPOSITION 3. Assume that n is 2-step nilpotent. Let X ∈ vµ and Z ∈ zλ be
unit vectors satisfying [X, j Z X] = | j Z X|2 Z . Then the parallel vector field U along
the geodesic γ X with U (0) = x 0 Z + y0 Y (Y = | jj ZZ XX | ) , x 02 + y02 = 1, is given by
U (t) = x(t)Z (t) + y(t)Y (t) where
x(t) = x 0 cos s(t) − y0 sin s(t) , y(t) = x 0 sin s(t) + y0 cos s(t) and
Z t
1
du
s(t) = | j Z X|
.
2
cosh
µu
0
64
M. J. Druetta
Proof. Note that the parallel displacement U (t) of U (0) along γ X (t) is expressed as
U (t) = x(t)Z (t)+ y(t)Y (t), since (dL γ X (t ) )e (span{Z , j Z X}) is invariant under ∇γ X0 (t ) .
In the same way as in Proposition 2, the equation ∇γ X0 (t )U (t) = 0 gives
x 0 (t)Z − x(t)
| j Z X|
| j Z X|
Y + y 0 (t)Y + y(t)
Z = 0 for all real t.
2 cosh µt
2 cosh µt
Hence, x(t) and y(t) are solutions of the differential equations
x 0 (t) +
| j Z X|
| j Z X|
y(t) = 0, y 0 (t) −
x(t) = 0,
2 cosh µt
2 cosh µt
which are given by
x(t) = x 0 cos s(t) − y0 sin s(t), y(t) = x 0 sin s(t) + y0 cos s(t)
R t du
with s(t) = 21 | j Z X| 0 cosh
µu .
4. Condition (P) on Lie algebras of Iwasawa type and rank one
In this section we will assume that s is a metric Lie algebra of Iwasawa type with
non-abelian n and algebraic rank one satisfying condition (P). We summarize in the
proposition below some properties of the algebraic structure of the Lie algebra n = z⊕v
(Similar ones were obtained in [4], Proposition 1.3). The following lemma will be
useful in what follows.
L EMMA 3. If S0 is a totally geodesic subgroup of a Lie group S which is a Pspace, then S0 is a P-space. Equivalently, a totally geodesic subalgebra s0 of a Lie
algebra s satisfying condition (P) satisfies condition (P).
Proof. Let s and s0 denote the Lie algebras of S and S0 , respectively, with associated
curvature
operator
R 0X =
tensors
R and R0 . Note that for a unit X ∈ s the symmetric
is defined along γ X (t) by Rγ0 0 (t ) = ∇γ X0 (t ) Rγ X0 (t ) on s(t) =
∇γ X0 (t ) Rγ X0 (t ) t =0
X
(dL γ X (t ) )e s, and for any orthonormal parallel basis {ei (t)} satisfying Rγ X0 (t ) ei (t) =
ai (t)ei (t), we have
Rγ0 0 (t )ei (t) = ∇γ X0 (t ) Rγ X0 (t )ei (t) − Rγ X0 (t ) ∇γ X0 (t ) ei (t) = ai0 (t)ei (t).
X
Let X ∈ s0 be a unit vector, and note that s0 and s⊥
0 are invariant under the symmetric
operators R X and R 0X in s since s0 is a totally geodesic subalgebra of s and R0 = R|s0 .
0
0
Hence, s0 and s⊥
0 are also invariant by the skew-symmetric operator R X ◦ R X − R X ◦ R X .
0
0
Using that condition (P) is equivalent to R X ◦ R X − R X ◦ R X = 0 for all unit vectors
0 − R0 ◦ R
0
X in s, it follows that R0X ◦ R0X
0X = 0 (R0X and R0X are the restrictions
0X
0
of R X and R X to s0 , respectively). Thus S0 is a P-space.
65
P-spaces of Iwasawa type
P ROPOSITION 4. Let s = n ⊕ RH, |H | = 1, be a metric Lie algebra of Iwasawa
type that satisfies condition (P). Then
(i) ad H |z = λId.
(ii) If n is non-abelian, λ > µ1 , the maximum eigenvalue of ad H |v. In particular
nλ = z and [nµ1 , v] ⊆ z.
(iii) For any Z ∈ z the linear
map j Z : v → v is an isomorphism with the properties
j Z |vµ : vµ → vλ−µ and j Z2 v : vµ → vµ (isomorphically). In particular, if µ is an
µ
eigenvalue of ad H |v, then λ − µ is also an eigenvalue of ad H |v.
Proof. (i) Let s0 = z ⊕ RH be the subalgebra of s with associated simply connected
Lie group S0 . It follows from the above lemma that S0 is a P-space since s0 is a
totally geodesic subalgebra of s. Moreover, if Z ⊥ Z ∗ are eigenvectors of ad H |z with
associated eigenvalues λ and λ∗ , respectively, the Lie algebra spanned by {Z , Z ∗ , H }
is a three-dimensional totally geodesic subalgebra of s0 , whose associated Lie group is
a P-space, by the previous lemma. Then we can assume that s = s0 and it is spanned
by {Z , Z ∗ , H }. Using Lemma 7 and the Remark on page 73 of [3], it follows that the
condition R X ◦ R 0X − R 0X ◦ R X = 0 implies that
(∇ Z Ric) (Z , H ) = (∇ Z ∗ Ric) (Z ∗ , H ),
where the Ricci tensor associated to S is defined by Ric(X, Y ) = tr(V → R(V, X)Y ).
We compute
Ric(Z , Z ∗ )
Ric(Z ∗ , Z ∗ )
=
=
Ric(Z , H ) = Ric(Z ∗ , H ) = 0, Ric(Z , Z ) = −λ(λ + λ∗ ),
−λ∗ (λ + λ∗ ) and Ric(H, H ) = −(λ2 + λ∗2 ).
As a consequence, it is easy to see that
(∇ Z Ric) (Z , H ) = −Ric(∇ Z Z , H ) − Ric(Z , ∇ Z H )
= λ (Ric(Z , Z ) − Ric(H, H )) = λλ∗ (λ∗ − λ)
and similarly,
(∇ Z ∗ Ric) (Z ∗ , H ) = λλ∗ (λ − λ∗ ).
Hence λ∗ = λ and there is a unique eigenvalue of ad H |z .
(ii) Assume that n is non-abelian. Then we have that [nµ1 , nµ ] 6= 0 for some eigenvalue µ of ad H |v , which implies that µ1 + µ = λ since µ1 is the maximum. Hence,
λ > µ1 ≥ µ for all eigenvalues µ of ad H in v, and it also follows that [nµ1 , v] ⊆ z
from the definition of µ1 .
(iii) We recall that j Z |nµ : nµ → nλ−µ . Hence j Z2 preserves nµ for all µ (see
the beginning of Section 3). Moreover, ker j Z is invariant by ad H since the condition
j Z X = 0 implies that
h j Z ([H, X]), Y i = h[[H, X], Y ], Z i = −h[[X, Y ], H ], Z i − h[[Y, H ], X], Z i
= h[X, Y ], [H, Z ]i + h j Z ([H, Y ]), Xi
= λh j Z X, Y i − h[H, Y ], j Z Xi = 0 for all Y ∈ v.
66
M. J. Druetta
We will show that j Z X 6= 0 for all unit Z in z and X ∈ v. If j Z X = 0, then by
the previous remark we can assume that X ∈ nµ and |X| = 1. Then s0 , the Lie
algebra spanned by {Z , X, H }, is a totally geodesic subalgebra of s since ∇ Z Z = λH ,
∇ X X = µH , ∇ Z X = ∇ X Z = 0 and ∇ Z H = −λZ , ∇ X H = −µX, ∇ H = 0. Hence,
s0 satisfies condition (P) and applying the same argument used to show that λ = λ∗
in (i) above, we obtain that λ = µ, contradicting (ii). Consequently, λ − µ is an
eigenvalue of ad H |v whenever µ is an eigenvalue since j Z |nµ : nµ → nλ−µ . Assertion
(iii) follows since dim vµ ≤ dim vλ−µ ≤ dim vµ ( j Z is an isomorphism).
Next we will show that under the hypothesis of condition (P), the number of eigenvalues of ad H |n can be reduced to at most two, namely λ and 21 λ, attained in z and v,
respectively. For any subspace u of the Lie algebra s and γ X (X ∈ s) a fixed geodesic
in S, we will denote by u(t) =(dLγ X (t ))e u.
R EMARK 1. If s0 ⊕ u is a direct sum decomposition into subspaces of the Lie
algebra s such that s0 (t) and u(t) are invariant under ∇γ X0 (t ) and Rγ X0 (t ) for all t, then
e(t) = e1 (t) + e2 (t), expressed according to the decomposition s0 (t) ⊕ u(t), is a
parallel eigenvector of Rγ X0 (t ) along γ X if and only if e1 (t) and e2 (t) are also parallel
eigenvectors of Rγ X0 (t ) .
In the case of an orthogonal direct sum decomposition, if s0 (t) is invariant under
∇γ X0 (t ) and Rγ X0 (t ), then u(t) = s0 (t)⊥ is also invariant under ∇γ X0 (t ) and Rγ X0 (t ) since
∇γ X0 (t ) is skew-symmetric and Rγ X0 (t ) is symmetric.
P ROPOSITION 5. If s = n⊕RH, |H | = 1, is a metric Lie algebra of Iwasawa type
with non-abelian nilpotent n satisfying condition (P), then the eigenvalues of ad H |n
are λ and 21 λ, when restricted to z and v, respectively. Then n is 2-step nilpotent.
Proof. Note that ad H |z = λId, by Proposition 4. Next we will show that ad H |v =
1
1
2 λId. For this purpose, we fix an eigenvalue µ of ad H |v and assume that µ 6 = 2 λ.
If Z ∈ z is a unit vector, it follows from the definition of ∇γ Z0 (t ) and the expression of
Rγ 0 (t ) given by Proposition 1 (i) that for any eigenvalue µ∗ of ad H |v , vµ∗ (t)⊕vλ−µ∗ (t)
Z
(or v 1 λ (t) in case µ∗ = 12 λ) is invariant under Rγ Z0 (t ) and ∇γ Z0 (t ) since vµ∗ ⊕ vλ−µ∗ is
2
1
R Z -invariant and ∇γ Z0 (t ) X (t) = − 2 cosh
λt j Z X (t).
Assume that the condition (P) is satisfied and let {ei (t) : i = 1, ..., dim s} be an orthonormal parallel basis that diagonalizes Rγ Z0 (t ). Therefore, ei (0) has a non-zero component e ∈ vµ ⊕ vλ−µ for
P some i = 1, ..., dim s (otherwise {ei (0) : i = 1, ..., dim s}
would be a basis of z ⊕ µ∗ 6=µ,λ−µ vµ∗ ⊕ RH ). It follows from the previous remark
that e(t), the parallel displacement of e along γ Z (t), is a parallel eigenvector of Rγ Z0 (t )
with e(t) ∈ vµ ⊕ vλ−µ (t).
Now, we choose a basis {X i : i = 1, ..., m = dim vµ } of vµ that diagonalizes
the symmetric operator j Z2 : vµ → vµ . Hence {X i , j Z X i : i = 1, ..., m} is an
orthogonal basis of vµ ⊕vλ−µ by Proposition 4, and vλ ⊕vλ−µ = ⊕m
i=1 span{X i , j Z X i },
where each span{X i , j Z X i }(t) is invariant under ∇γ Z0 (t ) and Rγ Z0 (t ) (X i and j Z X i are
67
P-spaces of Iwasawa type
eigenvectors of R Z ). Applying Remark 1 again, we can choose a unitP
X ∈ vµ such
that j Z2 X = − | j Z X|2 X (X = X i for some i = 1, ..., m so that e =
ei , ei 6= 0)
and a parallel eigenvector U (t) of Rγ Z0 (t ) along the geodesic γ Z (t) satisfying U (t) =
x(t)X (t) + y(t)Y (t) (Y = | jj ZZ XX | ), with x(t)2 + y(t)2 = 1.
We set U (0) = x 0 X + y0 Y and let a Z (t) be the associated eigenvalue of Rγ Z0 (t ) .
If x 0 6= 0 and y0 6= 0, we have that R Z X = a Z (0)X, R Z Y = a Z (0)Y. Therefore,
hR Z X, Xi = a Z (0) = hR Z Y, Y i and consequently µ = 21 λ since
1
1
| j Z X|2 − λµ = | j Z X|2 − λ(λ − µ).
4
4
Assume that x 0 6= 0 and y0 = 0, hence x 0 = 1, y0 = 0 and a Z (0) = hR Z X, Xi. It
follows from Lemma 2 (applied at t = 0) and Proposition 2 that
x(t)
1
a Z (0) = hR Z Y, Y i − ( λ − µ) | j Z X| lim
sinh λt
t →0 y(t)
2
where x(t) = cos s(t), y(t) = sin s(t) with s 0 (t) =
1
2
| j Z X| cosh1 λt . We compute
(sinh λt)0
(tan s(t))0
= λlimt →0
limt →0
x(t)
sinh λt
y(t)
=
limt →0
=
2λ
(s(t) → 0),
| j Z X|
cosh2 λt cosh2 s(t)
1
2
| j Z X|
which substituted in the above expression gives
a Z (0) =
1
| j Z X|2 − λ(2λ − 3µ).
4
From the equality a Z (0) = hR Z X, Xi, we obtain µ = 12 λ and get a contradiction. The
same argument is used in the case x 0 = 0, y0 = 1.
Now we observe that the conditions ad H |z = λId and ad H |v = 21 λId imply that the
eigenspaces associated to ad H |n are nλ = z and n 1 λ = v. Thus, [v, v] = [n 1 λ , n 1 λ ] ⊆
2
2
2
nλ = z, showing that n is 2-step nilpotent.
E XAMPLE 1. Consider the four-dimensional metric Lie algebra s of Iwasawa type
and algebraic rank one with nilpotent non-abelian n = z ⊕ v. Hence, s is spanned by
an orthogonal basis {Z , X, j Z X, H } with unit vectors Z ∈ z, X ∈ v and Lie bracket
[Z , X]
=
[H, Z ] =
0 = [Z , j Z X], [X, j Z X] = | j Z X|2 Z
1
1
λZ , [H, X] = λX, [H, j Z X] = λj Z X.
2
2
Note that j Z2 X = − | j Z X|2 X. We show that the Lie group S associated to s is a Pspace if and only if | j Z X| = λ. Thus, up to scaling, S is the 2-complex hyperbolic
space CH 2.
68
M. J. Druetta
For this purpose we see that
R Z +X ◦ R 0Z +X (H ) − R 0Z +X ◦ R Z +X (H ) = 0 ⇔ | j Z X| = λ.
Note that from the connection formulas,
∇ Z Z = λH, ∇ X X = 21 λH, ∇ j Z X j Z X = 12 λ | j Z X|2 H, ∇ Z X = − 21 j Z X, ∇ H = 0,
we get
1 1
2
2
| j Z X| − λ X,
RZ X =
2 2
1 1
2
2
| j Z X| − λ j Z X,
R Z ( j Z X) =
2 2
1 1
2
2
| j Z X| − λ Z ,
RX Z =
2 2
1
R X j Z X = − 3 | j Z X|2 + λ2 j Z X,
4
1
1
| j Z X|2
| j Z X|2 − λ2 Z ,
R jZ X Z =
2
2
1
R j Z X X = − | j Z X|2 3 | j Z X|2 + λ2 X.
4
Hence, taking into account that R Z +X (·) = R Z (·) + R X (·) + R(·, Z )X + R(·, X)Z , it
is a direct computation to see that
1
(1)
3λj Z X − 5λ2 H ,
R Z +X (H ) =
4
3 2
1
3
2
| j Z X| + λ j Z X + λ | j Z X|2 H ,
R Z +X ( j Z X) = −
2
4
4
1
| j Z X|2 − λ2 (Z − X),
R Z +X (Z − X) =
2
Recall that, by definition, R 0Z +X (·) =
[∇ Z +X , R Z +X ](·) − R(·, ∇ Z +X (Z + X))(Z + X) − R(·, Z + X)∇ Z +X (Z + X),
and by a straightforward computation, using the connection formulas and the definition
of R, we obtain the following expressions of R 0Z +X
(2)
R 0Z +X (H ) =
R 0Z +X ( j Z X)
=
1 λ | j Z X|2 − λ2 (Z − X),
2
1
− | j Z X|2 (| j Z X|2 − λ2 )(Z − X),
2
since
1
1
7 2
2
2
2
| j Z X| + λ Z + λ | j Z X| + λ X,
[∇ Z +X , R Z +X ](H ) = λ
2
4
2
69
P-spaces of Iwasawa type
1
R(H, ∇ Z +X (Z + X))(Z + X) = − λ | j Z X|2 (Z − X),
4
1
1
3 2
2
2
2
R(H, Z + X)∇ Z +X (Z + X) = λ (3λ + | j Z X| )Z + ( λ + | j Z X| )X ,
2
2
4
and
1
9
[∇ Z +X , R Z +X ]( j Z X) = − | j Z X|2 (| j Z X|2 + λ2 )Z + (| j Z X|2 + 3λ2 )X ,
4
2
R( j Z X, ∇ Z +X (Z + X))(Z + X)
R( j Z X, Z + X)∇ Z +X (Z + X)
3
= − λ2 | j Z X|2 (Z − X),
8
1
| j Z X|2 (| j Z X|2 − 5λ2 )Z
=
4
1
5
− | j Z X|2 (| j Z X|2 (3 | j Z X|2 + λ2 )X.
4
2
Finally, we get
[R Z +X , R 0Z +X ](H ) =
R Z +X ◦
R 0Z +X (H )
=
R 0Z +X ◦ R Z +X (H ) =
=
1 λ 5 | j Z X|2 + λ2 (| j Z X|2 − λ2 )(Z − X), since
8
1
1 2
2
2
2
|
|
λ j Z X| − λ
j Z X| − λ (Z − X) and
2
2
3 0
5
λR
( j Z X) − λ2 R 0Z +X (H )
4 Z +X
4
1 2
− λ 3 | j Z X| + 5λ2 (| j Z X|2 − λ2 )(Z − X),
8
which are computed using (1) and (2) above. The assertion follows as claimed.
T HEOREM 1. If S is a Lie group of Iwasawa type and algebraic rank one which is
a P-space, then S is a rank one symmetric space of noncompact type.
Proof. Let s = n⊕RH, |H | = 1, be the metric Lie algebra of Iwasawa type associated
to S. If n is abelian, then s = z ⊕ RH with λ as unique eigenvalue of ad H |z; thus S is,
up to scaling, the real hyperbolic space.
Assume that n is non-abelian, then by Proposition 5 n is 2-step nilpotent. We
show that | j Z X|2 = λ2 for unit vectors Z ∈ z and X ∈ v. For this purpose let X 6= 0, |X| = 1, be a vector in v. Let {Z 1 , ..., Z k } be an orthonormal basis of z diagonalizing the symmetric operator Z → [X, j Z X] on z. Hence,
2
h j Z i X, j Z l Xi = δil j Z i X and { j Z 1 X, ..., j Z k X} is an orthogonal basis of jz X. More
2
over, since [X, j Z i X] = j Z i X Z i (i = 1, ..., k), it follows that Z i and j Z i X are
eigenvectors of R X |z and R X | jz X , respectively (see Section 1, 1.3), and consequently,
z ⊕ jz X = ⊕ki=1 span{Z i , j Z i X} where span{Z i , j Z i X}(t) is invariant under Rγ X0 (t ) and
∇γ X0 (t ) for all i = 1, ..., k.
70
M. J. Druetta
Assume that condition (P) is satisfied and let {e j (t) : j = 1, ..., dim s} be an
orthonormal parallel basis diagonalizing Rγ X0 (t ). Applying the same argument as used
in the previous proposition, for each i = 1, ..., k, we can choose 1 ≤ ji ≤ dim s
such that Ui (t), the non-zero component of e ji (t) in span{Z i , j Z i X}(t) is a parallel
eigenvector of Rγ X0 (t ) along γ X (t) by Remark 1. We set Z = Z i , Y = | jj ZZ XX | , U =
Ui (0) = x 0 Z + y0 Y with x 02 + y02 = 1, and note that the unit Z ∈ z satisfies [X, j Z X] =
| j Z X|2 Z . By applying Proposition 3, the parallel eigenvector U along γ X with U (0) =
U is given by U (t) = x(t)Z (t) + y(t)Y (t), with
x(t) = x 0 cos s(t) − y0 sin s(t), y(t) = x 0 sin s(t) + y0 cos s(t)
1
,
cosh 12 λt
and s(t) =
1
2
hR X U, U i =
x 02 hR X Z ,
| j Z X|
Zi +
whose associated eigenvalue a X (t) satisfies a X (0) =
y02 hR X Y, Y i,
1
1
1
1
3
y(t)
1
a X (t) cosh2 λt = | j Z X|2 − λ2 − λ2 sinh2 λt + λ | j Z X|
sinh λt
2
4
2
2
4
x(t)
2
and
x(t)
3
1
1
3
1
1
sinh λt
a X (t) cosh2 λt = − | j Z X|2 − λ2 cosh2 λt + λ | j Z X|
2
4
4
2
4
y(t)
2
for all real t, by Lemma 2.
If x 0 6= 0 and y0 6= 0, then a X (0) = hR X Z , Z i = hR X Y, Y i (Z and Y are
eigenvectors of R X ) and we get | j Z X|2 = 14 λ2 since
1
1
3
1
| j Z X|2 − λ2 = − | j Z X|2 − λ2 .
4
2
4
4
If x 0 = 1 and y0 = 0, it follows from Proposition 3 that x(t) = cos s(t),
sin s(t) and in the same way that in the previous proposition,


1
cosh λt
x(t)
λ
1
1
limt →0
.
sinh λt = λlimt →0  s 0 (t2)  =
|
j
y(t)
2
2
Z X|
2
y(t) =
cosh s(t )
Substituting this limit in the last expression of a X (t) above, we get
1
x(t)
3
1
3
sinh λt
a X (0) = − | j Z X|2 − λ2 + λ | j Z X| limt →0
4
4
4
y(t)
2
1
3
= − | j Z X|2 + λ2 ,
4
2
and from the equality hR X Z , Z i = a X (0) it follows that | j Z X|2 = λ2 .
The same condition is obtained in the case x 0 = 0 and y0 = 1, since in this case
x(t) = − sin s(t), y(t) = cos s(t) and a X (0) = hR X Y, Y i with a X (0) computed as
!
sinh 21 λt
1
1
1 2 3
5
2
a X (0) = | j Z X| − λ − λ | j Z X| limt →0
= | j Z X|2 − λ2 .
4
2
4
tan s(t)
4
4
71
P-spaces of Iwasawa type
Finally, using the same argument on each span{Z i , j Z i X} (i = 1, ..., k), it follows
that for any unit vector X ∈ v, an orthonormal basis {Z 1 , ..., Z k } can be chosen so
2
that [X, j Z i X] = j Z i X Z i and { j Z i X : i = 1, ..., k} is an orthogonal basis of jz X
2
2
satisfying either j Z X = 1 λ2 or j Z X = λ2 for i = 1, ..., k. Hence,
4
i
i
1 2
λ |Z |2 ≤ | j Z X|2 ≤ λ2 |Z |2 for all Z ∈ z and X ∈ v, |X| = 1,
4
P
P
since j Z X = ki=1 ai j Z i X is an orthogonal sum for any Z = ki=1 ai Z i in z.
2
Next we show that the condition j Z X = 1 λ2 in the above basis is not possible.
(∗)
i
4
In fact, for a such Z i , it follows from (∗) above that 14 λ2 = −h j Z2 i X, Xi is the minimum
eigenvalue of the symmetric operator − j Z2 i with X as associated eigenvector. Thus
− j Z2 i X = 41 λ2 X and s0 , the Lie algebra spanned by {Z i , X, j Z i X, H }, is a totally
geodesic subalgebra of s. Applying Lemma 3, the associated Lie group to s0 is a Pspace, which is not possible by Example 1.
2
The condition j Z X = λ2 for all i = 1, ..., k, implies that | j Z X|2 = λ2 |Z |2
i
for all Z ∈ z and X ∈ v, |X| = 1, or equivalently j Z2 = −λ2 |Z |2 Id for all Z ∈ z.
Since ad H |z = λId and ad H |v = 12 λId it follows that S is, up to scaling of the metric,
a Damek-Ricci space. Theorem 2 of [2], Section 4.3 implies that S is a rank one
symmetric space of noncompact type.
C OROLLARY 1. If M is an irreducible, non-flat homogeneous Einstein P-space
with nonpositive curvature and algebraic rank one, then M is a symmetric space of
noncompact type and rank one.
Proof. Since M is irreducible and non-flat, by applying Corollary 1 of [8] it follows
that M is a simply connected homogeneous space with nonpositive curvature. Hence,
M can be represented as a solvable Lie group S of algebraic rank one with left invariant
metric of nonpositive curvature, whose associated metric Lie algebra s decomposes as
an orthogonal direct sum s = n ⊕ a where n = [s,s] and a is one-dimensional (see [1]).
By applying Proposition 4.9 and Theorem 4.10 of [6], S is isometric to a Lie group of
Iwasawa type and algebraic rank one. It follows from Theorem 1 that M is a symmetric
space of noncompact type and rank one.
References
[1] A ZENCOTT R. AND W ILSON E., Homogeneous manifolds with negative curvature I, Trans. Amer.Math. Soc. 215 (1976), 323–362.
[2] B ERNDT J., T RICERRI F. AND VANHECKE L.,Generalized Heisenberg groups
and Damek-Ricci harmonic spaces, Lecture Notes in Math. 1598, Springer,
Berlin 1995.
72
M. J. Druetta
[3] B ERNDT J., VANHECKE L., Two natural generalizations of locally symmetric
spaces, Differential Geometry and its Applications 2 (1992), 57–80.
[4] D OTTI I. AND D RUETTA M.J., Negatively curved homogeneous Osserman
spaces, Differential Geometry and Applications 11 (1999), 163–178.
[5] E BERLEIN P., Geometry of 2-step nilpotent groups with a left invariant metric,
Ann. Sci. Ecole. Norm. Sup. 27 (1994), 611–660.
[6] H EBER J., Noncompact homogeneous Einstein spaces, Invent. Math. 133 (1998),
279–352.
[7] H ELGASON S., Differential geometry, Lie groups, and symmetric spaces, Academic Press, New York 1978.
[8] W OLF J., Homogeneity and bounded isometries in manifolds of negative curvature, Illinois J. Math. 8 (1964), 14–18.
AMS Subject Classification: 53C30, 53C55.
Maria J. DRUETTA
Facultad de Matemática, Astronomı́a y Fı́sica
Universidad Nacional de Córdoba
Ciudad Universitaria
5000 Córdoba, ARGENTINA
e-mail: druetta@mate.uncor.edu
Lavoro pervenuto in redazione il 01.09.2001 e, in forma definitiva, il 29.05.2002.