DIFFERENTIAL GEOMETRY AND APPLICATIONS Satellite Conference of ICM in Berlin, Aug. 10 – 14, 1998, Brno Masaryk University in Brno (Czech Republic) 1999, 61–72

OSSERMAN-P SPACES OF IWASAWA TYPE

ISABEL G. DOTTI AND MARIA J. DRUETTA

Abstract. We show that a solvable Lie group of Iwasawa type, 2-step nilpotent nilradical with a left invariant metric satisfying the Osserman-p condition, (1 ≤ p < dim S) is either a space of constant negative sectional curvature or a Damek-Ricci space.

Let M^{n} be a riemannian manifold, R its curvature tensor and R_{X} the Jacobi operator, defined by R_{X}Y = R(Y, X)X, X a unit tangent vector. We say that M satisfies the Osserman-p condition, (Oss-p, for short), p a natural number, if for any set of p orthonormal vectors E = {X_{1}, X_{2}, …, X_{p}} in the tangent bundle, the symmetric operator

j_{E} = R_{X_{1}} + R_{X_{2}} + ⋯ + R_{X_{p}}

has constant eigenvalues, counting multiplicities.

This notion, introduced in [S-V], generalizes the Osserman condition (p = 1). It is immediate to see that if p = n then M^{n} has constant scalar curvature. Moreover, if p satisfies 1 ≤ p < n then the riemannian manifold is Einstein and there exist a duality between p and n − p, that is, M^{n} satisfies the Osserman-p condition if and only if it satisfies the Osserman-(n − p) condition.

We note that spaces of constant sectional curvature satisfy condition Oss-p for all p ≥ 1. It follows from the curvature formula for spaces of constant curvature c,

R(X, Y)Z = c (⟨Y, Z⟩X − ⟨Z, X⟩Y) ,

that R_{X}Y = cX for orthogonal tangent vectors X and Y. Hence, if E = {X_{1}, …, X_{p}} is an orthonormal set, we have that

j_{E} |_{E}= c(p − 1) Id, j_{E} |_{E^{⊥}}= cp Id ,

1991 Mathematics Subject Classification. 53C30.

Key words and phrases. Iwasawa type, Osserman-p condition.

Partially supported by grants from Conicor, SecytUNC, Conicet (Argentina). Part of this work was done during a stay at ICTP, Trieste, Italy; the first author as an associate member and the second supported by funds from IMPA (CNPq, Brasil).

This paper is in final form and no version of it will be submitted for publication elsewhere.

In this note we consider the following question raised by P. Gilkey: if a non positively curved homogeneous manifold M^{n} satisfies condition Oss-p, for some 1 < p < n then the curvature is constant. The purpose of the present work is to give an affirmative answer to this question in a subclass of that of homogeneous manifolds of nonpositive curvature. More precisely we prove: Theorem. If S is a solvable Lie group of Iwasawa type and dimension n, with [S, S] two step nilpotent and satisfying the Osserman-p condition for some 1 ≤ p < n, then S is either a Damek-Ricci space or a space of constant sectional curvature.

1. Curvature formulas on Iwasawa type Lie groups

A solvable Lie algebra 𝔰 with inner product ⟨, ⟩ is a metric Lie algebra of Iwasawa type, if it satisfies the conditions

(i) 𝔰 = 𝔞 ⊕ 𝔫 where 𝔫 = [𝔰, 𝔰] and 𝔞, the orthogonal complement of 𝔫, is abelian. (ii) All operators ad_{H}, H ∈ 𝔞 are symmetric. (iii) For some H_{0} ∈ 𝔞 ad_{H_{0}}|_{𝔫} has positive eigenvalues. The simply connected Lie group S with Lie algebra 𝔰 and left invariant metric

induced by the inner product ⟨, ⟩ will be called of Iwasawa type. The Levi Civita connection and the curvature tensor associated to the metric can be computed by,

2 ⟨∇_{X}Y, Z⟩ = ⟨[X, Y], Z⟩ − ⟨[Y, Z], X⟩ + ⟨[Z, X], Y ⟩
R(X, Y) = [∇_{X}, ∇_{Y}] − ∇_{[X,Y]}

for any X, Y, Z in 𝔰.

Let n = dim 𝔰 and 1 ≤ p < n. We will say that the metric Lie algebra 𝔰 satisfies the Oss-p condition, if the corresponding Lie group S of Iwasawa type satisfies it. As we recalled above, Oss-p spaces with 1 ≤ p < n are Einstein. Here, and from now on, n will stand for the dimension of S.

Let 𝔰 = 𝔫 ⊕ 𝔞 be a metric Lie algebra of Iwasawa type. Since 𝔞 is abelian, 𝔞 operates on 𝔫 by commuting symmetric maps {ad_{H}|_{𝔫} : H ∈ 𝔞} so 𝔫 has an orthogonal decomposition into root spaces 𝔫_{λ_{i}}, where λ_{i} ∈ 𝔞^{∗} is such that

ad_{H}X_{i} = λ_{i}(H)X_{i} for all X_{i} ∈ 𝔫_{λ_{i}}, H ∈ 𝔞.

Let 𝔷 denote the center of 𝔫 = [𝔰, 𝔰] and let 𝔳 be the orthogonal complement of 𝔷 with respect to the metric ⟨, ⟩ restricted to 𝔫. Thus 𝔫 decomposes 𝔫 = 𝔷 ⊕ 𝔳 and for all H ∈ 𝔞, ad_{H} : 𝔷 → 𝔷 hence ad_{H} : 𝔳 → 𝔳 (since ad_{H} is symmetric). Thus 𝔷 = ∑ 𝔷_{μ_{i}} and 𝔳 = ∑ 𝔳_{λ_{j}} (Note that the equality μ_{i} = λ_{j} is possible). For each element Z ∈ 𝔷 we define a skew symmetric linear transformation j_{Z} : 𝔳 → 𝔳 by

⟨j_{Z}X, Y ⟩ = ⟨[X, Y], Z⟩ for all X, Y ∈ 𝔳, all Z ∈ 𝔷 .

The transformations j_{Z} coincides with the usual ones considered in the case of a 2-step nilpotent 𝔫 (see [E]) and play an important role, since the geometry of 𝔫 and 𝔰 seems to be concentrated on their properties. We note that, by using the expression of the connection, one easily obtains

∇_{Z}X = ∇_{X}Z = −\frac{1}{2}j_{Z}X, Z ∈ 𝔷, X ∈ 𝔳 .

OSSERMAN-P SPACES OF IWASAWA TYPE 63

Computations. Now we express the curvature operators of the Lie group S in terms of the curvature of N, the Lie group associated to 𝔫, with induced metric from 𝔰 and second fundamental form α with respect to unit normal vectors H ∈ 𝔞. We will denote by R^{𝔫} the curvature tensor of N. We recall that if ∇^{𝔫} denotes the associated connection to the induced metric ⟨, ⟩ on 𝔫, then the Gauss’s formula is

∇_{X}Y = ∇_{X}^{𝔫}Y + ∑_{i=1}^{dim 𝔞} ⟨ad_{H_{i}}X, Y ⟩ H_{i}

since the second fundamental form α is defined by

⟨α(X, Y), H⟩ = ⟨−∇_{X}H, Y ⟩ = ⟨[H, X], Y ⟩, H ∈ 𝔞 .

Thus, for any X, Y, W in 𝔫 the equation of Gauss yields (see [K-N, Proposition 4.1, Chapter VII)]

⟨R(Y, X)X, W ⟩ = ⟨R^{𝔫}(Y, X)X, W ⟩

+ ∑_{i} ⟨ad_{H_{i}}Y, X⟩ ⟨ad_{H_{i}}X, W ⟩ − ∑_{i} ⟨ad_{H_{i}}Y, W ⟩ ⟨ad_{H_{i}}X, X⟩

where {H_{1}, …, H_{r}} is an orthonormal basis of 𝔞. Hence

(1) R_{X}Y = R_{X}^{𝔫}Y + ∑_{i}^{r} ⟨ad_{H_{i}}X, Y ⟩ ad_{H_{i}}X − ∑_{i}^{r} ⟨ad_{H_{i}}X, X⟩ ad_{H_{i}}Y

+ ∑_{i=1}^{r} ⟨R_{X}Y, H_{i}⟩ H_{i} .

One can easily verify that if Z, Z^{∗} ∈ 𝔷 and X ∈ 𝔳 then

R_{Z}^{𝔫}Z^{∗} = 0, R_{Z}^{𝔫}X = −\frac{1}{4}j_{Z}^{2}X . (2)

Furthermore, in the case when 𝔫 is 2-step nilpotent, we will use the following curvature formulas given in [E]

R_{X}^{𝔫}Z = \frac{1}{4}[X, j_{Z}X] and R_{X}^{𝔫}Y = −\frac{3}{4}j_{[X,Y]}X Z ∈ 𝔷, X, Y ∈ 𝔳 . (3)

In the next proposition we summarize the basic curvature formulas that will be used throughout the article.

Proposition 1.1. If 𝔰 is a Lie algebra of Iwasawa type then the following formulas involving curvatures hold. Assume {H_{1}, …, H_{r}} is an orthonormal basis of 𝔞.

(i) R_{H} = −ad_{H}^{2} for all H ∈ 𝔞.

(ii) If either Z ∈ 𝔷_{μ}, | Z |= 1 or X ∈ 𝔳_{λ}, | X |= 1 then

R_{Z}H = −μ(H) ∑_{i=1}^{r} μ(H_{i})H_{i} , R_{X}H = −λ(H) ∑_{i=1}^{r} λ(H_{i})H_{i} .

(iii) If Z ∈ 𝔷_{μ_{i}}, | Z |= 1, and Z^{∗} ∈ 𝔷_{μ_{j}} then

R_{Z}Z^{∗} = ∑_{k=1}^{r} μ_{i}(H_{k})μ_{j}(H_{k}) (⟨Z, Z^{∗}⟩Z − Z^{∗}) .

(iv) If Z ∈ 𝔷_{μ}, | Z |= 1, and X ∈ 𝔳_{λ}

R_{Z}X = −\frac{1}{4}j_{Z}^{2}(X) − ∑_{i=1}^{r} μ(H_{i})λ(H_{i})X .

If in addition 𝔫 is 2-step nilpotent, then

(v) If X ∈ 𝔳_{λ_{j}} and Z ∈ 𝔷_{μ}

R_{X}Z = \frac{1}{4}[X, j_{Z}X] − ∑_{i=1}^{r} λ_{j}(H_{i})μ(H_{i})Z .

(vi) If X ∈ 𝔳_{λ_{j}} and Y ∈ 𝔳_{λ_{k}}

R_{X}Y = −\frac{3}{4}j_{[X,Y]}X + ∑_{i=1}^{r} (λ_{j}(H_{i})^{2}⟨X, Y ⟩X − λ_{j}(H_{i})λ_{k}(H_{i})Y ) .

When 𝔰 = 𝔫 ⊕ RH with |H| = 1 and 2-step nilpotent 𝔫 one also has

(vii) Ric(H) = −trad_{H}^{2}. If Z ∈ 𝔷 and X ∈ 𝔳 are unit vectors then

Ric(Z) = −\frac{1}{4}trj_{Z}^{2}−⟨[H, Z], Z⟩trad_{H} , Ric(X) = \frac{1}{2}⟨∑_{i=1}^{k} j_{Z_{i}}^{2}X, X⟩−⟨[H, X], X⟩trad_{H} .

Proof. By a direct calculation, using the conexion formula, ∇_{H} = 0 if H ∈ 𝔞. Hence, R_{H} = −ad_{H}^{2} and (i) follows. Moreover, for Y ∈ 𝔫_{γ} and using again ∇_{H} = 0, one obtains R_{Y} H = −∇_{[H,Y]}Y = −γ(H)∇_{Y} Y. We observe next that ∇_{Z}Z and ∇_{X}X for Z ∈ 𝔷_{μ}, X ∈ 𝔳_{λ} have only components in 𝔞. Hence (ii) follows since

∇_{Z}Z = ∑_{i} μ(H_{i})H_{i}, ∇_{X}X = ∑_{i} λ(H_{i})H_{i} .

The proofs of (iii) and (iv) are immediately obtained by substituting, in each case, identity (2) into identity (1).

We note that (R_{X}Z)_{𝔞} = (R_{X}Y)_{𝔞} = 0, for any Z ∈ 𝔷_{μ}, X ∈ 𝔳_{λ_{j}} and Y ∈ 𝔫_{λ_{k}} since by (ii) above one has

⟨R_{X}Z, H⟩ = ⟨Z, R_{X}H⟩ = 0 , ⟨R_{X}Y, H⟩ = ⟨Y, R_{X}H⟩ = 0 .

Therefore, replacing (3) in (1), the proof of (v) and (vi) follows.

Since Ric(Y) = trR_{Y}, using (i) it follows Ric(H) = − tr ad_{H}^{2}. If Y ∈ 𝔫, |Y | = 1, identity (1) implies

Ric(Y) = Ric^{𝔫}(Y) − ⟨ad_{H}Y, Y ⟩ tr ad_{H} .

Finally, for unit vectors Z ∈ 𝔷 and X ∈ 𝔳, (2) and (3) give Ric^{𝔫}(Z) = −\frac{1}{4}tr j_{Z}^{2} and Ric^{𝔫}(X) = \frac{1}{2}⟨∑_{i=1}^{k} j_{Z_{i}}^{2}X, X⟩ (see [E] for a proof). Thus (vii) is obtained.

2. The Osserman-p condition on solvable Lie algebras of Iwasawa type and rank 1

In this section we obtain various results that hold if one imposes the Oss-p condition to Iwasawa type Lie algebras with dim 𝔞 = 1. We will call this class of algebras of rank one. We observe first

Lemma 2.1. Let 𝔰 be a solvable Lie algebra of Iwasawa type and Z ∈ 𝔷_{μ}, where μ is a root of ad_{H}|_{𝔷}. Then, j_{Z}|_{𝔳_{λ_{i}}} : 𝔳_{λ_{i}} → 𝔳_{μ−λ_{i}}. In particular j_{Z}^{2} preserves 𝔳_{λ_{i}}. Moreover, if 𝔫 is 2-step nilpotent and X ∈ 𝔳_{λ} then [X, j_{Z}X] ∈ 𝔷_{μ}.

Proof. We first show that j_{Z}|_{𝔳_{λ_{i}}} : 𝔳_{λ_{i}} → 𝔳_{μ−λ_{i}}. Let X ∈ 𝔳_{λ_{i}} and assume that j_{Z}X ≠ 0. Hence for some Y ∈ 𝔳_{λ_{j}} we have, ⟨j_{Z}X, Y ⟩ = ⟨[X, Y], Z⟩ ≠ 0 implying that [X, Y] has a non zero component in 𝔷_{μ}. Thus, λ_{i} + λ_{j} = μ. For the last assertion, if [X, j_{Z}X] ≠ 0, then [X, j_{Z}X] ∈ 𝔫_{μ} and j_{Z}X ≠ 0. Since [X, j_{Z}X] ∈ 𝔷 ∩ 𝔫_{μ}, it follows that [X, j_{Z}X] ∈ 𝔷_{μ}.

Proposition 2.2. If 𝔰 is a solvable Lie algebra of Iwasawa type with 2-step nilpotent 𝔫 and satisfying Oss-p for some p ≥ 1, then dim 𝔞 = 1.

Proof. We assume 𝔫 is 2-step nilpotent and dim 𝔞 = r > 1. Let H_{1} and H_{2} be orthonormal vectors in 𝔞 and

E_{s,t} = {sH_{1} + tH_{2}, H_{3}, …, H_{r}, Z_{1}, …, Z_{k}, X_{1}, …, X_{l}}

with s^{2} +t^{2} = 1 be the orthonormal set of p vectors in 𝔰, obtained after completing with H_{i} ∈ 𝔞, Z_{i} ∈ 𝔷 and X_{i} ∈ 𝔳, eigenvectors of ad_{𝔞}|_{𝔫}. Then

j_{E_{s,t}} = R_{sH_{1}+tH_{2}} − ∑_{i=3}^{r} ad_{H_{i}}^{2} + ∑_{i=1}^{k} R_{Z_{i}} + ∑_{i=1}^{l} R_{X_{i}} = −ad_{sH_{1}+tH_{2}}^{2} + T

where T = − ∑_{i=3}^{r}ad_{H_{i}}^{2} + ∑_{i=1}^{k} R_{Z_{i}} + ∑_{i=1}^{l} R_{X_{i}}. Since the eigenspaces 𝔷_{μ} are invariant under T (see the curvature formulas in Proposition 1.1 and previous lemma), by choosing an eigenvector Z ∈ 𝔷_{μ}, | Z |= 1, of T, we have that Z is also an eigenvector of j_{E_{s,t}} with associate eigenvalue

⟨j_{E_{s,t}}Z, Z⟩ = ⟨(−ad_{sH_{1}+tH_{2}}^{2} + T ) Z, Z⟩ = −μ(sH_{1} + tH_{2})^{2} + ⟨T Z, Z⟩

where ⟨T Z, Z⟩ does not depend on (s, t). By using the condition Oss-p, the continuous function from the circle (s^{2} + t^{2} = 1)

(s, t) → ⟨j_{E_{s,t}}Z, Z⟩

is constant since it take values in the finite set of eigenvalues of J_{E}, where E varies on p-dimensional subspaces. Consequently, μ(sH_{1} + tH_{2})^{2} is constant, implying μ(sH_{1} + tH_{2}) = sμ(H_{1}) + tμ(H_{2}) is also constant. Thus μ(H_{1}) = μ(H_{2}) hence μ(H_{i}) = μ(H_{j}) for all i, j = 1, …, r = dim 𝔞 which is impossible since μ ∈ 𝔞^{∗}−{0}.

In what follows we set dim 𝔞 = 1, choose a unit vector H ∈ 𝔞, and assume that all eigenvalues of ad_{H}|_{𝔫} are positive. It follows from the curvature formulas that

R_{Z}H = − | [H, Z] |^{2} H for any Z ∈ 𝔷 . (4)

In fact, if X ∈ 𝔫,

⟨R_{Z}H, X⟩ = − ⟨∇_{[H,Z]}Z, X⟩ = 0 and ⟨R_{Z}H, H⟩ = − |[H, Z]|^{2} .

Moreover, it is immediate from Proposition 1.1 that

R_{X}H = −λ^{2}H for X ∈ 𝔳_{λ} and R_{Z}H = −μ^{2}H if Z ∈ 𝔷_{μ} .

The next proposition shows that under the assumption that the Osserman-p condition is satisfied on an Iwasawa type algebra of rank 1, the center of 𝔫 is contained in an eigenspace of adH.

Proposition 2.3. If 𝔰 is a solvable Lie algebra of Iwasawa type and dim 𝔞 = 1 satisfying the Oss-p condition, p ≥ 1, then ad_{H}|_{𝔷} = λ Id. Furthermore, if 𝔫 is not abelian, let {λ_{1} > ⋯ > λ_{m}} denote the eigenvalues of ad_{H}|_{𝔳}, then

(i) λ > λ_{i} for all i = 1, …, m, in particular 𝔫_{λ} = 𝔷. Moreover, λ − λ_{1} is also an eigenvalue of ad_{H}|_{𝔳}, and λ_{1} ≥ \frac{1}{2}λ.

(ii) [𝔳_{λ_{1}}, 𝔳] ⊆ 𝔷.

Proof. Assume there exist Z_{1} and Z_{2} unit vectors in 𝔷_{μ} and 𝔷_{ν}, respectively, with μ ≠ ν eigenvalues of ad_{H}|_{𝔷}. Let

E_{r,s} = {rZ_{1} + sZ_{2}, X_{2}, …, X_{p}}, r^{2} + s^{2} = 1

be an orthonormal set of p vectors in 𝔰, obtained by completing with X_{i} ∈ 𝔫, eigenvectors of ad_{H}|_{𝔫}, with eigenvalues γ_{i}. Then the associated operator j_{E_{r,s}} is

j_{E_{r,s}} = R_{rZ_{1}+sZ_{2}} + ∑_{i=2}^{p} R_{X_{i}} .

From ∑_{i=2}^{p} R_{X_{i}}H = − (∑_{i=2}^{p} γ_{i}^{2}) H and R_{rZ_{1}+sZ_{2}}(H) = −|[H, rZ_{1} + sZ_{2}]|^{2}H, (see (4)), it follows that H is an eigenvector of j_{E_{r,s}} with associated eigenvalue

⟨j_{E_{r,s}}H, H⟩ = − | [H, rZ_{1} + sZ_{2}] |^{2} − ∑_{i=2}^{p} γ_{i}^{2} .

Since [H, rZ_{1} + sZ_{2}] = rμZ_{1} + sνZ_{2} , arguing as in the previous proposition, one obtains r^{2}μ^{2} + s^{2}ν^{2} is a constant (independent of (r, s)). Thus μ = ν, since all eigenvalues of ad_{H}|_{𝔫} are positive. We will denote by λ the eigenvalue of ad_{H}|_{𝔷}.

The proof of (i) and (ii) is the same as that of Proposition 1.2 in [D-D]. All is needed is ad_{H}|_{𝔷} = λ Id.

We summarize in the next proposition certain restrictions imposed by the Oss-p condition to the algebraic structure of the Lie algebra 𝔫, involving the transformations {j_{Z} : Z ∈ 𝔷}. We observe first

Remark. If Z ∈ 𝔷_{μ}, then ker j_{Z} is invariant by ad_{H}, H ∈ 𝔞.

The above assertion holds without restriction on dim 𝔞, since j_{Z}X = 0 implies

⟨j_{Z}([H, X]), Y ⟩ = ⟨[[H, X], Y], Z⟩ = −⟨[[X, Y], H], Z⟩ − ⟨[[Y, H], X], Z⟩
= ⟨[X, Y], [H, Z]⟩ + ⟨j_{Z}([H, Y]), X⟩ = ⟨j_{Z}X, μY − [H, Y]⟩ = 0 .

Proposition 2.4. If 𝔰 is a solvable Lie algebra of Iwasawa type with dim 𝔞 = 1 and 2-step nilpotent n, satisfying the Oss-p condition, p ≥ 1 then

(i) If Z ≠ 0, the linear map j_{Z} : 𝔳 → 𝔳 is an isomorphism. Hence, the linear map 𝔷 → 𝔳, Z → j_{Z}X, is one to one and ad_{X}|_{𝔳} : 𝔳 → 𝔷 is onto if X ∈ 𝔳, X ≠ 0.

(ii) If λ_{i} is an eigenvalue then λ − λ_{i} is also an eigenvalue. In particular λ − λ_{1} is the minimum eigenvalue of ad_{H}|_{𝔳}.

(iii) If X ∈ 𝔳_{λ_{i}} then j_{𝔷}X = {j_{Z}X : Z ∈ 𝔷} is the orthogonal complement of kerad_{X}|_{𝔳} in 𝔳. Consequently, 𝔳 = kerad_{X}|_{𝔳} ⊕ j_{𝔷}X and ad_{H}|_{j_{𝔷}X} = (λ − λ_{i}) Id.

Proof. Assume that for some Z in 𝔷 and X ∈ 𝔳, unit vectors, j_{Z}X = 0. By the remark above we may assume X ∈ 𝔳_{λ_{i}}. Computing R_{rZ+sX} with r^{2} + s^{2} = 1 we have,

R_{rZ+sX}(H) = r^{2}R_{Z}H + s^{2}R_{X}H + rs (R(H, Z)X + R(H, X)Z)
= r^{2}R_{Z}H + s^{2}R_{X}H

since R(H, Z)X = \frac{1}{2}λj_{Z}X = 0 and R(H, X)Z = \frac{1}{2}λ_{i}j_{Z}X = 0. Hence

R_{rZ+sX}(H) = − (r^{2}λ^{2} + s^{2}λ_{i}^{2}) H .

Choosing E_{r,s} = {H, rZ+sX, X_{2}, …, X_{p−1}} the p-space completed with orthonormal eigenvectors X_{i} (i = 2, …, p) of ad_{H}|_{𝔫}, we have that j_{E_{r,s}} = −ad_{H}^{2}+R_{rZ+sX}+ ∑_{i=2}^{p−1} R_{X_{i}} has H as common eigenvector, with associated eigenvalue

⟨j_{E_{r,s}}H, H⟩ = − (r^{2}λ^{2} + s^{2}λ_{i}^{2}) + ⟨∑_{i=2}^{p−1} R_{X_{i}}H, H⟩ .

Since the last term above is independent of (r, s) and we are assuming that the Oss-p condition holds, r^{2}λ^{2} + s^{2}λ_{i}^{2} is constant; this is impossible since λ_{i} < λ. Therefore j_{Z} is an isomorphism. It is straightforward now to complete the proof of (i), (ii) follows from Lemma 2.1 together with (i) above and finally (iii) follows as in Proposition 1.3 of [D-D].

3. Reduction of the eigenvalues when 𝔫 is 2-step nilpotent

We assume in this section 𝔫 is 2-step nilpotent, non abelian and obtain strong restrictions on the number of eigenvalues.

Proposition 3.1. Let V ∈ 𝔳_{λ_{1}}, | V |= 1. If 𝔰 is a solvable Lie algebra of Iwasawa type with non abelian 2-step nilpotent 𝔫 and satisfying Oss-p then kerad_{V} |_{𝔳} ⊂ 𝔳_{λ_{1}}.

Proof. We may assume that a unit X ∈ 𝔳_{λ_{i}} satisfies [X, V] = 0 (ker ad_{V} |_{𝔳} is ad_{H}- invariant). We compute, for (r, s) in the circle r^{2} + s^{2} = 1,

R_{rV +sX}(H) = r^{2}R_{V} H + s^{2}R_{X}H = − (r^{2}λ_{1}^{2} + s^{2}λ_{i}^{2}) H ,

since [𝔳, 𝔳] ⊂ 𝔷 gives R(H, X)V = −∇_{[H,X]}V = 0 , R(H, V)X = −∇_{[H,V]}X = 0 . Hence, the p-space E_{r,s} = {H, rV + sX, X_{2}, …, X_{p−1}} of orthonormal vectors, where X_{i} are eigenvectors of ad_{H}|_{𝔫}, has associated operator

j_{E_{r,s}} = −ad_{H}^{2} + R_{rV +sX} + ∑_{i=2}^{p−1} R_{X_{i}}

with H a common eigenvector. Thus the continuous map (r, s) → ⟨j_{E_{r,s}}H, H⟩ is constant. This implies that

− (r^{2}λ_{1}^{2} + s^{2}λ_{i}^{2}) + ⟨∑_{i=2}^{p−1} R_{X_{i}}, H⟩

is constant as function of (r, s), or equivalently, r^{2}λ_{1}^{2} + s^{2}λ_{i}^{2} is constant on the circle r^{2} + s^{2} = 1. Therefore λ_{i} = λ_{1}, and the assertion follows.

The following theorem shows that if 𝔳 ≠ {0}, only one eigenvalue occurs in 𝔳.

Theorem 3.2. If 𝔰 is a solvable Lie algebra of Iwasawa type and dimension n with non abelian 2-step nilpotent 𝔫 and satisfying the Oss-p condition, 1 ≤ p < n, then the eigenvalues of ad_{H}|_{𝔫} are λ and \frac{1}{2}λ.

Proof. It follows from Propositions 2.4 and 3.1 that if V ∈ 𝔳_{λ_{1}}

𝔳 = 𝔳_{λ_{1}} ⊕ (𝔳_{λ−λ_{1}} ∩ ker ad_{V}) ⊕ j_{𝔷}V .

Then, if 𝔳_{λ−λ_{1}} ∩ kerad_{V} ≠ 0, we have that λ_{1} = \frac{1}{2}λ. If 𝔳_{λ−λ_{1}} ∩ kerad_{V} = 0, it follows that 𝔳_{λ−λ_{1}} = j_{𝔷}V and 𝔳 has the direct sum decomposition 𝔳 = 𝔳_{λ_{1}} ⊕ j_{𝔷}V, with 𝔳_{λ−λ_{1}} = j_{𝔷}V isomorphic to 𝔳_{λ_{1}} by (i) of Proposition 2.4. In particular dim 𝔳_{λ_{1}} = dim 𝔳_{λ−λ_{1}} = dim 𝔷 = k. We assume next the last case holds.

Using that the associated riemannian manifold is Einstein and Proposition 1.1, (vii), from Ric(H) = Ric(Z), for any unit Z ∈ 𝔷 we obtain

− (kλ^{2} + kλ_{1}^{2} + k(λ − λ_{1})^{2}) = −\frac{1}{4}tr j_{Z}^{2} − 2kλ^{2} ,

or equivalently,

−\frac{1}{4}tr j_{Z}^{2} = 2kλ_{1}(λ − λ_{1}) . (5)

Let X be a unit vector in 𝔳_{λ_{1}}. Setting Ric(H) = Ric(X) and using the expressions for the Ricci curvatures obtained in Proposition 1.1, (vii), it follows that

kλ^{2} + kλ_{1}^{2} + k(λ − λ_{1})^{2} = \frac{1}{2} ∑_{i=1}^{k} | j_{Z_{i}}X |^{2} +2kλ_{1}λ ,

obtaining

\frac{1}{4} ∑_{i=1}^{k} | j_{Z_{i}}X |^{2}= k(λ − λ_{1})^{2}, hence − \frac{1}{4} ∑_{i=1}^{k} j_{Z_{i}}^{2}|_{𝔳_{λ_{1}}} = k(λ − λ_{1})^{2}Id . (6)

Similarly, from Ric(H) = Ric(Y) for a unit Y ∈ 𝔳_{λ−λ_{1}},

kλ^{2} + kλ_{1}^{2} + k(λ − λ_{1})^{2} = \frac{1}{2} ∑_{i=1}^{k} | j_{Z_{i}}Y |^{2} +2kλ(λ − λ_{1}) ,

gives

\frac{1}{4} ∑_{i=1}^{k} | j_{Z_{i}}Y |^{2}= kλ_{1}^{2} , hence − \frac{1}{4} ∑_{i=1}^{k} j_{Z_{i}}^{2}|_{𝔳_{λ−λ_{1}}} = kλ_{1}^{2}Id . (7)

If \{Z_{1}, …, Z_{k}\} is an orthonormal basis of 𝔷, then using (5), tr(−\frac{1}{4} ∑_{i=1}^{k} j_{Z_{i}}^{2}) =

2k^{2}λ_{1}(λ − λ_{1}). On the other hand, from (6) and (7)

tr (−\frac{1}{4} ∑_{i=1}^{k} j_{Z_{i}}^{2}) = k^{2}(λ − λ_{1})^{2} + k^{2}λ_{1}^{2}

gives the equality 2λ_{1}(λ − λ_{1}) = (λ − λ_{1})^{2} + λ_{1}^{2} with solution λ_{1} = \frac{1}{2}λ, as asserted.

4. The Osserman-p condition and Damek Ricci spaces

We recall that S is called a Damek-Ricci space (see for instance [B-T-V] ) when the associated metric Lie algebra of Iwasawa type 𝔰 = 𝔫 ⊕ RH, 𝔫 = 𝔷 ⊕ 𝔳 satisfies the extra conditions, ad_{H} acts as Id on 𝔷 and \frac{1}{2} Id on 𝔳 and when 𝔷 ≠ \{0\}, the skew symmetric transformations J_{Z}, Z ≠ 0 satisfy J_{Z}^{2} = −|Z|^{2} Id. It is known that Damek-Ricci spaces are Einstein manifolds of non positive curvature and contain as a subclass that of symmetric spaces of rank one and non compact type.

Lemma 4.1. Let 𝔰 = 𝔫⊕RH be a solvable Lie algebra of Iwasawa type and 2-step nilpotent 𝔫, satisfying the Osseman-p condition. If dim 𝔷 ≥ p − 1, then 𝔫 = 𝔷 is abelian. In particular S has constant negative sectional curvature.

Proof. Let E = {H, Z_{1}, …, Z_{p−1}} be an orthonormal set of p-vectors with Z_{i} ∈ 𝔷. We consider the operator j_{E} = −ad_{H}^{2} + ∑_{1=1}^{p−1} R_{Z_{i}}. First of all we compute,

j_{E}H = ∑_{1=1}^{p−1} R_{Z_{i}}H = −(p − 1)λ^{2}H , j_{E}Z_{i} = −(p − 1)λ^{2}Z_{i}, (i = 1, …, p − 1) .

Hence j_{E}|_{𝔷+RH} = −(p − 1)λ^{2} Id. Note that since ad_{H}|_{𝔳} = \frac{1}{2}λ Id, it follows that R_{Z_{i}}|_{𝔳} = −\frac{1}{4}j_{Z_{i}}^{2} − \frac{1}{2}λ^{2} Id (see Proposition 1.1). Thus 𝔳 is j_{E}-invariant and consequently the remaining eigenvalues of j_{E} have to be attained in 𝔳.

Assume there exists X ∈ 𝔳, with X ≠ 0 and consider the orthonormal set of p vectors E^{∗} = {Z_{1}, …, Z_{p−1}, X}. It follows easily

j_{E^{∗}}H = (−(p − 1)λ^{2} − \frac{1}{4}λ^{2})H = −\frac{1}{4}(4p − 3)λ^{2}H ,

hence −\frac{1}{4}(4p − 3)λ^{2} (≠ −(p − 1)λ^{2}), is an eigenvalue of j_{E} that has to be attained in 𝔳. Finally, any unit eigenvector Y of j_{E}|_{𝔳} has associated eigenvalue

⟨j_{E}Y, Y ⟩ = ⟨−ad_{H}^{2}Y + ∑_{1=1}^{p−1} R_{Z_{i}}Y, Y ⟩ = −\frac{1}{4}λ^{2} + ∑_{1=1}^{p−1} (\frac{1}{4} | j_{Z_{i}}Y |^{2} −\frac{1}{2}λ^{2})

= −\frac{1}{4} (1 + 2(p − 1)) λ^{2} + \frac{1}{4} ∑_{1=1}^{p−1} | j_{Z_{i}}Y |^{2} .

In particular the equality −\frac{1}{4}(4p − 3)λ^{2} = ⟨j_{E}Y, Y ⟩ for some Y ∈ 𝔳 gives

\frac{1}{4}(4p − 3)λ^{2} = \frac{1}{4}(2p − 1)λ^{2} − \frac{1}{4} ∑_{1=1}^{p−1} | j_{Z_{i}}Y |^{2}

which can not occur. Hence 𝔳 = 0 and in this case 𝔫 is abelian. This implies that S has constant sectional curvature −λ^{2}.

Theorem 4.2. ^{1} If S is a solvable Lie group of Iwasawa type, dimension n and 2-step nilpotent [S, S] satisfying the Osserman-p condition for some 1 ≤ p < n, then S is (up to a re-scaling of the metric) either a Damek-Ricci space or a space of constant sectional curvature −1.

Proof. Let 𝔰 = 𝔫 ⊕ RH be the Lie algebra of S and let 𝔫 = 𝔷 ⊕ 𝔳. We may assume that 1 < p ≤ dim 𝔳. In fact, if p = 1, it follows from [D-D] that S is a symmetric space of non compact type and rank one, hence a Damek-Ricci space. If p > dim 𝔳 = n − dim 𝔷 − 1, or equivalently dim 𝔷 > (n − p) − 1, it follows from the previous lemma that S has constant sectional curvature since the conditions Oss-p and Oss-(n − p) are equivalent.

Next we show that j_{Z}^{2} = −λ^{2} | Z |^{2}Id, for all nonzero Z ∈ 𝔷.

Let Z_{1} be a unit vector in 𝔷 and let {Z_{1}, Z_{2}, …, Z_{k}} be an orthonormal basis of 𝔷. For an arbitrary and unit X_{1} ∈ 𝔳, we consider the p-space E_{r,s} = {rZ_{1} + sH, Z_{2}, …, Z_{k}, X_{1}, …, X_{p−k}} with r^{2} + s^{2} = 1 and orthonormal vectors X_{i} ∈ kerad_{X_{1}}, i = 1, …, p − k. The last choice is possible since dim ker ad_{X_{1}} = dim 𝔳 − k ≥ p − k. By using the computation made in Proposition 2.1 of [D-D] (last part of the proof) and the fact that ad_{H}|_{𝔳} = \frac{1}{2}λId, we have that

R_{rZ_{1}+sH}|_{𝔳} = r^{2}R_{Z_{1}} − s^{2} \frac{1}{4}λ^{2}Id .

1Recently the authors have improved this result showing that the only Damek-Ricci space satisfying the Ossermann-p condition, for some p > 1 is the real hyperbolic space.

Hence X_{1} is an eigenvector of

j_{E_{r,s}} = R_{rZ_{1}+sH} + ∑_{i=2}^{k} R_{Z_{i}} + ∑_{i=1}^{p−k} R_{X_{i}} for all (r, s)

since ∑_{i=1}^{k} R_{Z_{i}}|_{𝔳} = −\frac{1}{4}kλ^{2}Id, (which follows from −\frac{1}{4} ∑_{i=1}^{k} j_{Z_{i}}^{2}|_{𝔳} = kλ^{2} Id) and R_{X_{i}}|_{ker ad_{X_{1}}} = −\frac{1}{4}λ^{2}Id. Thus the continuous map given by (r, s) → ⟨j_{E_{r,s}}X_{1}, X_{1}⟩ must be constant as a function of (r, s). Using the expression

⟨j_{E_{r,s}}X_{1}, X_{1}⟩ = r^{2} (\frac{1}{4} | j_{Z_{1}}X_{1} |^{2} −\frac{1}{2}λ^{2})−s^{2} \frac{1}{4}λ^{2}+⟨(∑_{i=2}^{k} R_{Z_{i}} + ∑_{i=1}^{p−k} R_{X_{i}}) X_{1}, X_{1}⟩ ,

and taking into account that the last term above does not depend on (r, s), we obtain that

r^{2} (\frac{1}{4}|j_{Z_{1}}X_{1}|^{2} − \frac{1}{2}λ^{2}) − s^{2} \frac{1}{4}λ^{2}

is independent of (r, s) in the circle r^{2} + s^{2} = 1. Therefore, |j_{Z_{1}}X_{1}|^{2} = λ^{2}. Since Z_{1} and X_{1} are arbitrary and the eigenvalues of j_{Z}^{2} are of the form |j_{Z}X|^{2}, it then follows that j_{Z}^{2} = −λ^{2}|Z|^{2}Id for any non zero Z ∈ 𝔷. Since ad_{H} acts as λ Id on 𝔷 and \frac{1}{2}λ Id on 𝔳, re-scaling the metric we obtain that S is a Damek-Ricci space.

Corollary 4.3. If S is the associated simply connected Lie group to a solvable Lie algebra of Iwasawa type of negative curvature and 2-step nilpotent nilradical satisfying the Osserman-p condition, 1 < p < dim S, then S has constant negative sectional curvature.

Proof. Since a Damek-Ricci space of negative curvature is a symmetric space of non compact type and rank one (see [D]) the corollary follows immediately after one checks that the real hyperbolic space is the only one satisfying Oss-p for some 1 < p < dim S.

If dim 𝔷 = 1 and dim 𝔰 = n = 2r + 2 we may assume 1 < p ≤ r + 1. In this case 𝔰 is spanned by the orthonormal basis

{H, Z, X_{1}, …, X_{r}, j_{Z}X_{1}, …, j_{Z}X_{r}}

with Z ∈ 𝔷, X_{i} ∈ 𝔳. Choosing E = {H, X_{1}, …, X_{p−1}} and computing the eigenvalues of j_{E} one obtains −(p − 1)\frac{1}{4}λ^{2}, −(p + 3)\frac{1}{4}λ^{2} both with multiplicity p and −p\frac{1}{4}λ^{2} with multiplicity 2(r−p+1). Next we consider the p-dimensional subspaces E^{∗} = {H, X_{1}, …, X_{l}, j_{Z}X_{1}, …, j_{Z}X_{l}} if p is odd and E^{∗} = {X_{1}, …, X_{l}, j_{Z}X_{1}, …, j_{Z}X_{l}} if p is even. In both cases X_{1} is an eigenvector of j_{E^{∗}} with eigenvalue −(p + 2)\frac{1}{4}λ^{2} contradicting constancy of the eigenvalues.

If dim 𝔷 = 3 and dim 𝔰 = n = 4r + 4 we may assume 1 < p ≤ 2r + 2. We fix an orthonormal basis of 𝔰 as follows

{H, Z_{1}, Z_{2}, Z_{3}, X_{1}, j_{Z_{1}}X_{1}, j_{Z_{2}}X_{1}, j_{Z_{3}}X_{1}, …, X_{r}j_{Z_{1}}X_{r}, j_{Z_{2}}X_{r}, j_{Z_{3}}X_{r}} ,

with Z_{i} ∈ 𝔷, X_{j} ∈ 𝔳. Choosing E = {Z_{1}, Z_{2}, X_{1}, …, X_{s}, j_{Z_{1}}X_{1}, …, j_{Z_{1}}X_{s}} if p is even and E = {Z_{1}, X_{1}, …, X_{s}, j_{Z_{1}}X_{1}, …, j_{Z_{1}}X_{s}} if p is odd it is possible to find all eigenvalues of j_{E}. They are given by −(p + 6)\frac{1}{4}λ^{2}, −(p + 2)\frac{1}{4}λ^{2} and −p\frac{1}{4}λ^{2} with multiplicities 2, 2 + 4s, 2(r − s) respectively, in the case p = 2s + 2. When p = 2s + 1, the eigenvalues are, −(p + 3)\frac{1}{4}λ^{2}, −(p − 1)\frac{1}{4}λ^{2}, −(p + 2)\frac{1}{4}λ^{2} and −p\frac{1}{4}λ^{2} with multiplicities 3, 1, 4s, 4(r − s) respectively.

Changing the p-subspaces by E^{∗} = {Z_{1}, Z_{2}, Z_{3}, X_{2}, …, X_{l}, j_{Z_{1}}X_{1}, …, j_{Z_{1}}X_{l}} if p is even and E^{∗} = {Z_{1}, Z_{2}, Z_{3}, X_{4}, …, X_{l}, j_{Z_{1}}X_{1}, …, j_{Z_{1}}X_{l}} if p is odd, one shows that H is an eigenvector with corresponding eigenvalue −(p + 9)\frac{1}{4}λ^{2}, p even and −(p + 8)\frac{1}{4}λ^{2}, p odd.

The case dim 𝔷 = 7 follows immediately from Lemma 4.1 since being dim 𝔰 = 16 we may assume p < 8.

References

[B-T-V] Berndt, J., Tricerri F., Vanhecke L., Generalized Heisenberg Groups and Damek-Ricci

harmonic spaces, Lecture Notes in Mathematics 1598, Springer, 1995.

[D] Dotti, I., On the curvature of certain extension of H-type groups, Proc. of the A.M.S.

125 (1997), 573-578. [D-D] Dotti, I., Druetta. M. J., Negatively curved homogeneous Osserman spaces, to appear in

Differential Geometry and Applications.

[E] Eberlein, P., Geometry of 2-step nilpotent groups with a left invariant metric, Ann.

Scient. Ec. Norm. Sup. 4 serie t. 27 (1994), 611-660. [K-N] Kobayashi, S., Nomizu, K. II, Foundations of Differential Geometry. [S-V] Stanislov, G., Videv, V., Four dimensional pointwise Osserman manifolds, to appear in

Hamburger Abhandlungen.

FaMAF, Universidad Nacional de Córdoba, Ciudad Universitaria, 5000 Córdoba, ARGENTINA E-mail: idotti@mate.uncor.edu, druetta@mate.uncor.edu