A Class of Nonsymmetric Harmonic Riemannian

Certain solvable extensions of H-type groups provide noncompact counterexamples to a conjecture of Lichnerowicz, which asserted that “harmonic” Riemannian spaces must be rank 1 symmetric spaces. A Riemannian space M with Laplace-Beltrami operator ∆ is called harmonic if, given any function f(x) on M depending only on the distance d(x, x0) from a given point x0, then also ∆f(x) depends only on d(x, x0). Equivalently, M is harmonic if for every p ∈ M the density function ωx0(x) expressed in terms of the normal coordinates around the point x0 is a function of d(x, x0) (see [1, 11]). In 1944 Lichnerowicz [10] proved that in dimensions not greater than 4 the harmonic spaces coincide with the rank-one symmetric spaces. He also raised the question of determining whether the same is true in higher dimensions. Among the most recent progress made on the so-called Lichnerowicz conjecture, Szabó [11] proved it to hold true in arbitrary dimension for compact manifolds with finite fundamental group. In this announcement we present a counterexample that arises in the noncompact case. It proves the Lichnerowicz conjecture not to be true in general for infinitely many dimensions, the smallest of them being 7. This example is based on the notion of H-type group due to Kaplan [8], and on the geometric properties of their one-dimensional extensions S introduced by Damek [5] and studied also in [2, 3, 4, 6]. The rank-one symmetric spaces of the noncompact type different from the hyperbolic spaces are special examples of such groups S. Even though symmetry is a main geometric difference that distinguishes some “good” S from other “bad” S, it turns out that a large part of the harmonic analysis on these groups can be worked out regardless of this distinction. A detailed account of this is given in our forthcoming paper [7]. We thank J. Faraut and A. Korányi for calling our attention to the Lichnerowicz conjecture. 1991 Mathematics Subject Classification. Primary 53C25, 53C30, 43A85, 22E25, 22E30. Received by the editors July 11, 1991 c ©1992 American Mathematical Society 0273-0979/92 $1.00 + $.25 per page 1 2 EWA DAMEK AND FULVIO RICCI 1. The extension S of an H-type group An H-type (or Heisenberg-type) algebra is a two-step nilpotent Lie algebra n with an inner product 〈 , 〉 such that if z is the center of n and v = z, then the map JZ : v → v given by 〈JZX,Y 〉 = 〈[X,Y ].Z〉 for X,Y ∈ v and Z ∈ z, satisfies the identity J Z = −|Z| I for every Z ∈ z. An H-type group is a connected and simply connected Lie group N whose Lie algebra is H-type. Let s = v ⊕ z ⊕ RT be the extension of n obtained by adding the rule [T,X + Z] = 1 2 X + Z for X ∈ v and Z ∈ z. Let S = NA be the corresponding connected and simply connected group extension of N , where A = expS(RT ). We parametrize the elements of S in terms of triples (X,Z, a) ∈ v ⊕ z ⊕ R by setting (X,Z, a) = expS(X + Z) expS(log a T ) . The product on S is then (X,Z, a)(X , Z , a) = (X + aX , Z + aZ ′ + 1 2 a[X,X ], aa) . We introduce the inner product 〈X + Z + tT |X ′ + Z ′ + tT 〉 = 〈X |X 〉+ 〈Z|Z 〉+ tt on s and endow S with the induced left-invariant Riemannian metric. Proposition 1 [4, 5, 9]. Let M = G/K be a rank-one symmetric space of the noncompact type different from SOe(n, 1)/SO(n), and let G = NAK be the Iwasawa decomposition of G. Then N is an H-type group and the map s 7→ sK is an isometry from S = NA to G/K. From the classification of symmetric spaces one knows that if the H-type group N appears in the Iwasawa decomposition of a rank-one symmetric space, then the dimension of its center equals 1, 3, or 7. On the other hand, there exist H-type groups with centers of arbitrary dimensions [8]. Direct proofs can be given [4, 5] to show that the space S is symmetric if and only if N is an “Iwasawa group.” We can then conclude that there are infinitely many S that are not symmetric. 2. The volume element in radial normal coordinates Given unit elements X0 ∈ v and Z0 ∈ z, we denote by SX0,Z0 the 4-dimensional subgroup of S generated by X0, Z0, JZ0X0, and T . Clearly, SX0,Z0 is also the extension of an H-type group, precisely of the 3-dimensional Heisenberg group. Proposition 2 [3]. SX0,Z0 is a totally geodesic submanifold of S, isometric to the Hermitian symmetric space SU(2, 1)/S(U(2)× U(1)). This observation suggests two alternative realizations of S [4, 7]: (1) the “Siegel domain” model: D = {(X,Z, t) ∈ v ⊕ z ⊕ R : t > 1 4 |X |} A CLASS OF NONSYMMETRIC HARMONIC RIEMANNIAN SPACES 3 with the metric transported from S via the map h(X,Z, a) = (X,Z, a + 1 4 |X |); (2) the “ball” model: B = {(X,Z, t) ∈ v ⊕ z ⊕ R : |X | + |Z| + t < 1} with the metric transported from D via the inverse of the “Cayley transform” C : B → D given by C(X,Z, t) = 1 (1− t)2 + |Z|2 ( 2(1− t+ JZ)X , 2Z , 1− t 2 − |Z| )