Quaternionic Structures in Mathematics and Physics SISSA - Trieste, 5-9 September 1994

WEYL STRUCTURES ON QUATERNIONIC MANIFOLDS

Liviu Ornea (*), Paolo Piccinni (**)

Abstract. We give a survey on the structure and the topology of a natural class of Einstein-Weyl manifolds, namely the quaternionic manifolds that admit a locally (and non globally) conformal quaternion Kähler metric.

1. Quaternion Hermitian-Weyl manifolds

A Weyl structure on a C^{∞} manifold M is a conformal class [g] of Riemannian metrics together with a torsionfree connection D preserving [g]. For any representative g of the conformal class [g] there is then an associated 1-form ω: Dg = ω ⊗ g, and the associated 1-forms of any two representatives of [g] differ by an exact form. If the symmetric part of the Ricci tensor of D is (pointwise) proportional to the metrics in [g], then the Weyl structure is said to be Einstein-Weyl.

We are here interested in the additional datum, on a 4n-dimensional M^{4n} (n ≥ 2), of a quaternion Hermitian structure (H, g). By H we denote the quaternionic structure of M, i.e. a 3-dimensional subbundle of End(T M), locally spanned by almost complex structures I_{1}, I_{2}, I_{3}, satisfying the quaternionic identities and related on the intersections of trivializing open sets by matrices of SO(3). We assume that the subbundle H is 1-integrable, i. e. parallel with respect to some torsionfree connection on M. The metric g is assumed to be Hermitian with respect to the local almost complex structures I_{α}, α = 1, 2, 3. If we endow (M^{4n}, [g]) with a structure of Weyl manifold, then it is natural to require the following compatibility condition:

Definition 1.1. (M^{4n}, H, g) is said to be quaternion-Hermitian-Weyl if a torsionfree connection D is given satisfying DH ⊂ H, Dg = ω ⊗ g.

It can be proved [Pe-Po-Sw] that (M^{4n}, H, g) is quaternion-Hermitian-Weyl if and only if g is locally conformal quaternion Kähler, i. e. g_{|_{U_{i}}} = e^{f_{i}}g′_{i}, where the g′_{i} are quaternion Kähler over open neighbourhoods {U_{i}} covering M. The connection D|_{U_{i}} is in fact the Levi Civita connection of the local quaternion Kähler metric g′_{i}, and it is related to the global Levi Civita connection ∇^{g} of g by the formula:

(*) Supported by C.N.R.

(**) Supported by M.U.R.S.T. and I.N.d.A.M.

(1.2)  D_{X}Y = ∇_{X}^{g}Y − \frac{1}{2}\{ω(X)Y + ω(Y)X − g(X, Y)ω^{♯}\}.

Moreover ω|_{U_{i}} = df_{i}. The Einstein property of quaternion Kähler metrics assures that quaternion Hermitian-Weyl manifolds are in particular Einstein-Weyl (we refer to [Fo], [Gau], [Hi], [Pe-Sw] for basic facts on Weyl and Einstein-Weyl manifolds).

The subclass of hyperhermitian-Weyl manifolds is then obtained when H is spanned by global D-parallel (and hence integrable) I_{α}, α = 1, 2, 3. Since the unique torsionfree compatible D is in this case the Obata connection, it follows that on a hypercomplex M^{4n} there is for each hyperhermitian metric g at most one compatible hyperhermitian-Weyl structure.

Examples of quaternion-Hermitian-Weyl manifolds may be obtained in the class of quaternionic Hopf manifolds M = (ℍ^{n} − {0})/G, where G is a discrete group acting without fixed points, preserving both the quaternionic structure and the globally conformal hyperkähler metric g = (∑_{α} h_{α}\overline{h}_{α})^{−1} ∑_{α} dh_{α} ⊗d\overline{h}_{α} of ℍ^{n} −\{0\}. The simplest such groups are the dicrete subgroups G ⊂ CO^{+}(4) of conformal transformations consisting of x̃^{i} = ρ ∑_{j} a_{j}^{i}x^{j} , a_{j}^{i} ∈ SO(4), ρ > 0 and not = 1 for all the elements in G, with the diagonal action, via the isomorphism CO^{+}(4) ≅ GL(1, ℍ)Ṡp(1), on the coordinates of ℍ^{n} .

The following result shows that the compact quaternion Hermitian-Weyl structures are in fact Ricci-flat-Weyl. (cf. [Gau], [Or-Pi 1]):

Theorem 1.3. Let (M, H, [g], D) be a compact quaternion Hermitian-Weyl manifold, and assume that no metric in [g] is quaternion Kähler. Then D is Ricci-flat.

It follows that on such M the local quaternion Kähler metrics, locally conformal related to the global metrics in the conformal class, are locally hyperkähler. This does not imply the existence of a global hypercomplex structure, but suggests the possibility of comparing compact quaternion Hermitian-Weyl manifolds with compact hyperhermitian-Weyl manifolds (cf. n. 3).

2. The fibrations

The fact that the associated 1-forms ω of any quaternion-Hermitian-Weyl manifold are closed allows to apply, for M compact, a result of [Gau] concerning compact Einstein-Weyl manifolds. Accordingly, it is possible to choose a representative g in the conformal class [g] so that the corresponding ω, if not exact, is ∇^{g}-parallel. The case ω exact corresponds of course to a connection D that is the Levi Civita connection of a quaternion Kähler metric in [g]. Hence, apart this case, on compact quaternion-Hermitian-Weyl manifolds there is a 1-dimensional foliation B generated by the vector field B = ω^{♯}. If we assume B to be a regular foliation, then the leaf space P = M/B inherits the structure defined below.

Definition 2.1. A Riemannian manifold (P, g) is said to be a locally 3-Sasakian manifold if a rank 3 vector subbundle K ⊂ T P is given, locally spanned by an orthonormal triple X_{1}, X_{2}, X_{3} of Killing vector fields satisfying:

(i) [X_{α}, X_{β}] = 2X_{γ} for (α, β, γ) = (1, 2, 3) and cyclic permutations.

(ii) Any two such triples X_{1}, X_{2}, X_{3} and X′_{1}, X′_{2}, X′_{3} are related on the intersections U ∩ U′ of their definition open sets by matrices of functions with values in SO(3).

(iii) If F_{α} = ∇^{g}X_{α} , η_{α} = X_{α}^{♯} (α = 1, 2, 3), then (∇_{Y}^{g} F_{α}) Z = η_{α}(Z)Y − g(Y, Z)X_{α}, for any local vector fields Y, Z.

Then, similarly to a result established in the complex case in [Va 1], the following fact can be proved:

Proposition 2.2. The class of compact quaternion Hermitian-Weyl manifolds M, that are not quaternion Kähler and with the foliation B regular, coincides with that of flat principal S^{1}-bundles over locally 3-Sasakian manifolds P = M/B.

The above correspondance between quaternion Hermitian-Weyl structures and locally 3-Sasakian structures is described through the projection π : M ⟶ P = M/B on the leaf space. The orthonormal triples of local Killing vector fields X_{1}, X_{2}, X_{3} defining the local 3-Sasakian structure of the leaf space P are obtained by projecting the local vector fields I_{1}B, I_{2}B, I_{3}B of M. Conversely if P is a locally 3-Sasakian manifold, a quaternion Hermitian-Weyl structure (H, [g], D) can be defined on any flat principal S^{1}-bundle π : M ⟶ P. In fact, if u is a closed 1-form on M giving a flat connection on such S^{1}-bundle, the metric g_{M} = π^{∗}g_{P} + u ⊗ u and the D defined by formula (1.2) with ω = u, define a Weyl structure on M. A compatible quaternionic structure is then given through the local almost complex structures I_{α}Y = −F_{α}Y − η_{α}(Y)B, I_{α}B = X_{α} (Y horizontal vector field and B = ω^{♯}).

The description (2.2) of compact regular quaternion Hermitian-Weyl manifolds supplements similar correspondances between complex generalized Hopf and Sasakian structures [Va 2], and between hyperhermitian-Weyl and (globally) 3-Sasakian structures [Pe-Po-Sw], [Bo-Ga-Ma].

Another foliation canonically defined on compact quaternion Hermitian-Weyl and non-quaternion Kähler manifolds is the 4-dimensional foliation D locally spanned by the vector fields B, I_{1}B, I_{2}B, I_{3}B. If all the leaves of D are compact, the leaf space is an orbifold. Then [Or-Pi 1]:

Theorem 2.3. Any compact quaternion Hermitian-Weyl manifold, that is not quaternion Kähler and such that all the leaves of D are compact, fibers over a compact quaternion Kähler orbifold of positive scalar curvature, and the fibers F are conformally flat 4-manifolds. Therefore F ≅ (ℍ − {0})/G, and G is a discrete subgroup of GL(1, ℍ) · Sp(1) inducing an integrable quaternionic structure.

Again the correspondance between the structures is given by projecting metric and quaternionic structure on the leaf space of D . The structure of the fibers is obtained from the vanishing of their Weyl curvature tensor and the expression S = g − ω ⊗ ω of their Ricci tensor, using a result by [La].

On the other hand also compact locally 3-Sasakian manifolds are endowed with a canonical foliation, namely the the 3-dimensional foliation F spanned by the local Killing vector fields X_{1}, X_{2}, X_{3}. The leaves of F, if compact, are spherical space forms. It follows:

Corollary 2.4. Let (M, H, [g], D) be a compact quaternion Hermitian-Weyl manifold that is not quaternion Kähler. Assume that B is a regular foliation and that all the leaves of the foliations D on M and F on M/B are compact. Then the fibration M ⟶ N = M/D of thm. 2.3 factorizes as:

M ⟶ P = M/B ⟶ N = P/F = M/D,

and the fibers of P → N are (generally inhomogeneous) 3-dimensional spherical space forms.

3. The hyperhermitian-Weyl covering

According to thm. 1.3, any compact M with a quaternion Hermitian-Weyl structure, and not quaternion Kähler, admits local compatible I_{1}, I_{2}, I_{3} satisfying DI_{1} = DI_{2} = DI_{3} = 0. The vector bundle H ⟶ M over such manifolds is therefore a flat bundle. This implies the flatness also of the vector bundle K ⟶ P of compatible Killing vector fields over the locally 3-Sasakian base P = M/B of M. On the other hand, any local property of (globally) 3-Sasakian manifolds can be trasfered to locally 3-Sasakian manifolds. Thus in particular, due to a result of T. Kashiwada completed in [Bo-Ga-Ma], locally 3-Sasakian manifolds are Einstein manifolds with positive scalar curvature. Myers’ theorem allows then to conclude that compact quaternion Hermitian-Weyl manifolds have finite fundamental group. Then, by looking at the universal covering of the locally 3-Sasakian base P of M, one gets [Or-Pi 2]:

Theorem 3.1. Let M be a compact quaternion Hermitian-Weyl manifold that is not quaternion Kähler, and assume that B is a regular foliation. Then M admits a finite covering \overline{M} that is hyperhermitian-Weyl and not hyperkähler.

\overline{M} can in fact be defined as the induced principal S^{1} bundle of M ⟶ P through the universal covering projection \overline{P} ⟶ P. The hyperhermitian-Weyl structure is constructed on \overline{M} similarly to how described after prop. 2.2.

Compact hyperhermitian-Weyl manifolds fiber in 1-dimensional complex tori over the twistor space Z of their quaternion Kähler base N = M/D. In this case, the fibers of M ⟶ N are complex Hopf surfaces equipped with an integrable hypercomplex structure, and the fibers of P ⟶ N are homogeneous 3-dimensional spherical space forms. By combining the Gysin sequences of the circle bundles M ⟶ P, P ⟶ Z with well known identities for Betti numbers of compact quaternion Kähler manifolds with positive scalar curvature [Be], [Le-Sa], the following restrictions are obtained:

Proposition 3.2. Let M^{4n} be a compact hyperhermitian-Weyl manifold with regular foliations B and D and let N = M/D be its quaternion Kähler base. Then the Betti numbers of M and N satisfy:

b_{2p}(M) = b_{2p+1}(M) = b_{2p}(N) − b_{2p−4}(N)
b_{2n}(M) = 0

( 0 ≤ 2p ≤ 2n − 2 and the Poincaré dual equalities for 2n + 2 ≤ 2p ≤ 4n).

Proposition 3.3. (Same hypotheses of prop. 3.2). In low dimensions:

(i) The Betti numbers of a M^{12} are: b_{0} = b_{1} = 1, b_{2} = b_{3} = b_{4} = b_{5} = b_{2}(N), b_{6} = 0 and Poincaré duals.

(ii) The Betti numbers of a M^{16} are: b_{0} = b_{1} = 1, b_{2} = b_{3} = b_{6} = b_{7} = b_{2}(N), b_{4} = b_{5} = b_{4}(N) − 1, b_{8} = 0 and Poincaré duals.

(iii) The Betti numbers of a M^{20} are: b_{0} = b_{1} = 1, b_{2} = b_{3} = b_{2}(N), b_{4} = b_{5} = b_{4}(N)−1, b_{6} = b_{7} = b_{6}(N)−b_{2}(N), 2b_{8} = 2b_{9} = −1+3b_{2}(N)+b_{4}(N)−b_{6}(N), b_{10} = 0 and Poincaré duals.

In the homogeneous case, all the above foliations are regular. By looking at the classification of the homogeneous 3-Sasakian manifolds [Bo-Ga-Ma] together with their H^{2} the following classification is obtained:

Proposition 3.4. The only compact homogeneous hyperhermitian-Weyl manifolds are the products (G/H) × S^{1}, where G/H is any of the globally 3-Sasakian homogeneous manifolds

S^{4n−1}, ℝP^{4n−1}, SU(m)/S(U(m − 2) × U(1)), m ≥ 3,
SO(k)/(SO(k − 4) × Sp(1)), k ≥ 7,
G_{2}/Sp(1), F_{4}/Sp(3), E_{6}/SU(6), E_{7}/Spin(12), E_{8}/E_{7},

and the unique non trivial principal S^{1}-bundle over ℝP^{4n−1} .

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Faculty of Mathematics Univ. of Bucharest 14, Academiei str 70109 Bucharest, Romania

Dipartimento di Matematica UniversitΦ ”La Sapienza” di Roma Piazzale A. Moro 2 I-00185 Roma, Italy