A Hyperbolic Twistor Space

David E. Blair

Dedicated to Prof.Dr. Constantin UDRIŞTE on the occasion of his sixtieth birthday

Abstract

In this lecture we introduce a twistor space over a manifold with an almost quaternionic structure of the second kind whose fibres are hyperbolic planes with a metric of constant curvature −1. We discuss almost complex structures and holomorphic functions on such a twistor space.

Mathematics Subject Classification: 53C15 53C26 53C28 53C50 Key words: almost quaternionic structure of the second kind, holomorphic function, hyperbolic twistor space, paraquaternionic Kähler manifold, self-dual metric.

This lecture is a preliminary announcement of continuing work with J. Davidov, B. Foreman and O. Mushkarov.

We begin with the following simple observation. In [8] P. Libermann introduced the notion of an almost quaternionic structure of the second kind (presque quaternioniennes de deuxième espèce). This consists of an almost complex structure J_{1} and an almost product structure, J_{2} such that J_{1}J_{2} + J_{2}J_{1} = 0. Setting J_{3} = J_{1}J_{2} one has a second almost product structure which also anti-commutes with J_{1} and J_{2}. Now on a manifold M with such a structure, set

j = y_{1}J_{1} + y_{2}J_{2} + y_{3}J_{3}.

Then j is an almost complex structure on M if and only if

−y_{1}^{2} + y_{2}^{2} + y_{3}^{2} = −1.

This suggests a hyperbolic twistor space π : Z ⟶ M with fibre the hyperbolic plane in the fibre of the subbundle E of End(T M) spanned by {J_{1}, J_{2}, J_{3}}.

Recall that the classical twistor space over a quaternionic Kähler manifold of dimension ≥ 8 is a bundle over the manifold with the fibre being a sphere in the subbundle of the bundle of endomorphisms determined by the three underlying local almost complex structures defining the quaternionic Kähler structure.

There are a number of examples of almost quaternionic structures of the second kind including the paraquaternionic projective space as described by Blaz̆ić [1]. Under certain holonomy assumptions almost quaternionic structures of the second kind become paraquaternionic Kähler (see e.g. Garcia-Rio, Matsushita and Vazquez-Lorenzo [3]). Even more strongly one has neutral hyperkähler structures and Kamada [7] showed that every primary Kodaira surface admits a neutral hyperkähler structure. We remark that neutral hyperkähler surfaces are Ricci flat and self-dual (Kamada [7]).

Balkan Journal of Geometry and Its Applications, Vol.5, No.2, 2000, pp. 9-16 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

Also the tangent bundle of a differentiable manifold carries an almost quaternionic structure of the second kind as studied by S. Ianuş and C. Udrişte [4,5]; this includes examples where the dimension of the manifold carrying the structure is not necessarily 4n.

The most natural setting for this kind of structure is however on a manifold M of dimension 4n with a neutral metric g, i.e. a semi-Riemannian metric of signature (2n, 2n). One reason for this is that such a metric may be given with respect to which J_{1} acts as an isometry on tangent spaces and J_{2}, J_{3} act as anti-isometries; the effect of this is that we may define three fundamental 2-forms Ω_{a}, a = 1, 2, 3, by Ω_{a}(X, Y) = g(X, J_{a}Y). Riemannian metrics can be chosen such that g(J_{a}X, J_{a}Y) = g(X, Y), but then Ω_{2} and Ω_{3} are symmetric tensor fields instead of 2-forms.

The neutral metric g induces a metric on the fibres of E by \frac{1}{4n}trA^{t}B where A

and B are endomorphisms of T_{p}M and A^{t} is the adjoint of A with respect to g. This metric on the fibre is of signature (+ − −), the norm of J_{1} being +1 and the norms of J_{2} and J_{3} being −1.

Alternatively one may choose a Lorentz metric ⟨ , ⟩ directly on the fibres of E such that ⟨J_{1}, J_{1}⟩ = −1, ⟨J_{2}, J_{2}⟩ = +1, ⟨J_{3}, J_{3}⟩ = +1. This metric is of signature (− + +) and has the advantage of inducing immediately a Riemannian metric of constant curvature −1 on the hyperbolic planes defined by −y_{1}^{2} + y_{2}^{2} + y_{3}^{2} = −1, y_{1} > 0, in each fibre. We adopt this metric for its geometric attractiveness but keep its negative in mind.

We will also use the following notation. For the metric ⟨ , ⟩ on the fibres of E we set ϵ_{1} = −1 and ϵ_{2} = ϵ_{3} = +1. For the neutral metric g of on the base, we set ε_{i} = ±1 according to the signature (+ ⋯+− ⋯ −). Further, denoting also by π the projection of E onto M, if x_{i} are local coordinates on M, set q_{i} = x_{i} ∘ π. For a section of E we denote its vertical lift to E as a vector field by the superscript ^{v} and frequently

utilize the natural identifications of J_{a}^{v} with J_{a} itself and with \frac{∂}{∂y_{a}} in terms of the

fibre coordinates y_{1}, y_{2}, y_{3}.

We remark that for Z as a hypersurface in E with the (−++) metric, the position vector is a time-like normal and the corresponding Weingarten map acts as the identity on vertical vectors and annihilates horizontal vectors.

As with the theory of twistor spaces over quaternionic Kähler manifolds, the theory of hyperbolic twistor spaces over paraquaternionic Kähler manifolds of dimension ≥ 8 developes nicely by virtue of the fact that the covariant derivatives of sections of the subbundle of the endomorphism bundle are again sections of the subbundle. To give this development we first need the natural machinery of horizontal lifts.

Let D denote the Levi-Civita connection of the neutral metric on M. Then the horizontal lift X^{h} of a vector field X to the bundle E is is given by

X^{h} = ∑_{i} X^{i} \frac{∂}{∂q^{i}} − ∑_{a,b=1}^{3} ϵ_{b}y_{a}(⟨D_{X}J_{a}, J_{b}⟩ ∘ π)\frac{∂}{∂y_{b}} .

It is straightforward to obtain the following at a point σ ∈ Z ⊂ E

[X^{h}, Y^{h}]_{σ} = [X, Y]_{σ}^{h} − (R_{XY} σ)^{v}.

For a section s of E, [X^{h}, s^{v}] = (D_{X}s)^{v}. Define a metric on E by h = π^{∗}g + ⟨ , ⟩, and denote its Levi-Civita connection by ¯∇. Then

( ¯∇_{X^{h}}Y^{h})_{σ} = (D_{X}Y)_{σ}^{h} − \frac{1}{2}(R_{XY} σ)^{v}, ( ¯∇_{X^{h}}s^{v})_{σ} = \frac{1}{2}(R̂_{σs}X)^{h} + (D_{X}s)_{σ}^{v}

where g(R̂_{σs}X, Y) = h((R_{XY} σ)^{v}, s^{v}), ( ¯∇_{s^{v}}X^{h})_{σ} = \frac{1}{2}(R̂_{σs}X)^{h}, ¯∇_{s^{v}}t^{v} = 0.

We now define two almost complex structures J_{1} and J_{2} on the hyperbolic twistor space Z as follows. Acting on horizontal vectors these are the same and given by J_{1}X^{h} = J_{2}X^{h} = (jX)^{h} where as before j = ∑ y_{a}J_{a}. For a vertical vector V tangent to Z, i.e. ⟨x, V ⟩ = 0, let

J_{1}V = (y_{3}V^{2} − y_{2}V^{3})\frac{∂}{∂y_{1}} + (y_{3}V^{1} − y_{1}V^{3})\frac{∂}{∂y_{2}} + (y_{1}V^{2} − y_{2}V^{1})\frac{∂}{∂y_{3}}

and let J_{2}V be the negative of this expression. In particular J_{k}V = (−1)^{k−1}σ × V, k = 1, 2, σ ∈ Z where × is the vector product determined by the paraquaternionic algebra.

Define a semi-Riemannian metric on Z by h = π^{∗}g+⟨ , ⟩_{v}, ⟨ , ⟩_{v} being the restriction of ⟨ , ⟩ to the fibres (hyperbolic planes) of Z. It is easy to check that this metric is Hermitian with respect to both J_{1} and J_{2}.

To get an idea as to what might be happening we first study the paraquaternionic Kähler case. The development of theory of paraquaternionic Kähler structures was carried out by Garcia-Rio, Matsushita and Vazquez-Lorenzo [3] when the dimension of the base manifold is 4n ≥ 8. As with the theory of quaternionic Kähler manifolds, dimension 4 is special. At the beginning however we will retain the 4-dimensional case and point out where the differences occur. The parallel to the present development in the quaternionic Kähler case can be found in Ishihara [6].

An almost quaternionic manifold of the second kind M of dimension 4n and neutral metric g is said to be paraquaternionic Kähler if the bundle E is parallel with respect to the Levi-Civita connection of g. This is equivalent to the existence of local 1-forms α, β and γ such that

D_{X}J_{1} = −γ(X)J_{2} − β(X)J_{3},
D_{X}J_{2} = −γ(X)J_{1} − α(X)J_{3},
D_{X}J_{3} = −β(X)J_{1} + α(X)J_{2}.

From the group theoretic point of view this structure corresponds to the linear holonomy group being a subgroup of Sp(n, R)·Sp(1, R), just as a quaternionic Kähler structure corresponds to the linear holonomy group being a subgroup of Sp(n)·Sp(1). For n = 1 this is not a restriction.

Setting

A = 2(dα − β ∧ γ), B = 2(dβ − α ∧ γ), C = 2(dγ + α ∧ β)

one can easily obtain the following central relation of the action of the curvature tensor:

[R_{XY} , J_{1}] = −C(X, Y)J_{2} − B(X, Y)J_{3},
[R_{XY} , J_{2}] = −C(X, Y)J_{1} − A(X, Y)J_{3},
[R_{XY} , J_{3}] = −B(X, Y)J_{1} + A(X, Y)J_{2}.

The theory now develops as in the quaternionic Kähler case and we have the following result quite analogous to the classical twistor space theory. Let τ denote the scalar curvature of the base manifold. Theorem 1. On the hyperbolic twistor space of a paraquaternionic Kähler manifold of dimension ≥ 8 we have the following: The almost complex structure J_{1} is integrable and the almost Hermitian structure (J_{1}, h) is indefinite Kähler if and only if τ = −4n(n + 2). The almost complex structure J_{2} is never integrable but (J_{2}, h) is indefinite almost Kähler if and only if τ = 4n(n + 2) and indefinite nearly Kähler if and only if τ = −2n(n + 2).

These numbers are the negatives of what one would have in the usual twistor space over a quaternionic Kähler manifold of dimension ≥ 8. This sign change is due to our choice of metric on the fibres of E. If we take < , > as the (+ − −) metric we would have the other values, but the fibres of Z would then have a negative definite metric.

We now turn our attention to the 4-dimensional case. Let M^{4} be an oriented 4dimensional manifold with a semi-Riemannian metric g of signature (+ + −−) and e_{1}, … e_{4} a local orthonormal basis with e_{1} ∧ e_{2} ∧ e_{3} ∧ e_{4} giving the orientation. The metric g induces a metric on bundle of bivectors, ⋀^{2} T M^{4}, by

g(e_{i} ∧ e_{j}, e_{k} ∧ e_{l}) = \frac{1}{2} | \begin{matrix}ε_{i}δ_{ik} & ε_{i}δ_{il} \\ ε_{j}δ_{jk} & ε_{j}δ_{jl} \end{matrix} | , ε_{1} = ε_{2} = 1, ε_{3} = ε_{4} = −1.

The Hodge star operator of the neutral metric acting on ⋀^{2} T M^{4} is given by

∗(e_{1} ∧ e_{2}) = e_{3} ∧ e_{4}, ∗(e_{1} ∧ e_{3}) = e_{2} ∧ e_{4}, ∗(e_{1} ∧ e_{4}) = −e_{2} ∧ e_{3}.

Let ⋀^{−} and ⋀^{+} denote the subbundles of ⋀^{2} T M^{4} determined by the corresponding eigenvalues of the Hodge star operator. The metrics induced on ⋀^{−} and ⋀^{+} have signature (+ − −).

Setting

s_{1} = −e_{1} ∧ e_{2} + e_{3} ∧ e_{4},  s̄_{1} = e_{1} ∧ e_{2} + e_{3} ∧ e_{4},
s_{2} = −e_{1} ∧ e_{3} + e_{2} ∧ e_{4},  s̄_{2} = e_{1} ∧ e_{3} + e_{2} ∧ e_{4},
s_{3} = −e_{1} ∧ e_{4} − e_{2} ∧ e_{3},  s̄_{3} = e_{1} ∧ e_{4} − e_{2} ∧ e_{3},

{s_{1}, s_{2}, s_{3}} and {s̄_{1}, s̄_{2}, s̄_{3}} are local oriented orthonormal frames for ⋀^{−} and ⋀^{+} respectively.

Following the Riemannian construction of the twistor space of a 4-dimensional Riemannian manifold, define the hyperbolic twistor space π : Z ⟶ M^{4} to be the bundle with fibres defined by −y_{1}^{2} + y_{2}^{2} + y_{3}^{2} = −1, y_{1} > 0, relative to the basis {s_{1}, s_{2}, s_{3}} of ⋀^{−}.

On the other hand identifying {s_{1}, s_{2}, s_{3}} with the matrices

( \begin{matrix}0 & −1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & −1 & 0 \end{matrix} ) , ( \begin{matrix}0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \end{matrix} ) , ( \begin{matrix}0 & 0 & 0 & −1 \\ 0 & 0 & −1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{matrix} )

and noting that

g = ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix} ) ,

J_{i} = s_{i}g, i = 1, 2, 3 defines an almost quaternionic structure of the second kind. Specifically

J_{1} = ( \begin{matrix}0 & −1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & −1 \\ 0 & 0 & 1 & 0 \end{matrix} ) , J_{2} = ( \begin{matrix}0 & 0 & 1 & 0 \\ 0 & 0 & 0 & −1 \\ 1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \end{matrix} ) , J_{3} = ( \begin{matrix}0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix} )

and we can identify Z with the space of almost complex structures and define almost complex structures J_{1} and J_{2} on Z as in the higher dimensional case.

The curvature operator R : ⋀^{2} T M^{4} ⟶ ⋀^{2} T M^{4} admits an SO(2, 2)-irreducible decomposition

R = \frac{τ}{6} I + B + W^{+} + W^{−}

similar to that in the 4-dimensional Riemannian case. Here B is the traceless Ricci tensor and W = W^{+} + W^{−} is the decomposition of the Weyl conformal curvature tensor as a curvature operator on ⋀^{2} T M^{4} respecting ∗W = W∗. The metric g is said to be self-dual if W^{−} = 0.

Computing the Nijenhuis tensor of J_{1} and J_{2} we obtain the following. Theorem 2. On the hyperbolic twistor space of a 4-dimensional manifold with a neutral metric g we have the following: The almost complex structure J_{1} on the hyperbolic twistor space of M^{4} in integrable if and only if the metric g is self-dual.The almost complex structure J_{2} is never integrable.

When one has a development of a theory analogous to an existing one, I generally believe that the differences are more interesting than the similarities. Let us then turn to a difference between the hyperbolic twistor space and the classical one.

On the classical twistor space over a Riemannian 4-manifold with either almost complex structure, there are no non-constant holomorphic functions, even when the base manifold is non-compact [2]. Review of the proof. In the classical case, since the fibre is compact and its tangent spaces are J_{k}-invariant, the restriction of a holomorphic function f on the twistor space to the fibre is a constant. Moreover X^{h}f must also be constant on fibres.

Now given a local oriented orthonormal basis e_{1}, … e_{4} about a point p ∈ M and consider local sections of the twistor space {s_{1}, s_{2}, s_{3}} defined by

s_{1} = e_{1} ∧ e_{2} − e_{3} ∧ e_{4}, s_{2} = e_{1} ∧ e_{3} + e_{2} ∧ e_{4}, s_{3} = e_{1} ∧ e_{4} − e_{2} ∧ e_{3}.

Since f is holomorphic and e_{k}^{h}f is constant on fibres, one can compare (J_{k}e_{k}^{h})f = ie_{k}^{h}f at s_{1}(p) and s_{2}(p) and see that df restricted to the horizontal subbundle vanishes. Therefore df vanishes on both vertical and horizontal vectors and hence is constant on Z.

In contrast we give an example of a hyperbolic twistor space with non-trivial holomorphic functions with respect to the (almost) complex structure J_{1}. Let M be simply R^{4} ≅ C^{2} with standard almost quaternionic structure of the second kind given by the matrices

J_{1} = ( \begin{matrix}0 & −1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & −1 \\ 0 & 0 & 1 & 0 \end{matrix} ) , J_{2} = ( \begin{matrix}0 & 0 & 1 & 0 \\ 0 & 0 & 0 & −1 \\ 1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \end{matrix} ) , J_{3} = ( \begin{matrix}0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix} )

and let g be the neutral metric given by dx_{1}^{2} +dx_{2}^{2} −dx_{3}^{2} −dx_{4}^{2}. Then the twistor space Z = R^{4} × H^{2} where H^{2} is the hyperboloid model of the hyperbolic plane defined by −y_{1}^{2} + y_{2}^{2} + y_{3}^{2} = −1, y_{1} > 0. J_{1} is integrable and its action is as follows. Since the

horizontal lift of \frac{∂}{∂x_{i}} is just \frac{∂}{∂q_{i}}, we identify x_{i} and q_{i}.

J_{1}(\frac{∂}{∂x_{1}} ) = y_{1}(\frac{∂}{∂x_{2}} ) + y_{2}(\frac{∂}{∂x_{3}} ) + y_{3}(\frac{∂}{∂x_{4}} ),

J_{1}(\frac{∂}{∂x_{2}} ) = −y_{1}(\frac{∂}{∂x_{1}} ) − y_{2}(\frac{∂}{∂x_{4}} ) + y_{3}(\frac{∂}{∂x_{3}} ),

J_{1}(\frac{∂}{∂x_{3}} ) = y_{1}(\frac{∂}{∂x_{4}} ) + y_{2}(\frac{∂}{∂x_{1}} ) + y_{3}(\frac{∂}{∂x_{2}} ),

J_{1}(\frac{∂}{∂x_{4}} ) = −y_{1}(\frac{∂}{∂x_{3}} ) − y_{2}(\frac{∂}{∂x_{2}} ) + y_{3}(\frac{∂}{∂x_{1}} ).

The action of J_{1} on vertical vectors can be described by setting U = y_{2} \frac{∂}{∂y_{1}} + y_{1} \frac{∂}{∂y_{2}} ;

then

J_{1}U = −y_{1}y_{3} \frac{∂}{∂y_{1}} − y_{2}y_{3} \frac{∂}{∂y_{2}} + (1 + y_{3}^{2})\frac{∂}{∂y_{3}} .

The map φ of H^{2} onto the unit disk Δ in the y_{1} = 0 plane given by

φ(y_{1}, y_{2}, y_{3}) = \frac{y_{2} + iy_{3}}{1 + y_{1}}

maps the hyperboloid model H^{2} of the hyperbolic plane onto the Poincaré disk model Δ. It is easy to check that (U + iJ_{1}U)φ = 0 and hence that φ is a biholomorphism. Set

(1)  z = \frac{y_{2} + iy_{3}}{1 + y_{1}} .

One may readily check that

(2)  y_{1} = \frac{1 + |z|^{2}}{1 − |z|^{2}} , y_{2} = \frac{z + z̄}{1 − |z|^{2}} , y_{3} = \frac{z − z̄}{i(1 − |z|^{2})}.

Now the restriction of any J_{1}-holomorphic function to the fibre is a holomorphic function on H^{2}. By the above identification of H^{2} and Δ, we may identify Z with C^{2} × Δ. We will use the notation x = (x_{1}, x_{2}, x_{3}, x_{4}) as well as u = x_{1} + ix_{2}, v = x_{3} + ix_{4}.

Suppose now that f(x, z) is a J_{1}-holomorphic function on Z, then we may write f as

f(x, z) = ∑_{k=0}^{∞} f_{k}(x)z^{k}.

Moreover f satisfies

(3)  (\frac{∂}{∂x_{i}} )f + i(J_{1} \frac{∂}{∂x_{i}} )f = 0.

Using (1) and (2), (3) yields the following conditions:

(4)  \frac{∂f_{0}}{∂x_{1}} + i\frac{∂f_{0}}{∂x_{2}} = 0, \frac{∂f_{0}}{∂x_{3}} + i\frac{∂f_{0}}{∂x_{4}} = 0

(5) \frac{∂f_{k+1}}{∂x_{1}} + i\frac{∂f_{k+1}}{∂x_{2}} = −i(\frac{∂f_{k}}{∂x_{3}} − i\frac{∂f_{k}}{∂x_{4}} ), \frac{∂f_{k+1}}{∂x_{3}} + i\frac{∂f_{k+1}}{∂x_{4}} = −i(\frac{∂f_{k}}{∂x_{1}} − i\frac{∂f_{k}}{∂x_{2}} ),

k = 0, 1, 2, . . ..

Equation (4) shows that f_{0} is a holomorphic function of u and v and (5) takes the form

(6)  \frac{∂f_{k+1}}{∂ū} = −i\frac{∂f_{k}}{∂v} , \frac{∂f_{k+1}}{∂v̄} = −i\frac{∂f_{k}}{∂u} , k = 0, 1, 2, …

Taking f_{0} = αu+βv for complex constants α, β, f_{1} = −i(βū+αv̄) and the remaining f_{k} equal to zero, the conditions (6) are satisfied and hence

(7)  f(x, z) = f_{0} + f_{1}z = αu + βv − iz(βū + αv̄)

is a J_{1}-holomorphic function. Also z itself is J_{1}-holomorphic, though not of the form (7). Theorem 3. (Z(R^{4}), J_{1}) is biholomorphic to C^{2} × Δ with the standard complex structure. Proof. Set

F_{1}(x, z) = u − izv̄, F_{2}(x, z) = v − izū, F_{3}(x, z) = z

and define F : R^{4} × Δ ⟶ C^{2} × Δ by

F (x, z) = (F_{1}(x, z), F_{2}(x, z), F_{3}(x, z)).

This mapping is J_{1}-holomorphic and is 1:1 as can be seen by noting that if (w_{1}, w_{2}, ζ) is in C^{2} × Δ, then

u − izv̄ = w_{1}, v − izū = w_{2}, z = ζ

has a unique solution (u, v, z). Indeed

ū + iz̄v = w̄_{1}, −izū + v = w_{2}

has a unique solution since | \begin{matrix}1 & iz̄ \\ −iz & 1 \end{matrix} | = 1 − |z|^{2} > 0.

As a consequence we have the following result. Theorem 4. Any J_{1}-holomorphic function on (Z(R^{4}), J_{1}) has the form

h(u − izv̄, v − izū, z)

where h(w_{1}, w_{2}, ζ) is a holomorphic function on C^{2} × Δ.

When the base manifold is compact, the story is not quite as good; we give the following Theorem. Theorem 5. Let T^{4} be the 4-torus with the flat metric of signature (2, 2). Then Z(T^{4}) ≅ T^{4} × Δ and any J_{1}-holomorphic function f(x, z) on Z(T^{4}) has the form f(x, z) = h(z) where h(z) is a holomorphic function on Δ. Aknowledgements. Lecture given at the Third Conference of the Balkan Society of Geometers, 31 July-3 August 2000, University POLITEHNICA of Bucharest, Romania.

Research supported in part by NSF grant INT-9903302.

References

[1] N. Blaz̆ić, Paraquaternionic projective space and pseudo-Riemannian geometry Pub. Inst. Math. 60 (1996), 101-107.

[2] J. Davidov, O. Mus̆karov and G. Grantcharov, Almost complex structures on twistor spaces, Almost Complex Structures, World Scientific, Singapore, 1994, 113-149.

[3] E. Garcia-Rio, Y. Matsushita and R. Vázquez-Lorenzo, Paraquaternionic Kähler manifolds, to appear.

[4] S. Ianuş, Sulle strutture canoniche dello spazio fibrato tangente di una varietà riemanniana, Rend. Mat. 6 (1973), 1-22.

[5] S. Ianuş and C. Udrişte, Asupra spatiului fibrat tangent al unei varietati diferentiabile, St. Cerc. Mat. 22 (1970), 599-611.

[6] S. Ishihara, Quaterion Kählerian manifolds, J. Differential Geometry 9 (1974), 483-500.

[7] H. Kamada, Neutral hyperkähler structures on primary Kodaira surfaces, Tsukuba J. Math. 23 (1999), 321-332.

[8] P. Libermann, Sur les structures presque quaternioniennes de deuxième espèce, C. R. Acad. Sci Paris 234 (1952), 1030-1032.

Department of Mathematics

Michigan State University

East Lansing, MI 48824