PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 94 (108) (2013), 91–98 DOI: 10.2298/PIM1308091G

4-DIMENSIONAL (PARA)-KÄHLER–WEYL STRUCTURES

Peter Gilkey and Stana Nikčević

Abstract. We give an elementary proof of the fact that any 4-dimensional para-Hermitian manifold admits a unique para-Kähler–Weyl structure. We then use analytic continuation to pass from the para-complex to the complex setting and thereby show that any 4-dimensional pseudo-Hermitian manifold also admits a unique Kähler–Weyl structure.

1. Introduction

1.1. Weyl manifolds. Let (M, g) be a pseudo-Riemannian manifold of dimension m. A triple (M, g, ∇) is said to be a Weyl manifold and ∇ is said to be a Weyl connection if ∇ is a torsion free connection with ∇g = −2ϕ ⊗ g for some smooth 1-form ϕ. This is a conformal theory; if g̃ = e^{2f} g is a conformally equivalent metric, then (M, g̃, ∇) is a Weyl manifold with associated 1-form ϕ̃ = ϕ − df. If ∇^{g} is the Levi-Civita connection, we may then express ∇ = ∇^{ϕ} in the form:

(1.1)  ∇_{x}^{ϕ}y = ∇_{x}^{g}y + ϕ(x)y + ϕ(y)x − g(x, y)ϕ^{#}

where ϕ^{#} is the dual vector field. Thus ϕ determines ∇. Conversely, if ϕ is given and if we use Equation (1.1) to define ∇, then ∇ is a Weyl connection with associated 1-form ϕ. We refer to [5] for further details concerning Weyl geometry.

1.2. Para-Hermitian manifolds. Let m = 2m̄. A triple (M, g, J_{+} ) is said to be an almost para-Hermitian manifold with an almost para-complex structure J_{+} if g is a pseudo-Riemannian metric on M of neutral signature (m̄, m̄) and if J_{+} is an endomorphism of the tangent bundle T M so that J_{+}^{2} = Id and so that J_{+}^{∗} g = −g; (M, g, J_{+} ) is said to be para-Hermitian with an integrable complex structure J_{+} if the para-Nijenhuis tensor

N_{J_{+}} (x, y) := [x, y] − J_{+} [J_{+} x, y] − J_{+} [x, J_{+} y] + [J_{+} x, J_{+} y]

2010 Mathematics Subject Classification: Primary 53B05; Secondary 15A72, 53A15, 53C07.

Key words and phrases: Weyl geometry, Hermitian manifold, para-Hermitian manifold, pseudo-Hermitian manifold, Kähler manifold, para-Kähler manifold, 4-dimensional geometry.

Research of the authors partially supported by project MTM2009-07756 (Spain), and by project 174012 (Serbia).

vanishes or, equivalently, if there are local coordinates (u^{1}, …, um̄, v^{1}, …, vm̄) centered on an arbitrary point of M so that J_{+} ∂_{u_{i}} = ∂_{v_{i}} and J_{+} ∂_{v_{i}} = ∂_{u_{i}} .

1.3. Pseudo-Hermitian manifolds. Let m = 2m̄. A triple (M, g, J_{−} ) is said to be an almost pseudo-Hermitian manifold with an almost complex structure J_{−} if (M, g) is a pseudo-Riemannian manifold, if J_{−} is an endomorphism of the tangent bundle so that J_{−}^{2} = − id and so that J_{−}^{∗} g = g; (M, g, J_{−} ) is said to be a pseudo-Hermitian manifold with an integrable complex structure J_{−} if the Nijenhuis tensor

N_{J_{−}} (x, y) := [x, y] + J_{−} [J_{−} x, y] + J_{−} [x, J_{−} y] − [J_{−} x, J_{−} y]

vanishes or, equivalently, if there are local coordinates (u^{1}, …, um̄, v^{1}, …, vm̄) centered on an arbitrary point of M so that J_{−} ∂_{u_{i}} = ∂_{v_{i}} and J_{−} ∂_{v_{i}} = −∂_{u_{i}} .

1.4. Para-Kähler and Kähler manifolds. One says that a Weyl connection ∇ on a para-Hermitian manifold (M, g, J_{+} ) is a para-Kähler–Weyl connection if ∇J_{+} = 0. Similarly, one says that a Weyl connection ∇ on a pseudo-Hermitian manifold (M, g, J_{−} ) is a Kähler–Weyl connection if ∇J_{−} = 0. Since ∇J_{±} = 0 implies J_{±} to be integrable, we assume this condition henceforth. If ∇ = ∇^{g} is the Levi-Civita connection, then (M, g, J_{±} ) is said to be (para)-Kähler.

Let ⋆ be the Hodge operator and let Ω_{±} (x, y) := g(x, J_{±} y) be the (para)-Kähler form. The co-derivative δΩ_{±} is given, see [2] for example, by δΩ_{±} = − ⋆ d ⋆ Ω_{±} .

The following is well known – see, for example, the discussion in [9] of the Riemannian setting (which uses results of [10, 11]) and the generalization given in [3] to the more general context.

Theorem 1.1. Let m ⩾ 6. If (M, g, J_{±} , ∇) is a (para)-Kähler–Weyl structure, then the associated Weyl structure is trivial, i.e., there is always locally a conformally equivalent metric g̃ = e^{2f} g so that (M, g̃, J_{±} ) is (para)-Kähler and so that ∇ = ∇g̃.

By Theorem 1.1, only the 4-dimensional setting is relevant. The following is the main result of this short note; it plays a central role in the discussion of [1].

Theorem 1.2. (1) If M = (M, g, J_{+} ) is a para-Hermitian manifold of signature (2, 2), then there is a unique para-Kähler–Weyl structure on M with ϕ = \frac{1}{2}J_{+} δΩ_{+} .

(2) If M = (M, g, J_{−} ) is a pseudo-Hermitian manifold of signature (2, 2), then there is a unique Kähler–Weyl structure on M with ϕ = −\frac{1}{2}J_{−} δΩ_{−} .

(3) If M = (M, g, J_{−} ) is a Hermitian manifold of signature (0, 4), then there is a unique Kähler–Weyl structure on M with ϕ = −\frac{1}{2}J_{−} δΩ_{−} .

Assertion (3) of Theorem 1.2, which deals with the Hermitian setting, is well known – see, for example, the discussion in [8]. Subsequently, Theorem 1.2 was established in full generality (see [3, 4]) by extending the Higa curvature decomposition [6, 7] from the real to the Kähler–Weyl and to the para-Kähler Weyl contexts.

Here is a brief outline to this paper. In Section 2, we show that if a (para)- Kähler–Weyl structure exists, then it is unique. In Section 3, we give a direct proof of Assertion (1) of Theorem 1.2 in the para-Hermitian setting. In Section 4, we use analytic continuation to derive Assertions (2) and (3), which deal with the complex setting, from Assertion (1). This reverses the usual procedure of viewing para-complex geometry setting as an adjunct to complex geometry and is a novel feature of this paper.

2. Uniqueness of the (para)-Kähler–Weyl structure

This section is devoted to the proof of the following uniqueness result.

Lemma 2.1. (1) If ∇^{ϕ_{1}} and ∇^{ϕ_{2}} are two para-Kähler–Weyl connections on a 4-dimensional para-Hermitian manifold (M, g, J_{+} ), then ϕ_{1} = ϕ_{2}.

(2) If ∇^{ϕ_{1}} and ∇^{ϕ_{2}} are two Kähler–Weyl connections on a 4-dimensional pseudo-Hermitian manifold (M, g, J_{−} ), then ϕ_{1} = ϕ_{2}.

Proof. Let ϕ = ϕ_{1} − ϕ_{2} and let Θ_{X} (Y) = ϕ(X)Y + ϕ(Y)X − g(X, Y)ϕ^{#} . By

Equation (1.1), ∇_{X}^{ϕ_{1}} − ∇_{X}^{ϕ_{2}} = Θ_{X} ∈ End(T M). Consequently, {∇^{ϕ_{1}} − ∇^{ϕ_{2}} }J_{±} = 0 implies [Θ_{X} , J_{±} ] = 0 for all X.

We first deal with the para-Hermitian case. This is a purely algebraic computation. Let {e_{1}, e_{2}, e_{3}, e_{4}} be a local frame for T M so that

(2.1)  \begin{matrix}J_{+} e_{1} = e_{1}, J_{+} e_{2} = e_{2}, J_{+} e_{3} = −e_{3}, J_{+} e_{4} = −e_{4}, \\ g(e_{1}, e_{3}) = g(e_{2}, e_{4}) = 1. \end{matrix}

We expand ϕ = a_{1}e^{1} + a_{2}e^{2} + a_{3}e^{3} + a_{4}e^{4} and compute:

Θ_{e_{1}} e_{4} = a_{1}e_{4} + a_{4}e_{1}, J_{+} Θ_{e_{1}} e_{4} = −a_{1}e_{4} + a_{4}e_{1}, Θ_{e_{1}} J_{+} e_{4} = −a_{1}e_{4} − a_{4}e_{1}, Θ_{e_{2}} e_{3} = a_{2}e_{3} + a_{3}e_{2}, J_{+} Θ_{e_{2}} e_{3} = −a_{2}e_{3} + a_{3}e_{2}, Θ_{e_{2}} J_{+} e_{3} = −a_{2}e_{3} − a_{3}e_{2}, Θ_{e_{4}} e_{1} = a_{4}e_{1} + a_{1}e_{4}, J_{+} Θ_{e_{4}} e_{1} = a_{4}e_{1} − a_{1}e_{4}, Θ_{e_{4}} J_{+} e_{1} = a_{4}e_{1} + a_{1}e_{4}, Θ_{e_{3}} e_{2} = a_{3}e_{2} + a_{2}e_{3}, J_{+} Θ_{e_{3}} e_{2} = a_{3}e_{2} − a_{2}e_{3}, Θ_{e_{3}} J_{+} e_{2} = a_{3}e_{2} + a_{2}e_{3}.

Equating Θ_{e_{i}} J_{+} e_{j} with J_{+} Θ_{e_{i}} e_{j} then implies a_{1} = a_{2} = a_{3} = a_{4} = 0 so ϕ = 0 and ϕ_{1} = ϕ_{2}. This establishes Assertion (1).

Next assume we are in the pseudo-Hermitian setting. Complexify and extend g to be complex bilinear. Choose a local frame {Z_{1}, Z_{2}, Z̄_{1}, Z̄_{2}} for T M ⊗_{ℝ} ℂ so

J_{−} Z_{1} = \sqrt{−1} Z_{1},  J_{−} Z_{2} = \sqrt{−1} Z_{2},
J_{−} Z̄_{1} = −\sqrt{−1} Z̄_{1}, J_{−} Z̄_{2} = −\sqrt{−1} Z̄_{2},
g(Z_{1}, Z̄_{1}) = 1,  g(Z_{2}, Z̄_{2}) = ε_{2}

where we take ε_{2} = +1 in signature (0, 4) and ε_{2} = −1 in signature (2, 2). We set J_{+} := −\sqrt{−1} J_{−} , e_{1} := Z_{1}, e_{2} := Z_{2}, e_{3} := Z̄_{1}, and e_{4} := ε_{2} Z̄_{2} and apply the argument given to prove Assertion (1) (where the coefficients a_{i} are now complex) to derive Assertion (2). □

3. Para-Hermitian geometry

3.1. The algebraic context. Let (V, ⟨·, ·⟩, J_{+} ) be a para-Hermitian vector space of dimension 4. Here ⟨·, ·⟩ is an inner product on V of signature (2, 2) and J_{+} is an endomorphism of V satisfying J_{+}^{2} = Id and J_{+}^{∗} ⟨·, ·⟩ = −⟨·, ·⟩. We may then choose a basis {e_{1}, e_{2}, e_{3}, e_{4}} for V = ℝ^{4} so that the relations of Equation (2.1) are satisfied. The Kähler form and orientation μ are then given by

Ω_{+} = −e^{1} ∧ e^{3} − e^{2} ∧ e^{4} and μ = \frac{1}{2}Ω_{+} ∧ Ω_{+} = e^{1} ∧ e^{3} ∧ e^{2} ∧ e^{4}.

Let ⋆ be the Hodge operator, characterized by

ω_{1} ∧ ⋆ω_{2} = ⟨ω_{1}, ω_{2}⟩e^{1} ∧ e^{3} ∧ e^{2} ∧ e^{4} for all ω_{i}.

Consequently:

(3.1)  \begin{matrix}⋆ e^{1} ∧ e^{3} = −e^{2} ∧ e^{4}, & ⋆ e^{2} ∧ e^{4} = −e^{1} ∧ e^{3}, \\ ⋆ e^{1} ∧ e^{2} ∧ e^{3} = −e^{2}, & ⋆ e^{1} ∧ e^{2} ∧ e^{4} = e^{1}, \\ ⋆ e^{1} ∧ e^{3} ∧ e^{4} = −e^{4}, & ⋆ e^{2} ∧ e^{3} ∧ e^{4} = e^{3}. \end{matrix}

3.2. Example. We begin the proof of Theorem 1.2 by considering a very specific example. Let (x^{1}, x^{2}, x^{3}, x^{4}} be the usual coordinates on ℝ^{4}, let ∂_{i} := ∂_{x_{i}} , and let J_{+} be the standard para-complex structure:

J_{+} ∂_{1} = ∂_{1}, J_{+} ∂_{2} = ∂_{2}, J_{+} ∂_{3} = −∂_{3}, J_{+} ∂_{4} = −∂_{4}.

Let f(0) = 0. We take the metric to have non-zero components determined by g(∂_{1}, ∂_{3}) = 1 and g(∂_{2}, ∂_{4}) = e^{2f} . Let f_{i} := {∂_{i}f}(0). The (possibly) non-zero Christoffel symbols of ∇^{g} at the origin are given by:

g(∇_{∂_{1}}^{g} ∂_{2}, ∂_{4}) = g(∇_{∂_{2}}^{g} ∂_{1}, ∂_{4}) = g(∇_{∂_{1}}^{g} ∂_{4}, ∂_{2}) = g(∇_{∂_{4}}^{g} ∂_{1}, ∂_{2}) = f_{1},
g(∇_{∂_{3}}^{g} ∂_{2}, ∂_{4}) = g(∇_{∂_{2}}^{g} ∂_{3}, ∂_{4}) = g(∇_{∂_{3}}^{g} ∂_{4}, ∂_{2}) = g(∇_{∂_{4}}^{g} ∂_{3}, ∂_{2}) = f_{3},
g(∇_{∂_{4}}^{g} ∂_{4}, ∂_{2}) = 2f_{4}, g(∇_{∂_{2}}^{g} ∂_{2}, ∂_{4}) = 2f_{2},

g(∇_{∂_{2}}^{g} ∂_{4}, ∂_{1}) = g(∇_{∂_{4}}^{g} ∂_{2}, ∂_{1}) = −f_{1}, g(∇_{∂_{2}}^{g} ∂_{4}, ∂_{3}) = g(∇_{∂_{4}}^{g} ∂_{2}, ∂_{3}) = −f_{3}.

Consequently the (possibly) non-zero covariant derivatives at the origin are:

∇_{∂_{1}}^{g} ∂_{2} = ∇_{∂_{2}}^{g} ∂_{1} = f_{1}∂_{2}, ∇_{∂_{1}}^{g} ∂_{4} = ∇_{∂_{4}}^{g} ∂_{1} = f_{1}∂_{4},
∇_{∂_{3}}^{g} ∂_{2} = ∇_{∂_{2}}^{g} ∂_{3} = f_{3}∂_{2}, ∇_{∂_{3}}^{g} ∂_{4} = ∇_{∂_{4}}^{g} ∂_{3} = f_{3}∂_{4},
∇_{∂_{4}}^{g} ∂_{4} = 2f_{4}∂_{4}, ∇_{∂_{2}}^{g} ∂_{2} = 2f_{2}∂_{2},
∇_{∂_{2}}^{g} ∂_{4} = ∇_{∂_{4}}^{g} ∂_{2} = −f_{1}∂_{3} − f_{3}∂_{1}.

Since ∇_{∂_{1}}^{g} and ∇_{∂_{3}}^{g} are diagonal, they commute with J_{+} so ∇_{∂_{1}}^{g} (J_{+} ) = ∇_{∂_{3}}^{g} (J_{+} ) = 0. We compute

(∇_{∂_{2}}^{g} J_{+} )∂_{1} = (1 − J_{+} )∇_{∂_{2}}^{g} ∂_{1} = (1 − J_{+} )f_{1}∂_{2} = 0,
(∇_{∂_{2}}^{g} J_{+} )∂_{2} = (1 − J_{+} )∇_{∂_{2}}^{g} ∂_{2} = (1 − J_{+} )2f_{2}∂_{2} = 0,
(∇_{∂_{2}}^{g} J_{+} )∂_{3} = (−1 − J_{+} )∇_{∂_{2}}^{g} ∂_{3} = (−1 − J_{+} )f_{3}∂_{2} = −2f_{3}∂_{2},
(∇_{∂_{2}}^{g} J_{+} )∂_{4} = (−1 − J_{+} )∇_{∂_{2}}^{g} ∂_{4} = (−1 − J_{+} )(−f_{1}∂_{3} − f_{3}∂_{1}) = 2f_{3}∂_{1},

(∇_{∂_{4}}^{g} J_{+} )∂_{1} = (1 − J_{+} )∇_{∂_{4}}^{g} ∂_{1} = (1 − J_{+} )f_{1}∂_{4} = 2f_{1}∂_{4},
(∇_{∂_{4}}^{g} J_{+} )∂_{2} = (1 − J_{+} )∇_{∂_{4}}^{g} ∂_{2} = (1 − J_{+} )(−f_{1}∂_{3} − f_{3}∂_{1}) = −2f_{1}∂_{3},
(∇_{∂_{4}}^{g} J_{+} )∂_{3} = (−1 − J_{+} )∇_{∂_{4}}^{g} ∂_{3} = (−1 − J_{+} )f_{3}∂_{4} = 0,
(∇_{∂_{4}}^{g} J_{+} )∂_{4} = (−1 − J_{+} )∇_{∂_{4}}^{g} ∂_{4} = (−1 − J_{+} )2f_{4}∂_{4} = 0.

We apply Equation (3.1). We have ⋆Ω_{+} = −Ω_{+} . Setting e^{1} = dx^{1}, e^{2} = e^{f} dx^{2}, e^{3} = dx^{3}, and e^{4} = e^{f} dx^{4} and recalling f(0) = 0 yields

⋆Ω_{+} = −Ω_{+} = dx^{1} ∧ dx^{3} + e^{2f} dx^{2} ∧ dx^{4},
d ⋆ Ω_{+} = 2f_{1}dx^{1} ∧ dx^{2} ∧ dx^{4} − 2f_{3}dx^{2} ∧ dx^{3} ∧ dx^{4},
δ_{g}Ω_{+} (0) = − ⋆ d ⋆ Ω_{+} (0) = −2f_{1}dx^{1} + 2f_{3}dx^{3},
ϕ(0) = \frac{1}{2}Jδ_{g}Ω_{+} = −f_{1}dx^{1} − f_{3}dx^{3}, and ϕ^{#} = −f_{1}∂_{3} − f_{3}∂_{1}.

Let Θ_{ij} := ϕ(∂_{i})∂_{j} + ϕ(∂_{j})∂_{i} − g(∂_{i}, ∂_{j})ϕ^{#} = (∇^{ϕ} − ∇^{g})_{∂_{i}} ∂_{j} at 0. Then:

Θ_{11} = −2f_{1}∂_{1}, Θ_{12} = −f_{1}∂_{2}, Θ_{13} = (−f_{1}∂_{3} − f_{3}∂_{1}) + (f_{1}∂_{3} + f_{3}∂_{1}) = 0,
Θ_{14} = −f_{1}∂_{4}, Θ_{22} = 0, Θ_{23} = −f_{3}∂_{2}, Θ_{24} = (f_{1}∂_{3} + f_{3}∂_{1}),
Θ_{33} = −2f_{3}∂_{3}, Θ_{34} = −f_{3}∂_{4}, Θ_{44} = 0.

Since Θ(∂_{1}) and Θ(∂_{3}) are diagonal, [Θ(∂_{1}), J_{+} ] = [Θ(∂_{3}), J_{+} ] = 0. We compute:

[Θ(∂_{2}), J_{+} ]∂_{1} = (1 − J_{+} )Θ_{12} = 0,
[Θ(∂_{2}), J_{+} ]∂_{2} = (1 − J_{+} )Θ_{22} = 0,
[Θ(∂_{2}), J_{+} ]∂_{3} = (−1 − J_{+} )Θ_{23} = 2f_{3}∂_{2},
[Θ(∂_{2}), J_{+} ]∂_{4} = (−1 − J_{+} )Θ_{24} = −2f_{3}∂_{1},
[Θ(∂_{4}), J_{+} ]∂_{1} = (1 − J_{+} )Θ_{14} = −2f_{1}∂_{4},
[Θ(∂_{4}), J_{+} ]∂_{2} = (1 − J_{+} )Θ_{24} = 2f_{1}∂_{3},
[Θ(∂_{4}), J_{+} ]∂_{3} = (−1 − J_{+} )Θ_{34} = 0,
[Θ(∂_{4}), J_{+} ]∂_{4} = (−1 − J)Θ_{44} = 0.

We now observe that [∇^{g}, J_{+} ] + [Θ, J_{+} ] = 0. Consequently ∇^{ϕ}J_{+} = 0 for this metric and Assertion (1) of Theorem 1.2 holds in this special case.

Proof of Theorem 1.2 (1). Let V = ℝ^{4}, let S_{−}^{2} be the vector space of symmetric 2-cotensors ω so that J_{+}^{∗} ω = −ω, and let ε ∈ C^{∞} (S^{2}) satisfy ε(0) = 0. We use ε to define a perturbation of the flat metric by setting:

g = dx^{1} ∘ dx^{3} + dx^{2} ∘ dx^{4} + ε.

This is non-degenerate near the origin. Since only the 1-jets of ε are relevant in examining ∇^{ϕ}(J_{+} )(0), this is a linear problem and we may take ε ∈ S_{−}^{2} ⊗ V^{⋆} so:

g = g_{0} + ∑_{i} x^{i}ε(e_{i}).

Then ε → (∇^{ϕ}J_{+} )(0) defines a linear map

E : S_{−} (V) ⊗ V^{∗} → End(V) ⊗ V^{∗} or equivalently
E : S_{−} (V) → Hom(V^{∗}, End(V) ⊗ V^{∗}).

The analysis of Section 3.2 shows that E(dx^{2} ∘ dx^{4}) = 0. Permuting the indices 1 ↔ 2 and 3 ↔ 4 then yields E(dx^{1} ∘ dx^{3}) = 0. The question is invariant under the action of the para-unitary group; we must preserve J_{+} and we must preserve the inner product at the origin. Define a unitary transformation T by setting:

T (e^{1}) = e^{1} + ae^{2}, T (e^{2}) = e^{2}, T (e^{3}) = e^{3}, T (e^{4}) = e^{4} − ae^{3}.

Then

T (e^{1} ∘ e^{3}) = e^{1} ∘ e^{3} + ae^{2} ∘ e^{3}.

Consequently, E(e^{2} ∘ e^{3}) = 0. Permuting the indices 1 ↔ 2 and 3 ↔ 4 then yields E(e^{1} ∘ e^{4}) = 0. Since S_{−} = Span{e^{1} ∘ e^{3}, e^{1} ∘ e^{4}, e^{2} ∘ e^{3}, e^{2} ∘ e^{4}}, we see that E = 0 in general; this completes the proof of Assertion (1) of Theorem 1.2. □

4. Hermitian and pseudo-Hermitian manifolds

In this section, we will use analytic continuation to derive Theorem 1.2 in the complex setting from Theorem 1.2 in the para-complex setting. Let V = ℝ^{4} with the usual basis {e_{1}, e_{2}, e_{3}, e_{4}} and coordinates {x^{1}, x^{2}, x^{3}, x^{4}}, where we expand v = x^{1}e_{1} + x^{2}e_{2} + x^{3}e_{3} + x^{4}e_{4}. Let S^{2} denote the space of symmetric 2-tensors. We complexify and consider

S := {S^{2} ⊗_{ℝ} ℂ} ⊕ {(V^{∗} ⊗_{ℝ} S^{2}) ⊗_{ℝ} ℂ} .

Let J_{+} ∈ M_{2}(ℂ) be a complex 2 × 2 matrix with J_{+}^{2} = Id and Tr(J_{+} ) = 0. Let

(4.1)  S(J_{+} ) := {(g_{0}, g_{1}) ∈ S : det(g_{0} − J_{+}^{∗} g_{0}) ≠ 0}.

For (g_{0}, g_{1}) ∈ S(J_{+} ), define:

g(x)(X, Y) := \frac{1}{2}\{g_{0}(X, Y) − g_{0}(J_{+} X, J_{+} Y)\}

+ ∑_{i=1}^{4} x^{i} · \frac{1}{2}\{g_{1}(e_{i}, X, Y) − g_{1}(e_{i}, J_{+} X, J_{+} Y)\}.

By Equation (4.1), this is non-degenerate at 0 and defines a complex metric on some neighborhood of 0 so J_{+}^{∗} g = −g. Let ∇^{g} be the complex Levi-Civita connection:

∇_{∂_{i}}^{g} ∂_{j} = \frac{1}{2}g^{kl}\{∂_{i}g_{jl} + ∂_{j} g_{il} − ∂_{x_{l}} g_{ij} \}∂_{k}.

Then ∇^{g} is a torsion free connection on T_{ℂ}M := T_{M} ⊗_{ℝ} ℂ. The para-Kähler form is defined by setting Ω_{+} (x, y) = g(x, J_{+} y) and we have

δΩ_{+} = ⋆dΩ_{+} and ϕ := \frac{1}{2}J_{+} δ_{g}Ω.

We then use ϕ to define a complex Weyl connection ∇^{ϕ} on T_{ℂ}M and define a holomorphic map from S(J_{+} ) to 𝔙 := V^{∗} ⊗ M_{4}(ℂ) by setting

E(g_{0}, g_{1}; J_{+} ) := ∇^{ϕ}(J_{+} )|_{x=0} .

Lemma 4.1. Let J_{+} ∈ M_{4}(ℂ) with J_{+}^{2} = id and Tr(J_{+} ) = 0. Suppose that (g_{0}, g_{1}) ∈ S(J_{+} ).

(1) If J_{+} is real and if (g_{0}, g_{1}) is real, then E(g_{0}, g_{1}; J_{+} ) = 0.

(2) If J_{+} is real and if (g_{0}, g_{1}) is complex, then E(g_{0}, g_{1}; J_{+} ) = 0.

(3) If J_{+} is complex and if (g_{0}, g_{1}) is complex, then E(g_{0}, g_{1}; J_{+} ) = 0.

Proof. Assertion (1) follows from Theorem 1.2 (1). We argue as follows to prove Assertion (2). S(J_{+} ) is an open dense subset of S and inherits a natural holomorphic structure thereby. Assume that J_{+} is real. The map E is a holomorphic map from S(J_{+} ) to 𝔙. By Assertion (1), E(g_{0}, g_{1}; J_{+} ) vanishes if (g_{0}, g_{1}) is real. Thus, by the identity theorem, E(g_{0}, g_{1}; J_{+} ) vanishes for all (g_{0}, g_{1}) ∈ S_{J_{+}} . This establishes Assertion (2) by removing the assumption that (g_{0}, g_{1}) is real.

We complete the proof by removing the assumption that J_{+} is real. The general linear group GL_{4}(ℂ) acts on the structures involved by change of basis (i.e., conjugation). Let (g_{0}, g_{1}) ∈ S(J_{+} ) where J_{+} is real and Tr(J_{+} ) = 0. We consider the real and complex orbits

O_{ℝ}(g_{0}, g_{1}; J_{+} ) := GL_{4}(ℝ) · (g_{0}, g_{1}; J_{+} ),
O_{ℂ}(g_{0}, g_{1}; J_{+} ) := GL_{4}(ℂ) · (g_{0}, g_{1}; J_{+} ).

Let F(A) := E(A · (g_{0}, g_{1}; J_{+} )) define a holomorphic map from GL_{4}(ℂ) to 𝔙. By Assertion (2), F vanishes on GL_{4}(ℝ). Thus by the identity theorem, F vanishes on GL_{4}(ℂ) or, equivalently, E vanishes on the orbit space O_{ℂ}(g_{0}, g_{1}; J_{+} ). Given any J_{+} ∈ M_{4}(ℂ) with J_{+}^{2} = Id and Tr(J_{+} ) = 0, we can choose A ∈ GL_{4}(ℂ) so that A · J_{+} is real. The general case now follows from Assertion (2). □

Proof of Theorem 1.2 (2,3). Let (M, g, J_{−} ) be a 4-dimensional pseudo-Her-mitian manifold of dimension 4. Fix a point P of M. Since J_{−} is integrable, we may choose local coordinates (x^{1}, x^{2}, x^{3}, x^{4}) so the matrix of J_{−} relative to the coordinate frame {∂_{i}} is constant. Define a Weyl connection with associated 1form given by ϕ = −\frac{1}{2}J_{−} δΩ_{−} . Only the 0 and the 1-jets of the metric play a role in the computation of (∇^{ϕ}J_{−} )(P). So we may assume g = g(g_{0}, g_{1}). We set J_{+} = \sqrt{−1} J_{−} . We have that

J_{+}^{2} = \sqrt{−1} J_{−} \sqrt{−1} J_{−} = −J_{−}^{2} = id, Tr(J_{+} ) = \sqrt{−1} Tr(J_{−} ) = 0,
J_{+}^{⋆} (g)(X, Y) = g(\sqrt{−1} J_{−} X, \sqrt{−1} J_{−} Y ) = −g(J_{−} X, J_{−} Y) = −g(X, Y)

so J_{+}^{⋆} (g) = −g and (g_{0}, g_{1}) ∈ S_{J_{+}} . Finally, since J_{−} = −\sqrt{−1} J_{+} , we have

Ω_{−} = −\sqrt{−1} Ω_{+} ,
ϕ_{J_{−}} = −\frac{1}{2}J_{−} δ_{g}Ω_{−} = −\frac{1}{2}( − \sqrt{−1} J_{+} )δ_{g}( − \sqrt{−1} Ω_{+} ) = \frac{1}{2}J_{+} δ_{g}Ω_{+} = ϕ_{J_{+}} .

We apply Lemma 4.1 to complete the proof. □

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Mathematics Department University of Oregon Eugene OR 97403 USA

 gilkey@uoregon.edu

Mathematical Institute SANU Beograd Serbia

 stanan@mi.sanu.ac.rs