Abstract
We study the Einstein-Dirac equation as well as the weak Killing equation on Riemannian spin manifolds with codimension one foliation. We prove that, for any manifold M n admitting real Killing spinors (resp. parallel spinors), there exist warped product metrics η on M n × R such that (M n × R, η) admit Einstein spinors (resp. weak Killing spinors). To prove the result we split the Einstein-Dirac equation into evolution equations and constraints, by means of Cartan's frame formalism, and apply the local preservation property of constraints.
Introduction
Let (P m , η) be an m-dimensional smooth oriented Riemannian spin manifold and denote by Ric and S the Ricci tensor and the scalar curvature, respectively. Let ( , ) = Re , be the real part of the standard Hermitian product , on the spinor bundle Σ(P ) over P m . Let D be the Dirac operator acting on sections ψ ∈ Γ(Σ(P )) of the spinor bundle Σ(P ). The Einstein-Dirac equation is a minimal coupling of the Dirac equation to the Einstein equation and defined by (see [11] )
Introduction
where λ ∈ R is some real number and the energy-momentum tensor T is given by
Introduction
A non-trivial spinor field ψ solving this Einstein-Dirac system is called an Einstein spinor to eigenvalue λ ∈ R. In case that the scalar curvature S does not vanish at any point, one defines the weak Killing equation by
Introduction
where λ ∈ R is some real number. A non-trivial solution ψ to the equation is called a weak Killing spinor to weak Killing number λ (shortly, WK-spinor to WK-number λ). Since rescaling the length of any WK-spinor provides an Einstein spinor, the WK-equation is stronger than the Einstein-Dirac equation (In dimension n = 3, the considered two equations are essentially equivalent). Moreover, the WK-equation reduces to the Killing equation [3, 8] ,
Introduction
if the metric η is Einstein.
Introduction
Till now, the known examples of the Einstein spinors on Riemannian manifolds are as follows:
Introduction
(i) Real Killing spinors [2, 3, 9, 15] .
Introduction
(ii) WK-spinors on quasi-Einstein Sasakian manifolds [11] .
Introduction
(iii) Einstein spinors on product manifolds M 6 × N r , where M 6 is a six-dimensional simply connected nearly Kähler manifold and N r is a manifold of general dimension r admitting Killing spinors [11] .
Introduction
(iv) WK-spinors on the three-dimensional sphere S 3 with non-standard merics [4, 10, 11] .
Introduction
(v) WK-spinors on the three-dimensional Euclidean space R 3 with non-constant scalar curvature [11, 13] .
Introduction
The object of this paper is to establish a special existence theorem for WK-as well as Einstein spinors. Namely, we prove the following theorem (see Theorem 5.1, Theorem 7.1 and Corollary 7.1). Interestingly, we find that the WK-spinors constructed on R 3 with non-constant scalar curvature (see [11] , p.171) are a special case of this theorem.
Introduction
that, in Riemannian signature the Einstein equations are generally of elliptic type, making the initial-value problem (the Cauchy problem) for general smooth data inappropriate. However, when the considered Riemannian manofolds admit a codimension one foliation, one can represent the Einstein equations to be of hyperbolic type, just as one does over Lorentzian manifolds, and can indeed formulate the initial-value problem in a natural way.
Introduction
So far, not much has been studied about the initial-value problem for the Einstein-Dirac equation. In Lorentzian signature, the spacelike initial-value problem for the EinsteinDirac system was considered by Bao/Isenberg/Yasskin [1] in terms of 3+1 Hamiltonian formalism, but no existence theorem was proved there. Recently, Friedrich/Rendall indicated [7] , in terms of Penrose's two-spinor formalism, that the Einstein-Dirac equation may be reduced to symmetric hyperbolic evolution equations, illustrating some questions arising in the reduction.
Introduction
In this paper we give an invariant description of the initial-value formulation for the Einstein-Dirac equation on Riemannian manifolds with codimension one foliation, in an explicit form and in complete generality. The splitting of the Einstein-Dirac equation into evolution equations and constraints will be achieved in terms of Cartan's frame formalism, and hence our formulation is valid on Riemannian manifolds M n × R of general dimension n + 1. The first three sections (Section 2,3,4) of the paper are devoted to establishing the basic framework, the hyperbolic representation of curvatures and the Dirac equation, on (possibly compact) manifolds with codimension one foliation, and the framework may be of independent interest for further study of the behaviour of spinor field equations under global change of metrics.
Representation of curvatures and the Dirac equation with respect to reference metric
Let P m be an m-dimensional simply-connected smooth oriented manifold allowing spin structure, and let η, η be two Riemannian metrics on P m . Henceforth we fix the notation η to denote a reference metric. Then there exists a unique (1, 1)-tensor field K on P m that is positive definite with respect to η and satisfies
Representation of curvatures and the Dirac equation with respect to reference metric
for all vector fields X, Y . Recall that the Levi-Civita connection ∇ η of (P m , η) is characterized by the Koszul formula
Representation of curvatures and the Dirac equation with respect to reference metric
Letting (E 1 , . . . , E m ) be a local η-orthonormal frame field on P m , for which
Representation of curvatures and the Dirac equation with respect to reference metric
is η-orthonormal, and inserting X = F i , Y = F j , Z = F k into the Koszul formula, we have
Representation of curvatures and the Dirac equation with respect to reference metric
Remark 2.1 (i) The exact difference between the Levi-Civita connections ∇ η , ∇ η is related to the tensor Λ η by
Representation of curvatures and the Dirac equation with respect to reference metric
is valid for all vector fields X, Y, Z.
Representation of curvatures and the Dirac equation with respect to reference metric
We will often use the shorthand notation Λ η = Λ, if there is no possibility of confusion. Proposition 2.1 enables us to describe the behaviour of curvatures under global change of metrics in a nice way : A direct computation gives
Representation of curvatures and the Dirac equation with respect to reference metric
where R η (resp. R η ) is the Riemann tensor of η (resp. η). Contracting both sides of the equation, we can now represent the Ricci curvature Ric η as well as the scalar curvature S η with respect to the reference metric η.
Proposition 2.2
Ric
Proposition 2.2
In particular,
Proposition 2.2
Next, we review briefly the behaviour of the Dirac operator under change of metrics. Let T (P ) be the tangent bundle of P m , and let Σ(P ) η (resp. Σ(P ) η ) be the spinor bundle of (P, η) (resp. (P, η)) equipped with the standard Hermitian product , η (resp. , η ). We know that there exists a natural isomorphism K : Σ(P ) η −→ Σ(P ) η with
Proposition 2.2
for all Z ∈ T (P ), ϕ, ψ ∈ Σ(P ) η , where the dots "·" in the latter relation indicate the Clifford multiplication with respect to η and η, respectively. In terms of local η-orthonormal frame field (F 1 , . . . , F m ), the spin derivative ∇ η ϕ is expressed as
Proposition 2.2
and the Dirac operator D η ϕ as
Proposition 2.2
Making use of the formula in Proposition 2.1, one finds now readily that the spinor derivatives ∇ η , ∇ η and the Dirac operators D η , D η are related as follows.
Proposition 2.2
Proposition 2.3 (see [5] ) For all ψ ∈ Γ(Σ(P ) η ),
Proposition 2.2
where
Proposition 2.2
3 Representation of curvatures on manifolds with codimension one foliation
Proposition 2.2
In this section we establish an intrinsic setting of the formulas that constitute the wellknown evolution system for the Einstein (vacuum) equation (see [6, 7] ). The evolution system consists of two differential equations, describing the evolution of metrics (see Corollary 3.1) and the evolution of symmetric (0,2)-tensor fields (see Proposition 3.2), respectively. The main aim of this section is to represent the Ricci tensor Ric η hyperbolically with respect to codimension one foliation. We use the terminology "hyperbolic representation" in the sense that such representation of differential operators, on manifolds with codimension one foliation, transforms field equations of elliptic type involving metrics to hyperbolic systems in PDE theory. Note in this view that the formulas in Proposition 2.2 may be thought of as the elliptic representation of curvatures. Let (Q n+1 , η) be an (n+1)-dimensional smooth oriented Riemannian spin manifold. We assume that there exists a codimension one foliation on (Q n+1 , η) defined by a unit vector field E n+1 with dE n+1 = 0, where E n+1 = η(E n+1 , ·) is the dual 1-form of E n+1 . Letting E ⊥ n+1 denote the η-orthogonal complement of E n+1 in the tangent bundle T (Q), we note that dE n+1 = 0 implies the following facts(e.g. see [14] ):
Proposition 2.2
(i) For all vector fields V, W belonging to
Proposition 2.2
is compact, then all the slices of the foliation are diffeomorphic. (iv) If Q n+1 is simply-connected, then E n+1 = ds for some real-valued function s : Q n+1 −→ R (Q n+1 must be noncompact) and the foliation is defined by the level hypersurfaces s = constant.
Proposition 2.2
Let (E 1 , . . . , E n , E n+1 ) be a local η-orthonormal frame field on Q n+1 , with E j ∈ E ⊥ n+1 , j = 1, · · · , n, and (E 1 , . . . , E n , E n+1 ) the dual frame field. Denote ⊗ r s (E ⊥ n+1 ) the space of all (r, s)-tensor fields B on Q n+1 such that
Proposition 2.2
whenever either i k = n + 1 for some i k or j l = n + 1 for some j l . Now, consider a positive definite (1,1)-tensor field K on (Q n+1 , η). Letting η be the metric induced by K via η(X, Y ) = η(K(X), K(Y )) and identifying η with K 2 , we can express η as
Proposition 2.2
where
Proposition 2.2
) and ρ : Q n+1 −→ R is a positive function. This may be thought of as an intrinsic (Riemannian) version of the well-known ADMrepresentation of metrics in general relativity. ζ agrees with the shift vector field and ρ with the lapse function. Note that the (1,1)-tensor K 2 is related to the (1,1)-tensor L 2 by
Proposition 2.2
for all vector fields V ∈ ⊗ 1 0 (E ⊥ n+1 ) and
Proposition 2.2
Furthermore, L 2 is positive definite on each slice of the foliation and, on the slices, coincides with the metrics induced by K 2 . Certainly,
Proposition 2.2
is a local η-orthonormal frame field on Q n+1 , its dual frame field being given by
Proposition 2.2
Let Z be a vector field on Q n+1 , and let V, W be vector fields in E ⊥ n+1 . In what follows we fix the notations V, W to mean vector fields in E ⊥ n+1 . Then, one verifies easily the following basic identities:
Proposition 2.2
The identity η(V, F n+1 ) = 0 in the last line implies that E ⊥ n+1 coincides with the η-orthogonal complement of F n+1 in T (Q).
Proposition 2.2
We let
Proposition 2.2
denote the second fundamental form, on each slice, defined by the unit vector field F n+1 and E n+1 , respectively. Let g (resp. g) denote the metric, on each slice, induced by η (resp. η) and ∇ g (resp. ∇ g ) its Levi-Civita connection. In the notations, the tensor L satisfies
Proposition 2.2
. In order to represent curvatures of η hyperbolically with respect to codimension one foliation (Proposition 3.2), we must explicitly know how the connection ∇ η is related to the connections ∇ η and ∇ g . This is done in the following proposition, which may be thought of as the hyperbolic version of Proposition 2.1.
Proposition 2.2
Proof. One computes directly, substituting the identities
Proposition 2.2
in the Koszul formula.
Proposition 2.2
We can equivalently rewrite the third equation (iii) in Proposition 3.1 as follows.
Proposition 2.2
To prove Proposition 3.2 below, we need to recall the Gauss-Codazzi equations that relates the curvatures of (Q n+1 , η) to the curvatures of the slices.
Proposition 2.2
Proof. Via a direct computation, we have
Proposition 2.2
Using the equation (iv) in Proposition 3.1, we compute
Proposition 2.2
On the other hand, from the Koszul formula for ∇ η , we know that
Proposition 2.2
Rewriting yields,
Proposition 2.2
With the help of the equation (ii) in Lemma 3.1 and the identities,
Proposition 2.2
we obtain the asserted formula of the proposition.
Proposition 2.2
Remark 3.1 Contracting the equation in Proposition 3.2 and applying (iii) in Lemma 3.1, we obtain the following formula for Ric η (F n+1 , F n+1 ):
Representation of the Dirac equation on manifolds with codimension one foliation
In this section we will represent the Dirac equation on (Q n+1 , η) hyperbolically with respect to codimension one foliation (see Proposition 4.1 and Corollary 4.1). Let us fix a slice (M n , g) of the foliated manifold (Q n+1 , η). We will identify Σ(Q) η with Σ(Q) η and ψ ∈ Γ(Σ(Q) η ) with its pullback K(ψ), via the natural isomorphism K : Σ(Q) η −→ Σ(Q) η , and write simply as Σ(Q) and ψ, respectively. Depending on the dimension n + 1 of the manifold Q n+1 , we will use two different Clifford multiplications in the subbundle Σ(M ) ⊂ Σ(Q). For the realization of the Clifford algebra over R, we refer to [11] . Let Cl(M ) (resp. Cl(Q)) denote the Clifford bundle over M n (resp. Q n+1 ).
Representation of the Dirac equation on manifolds with codimension one foliation
(i) In case of n = 2m, we use the Clifford multiplication
Representation of the Dirac equation on manifolds with codimension one foliation
where π * : Σ(Q) −→ Σ(M ) is the restriction map and µ g is the volume element of (M 2m , g). The second relation is an immediate consequence of the algebraic relation
Representation of the Dirac equation on manifolds with codimension one foliation
(ii) In the other case n = 2m − 1, we identify the spinor bundle Σ(M ) with the positive part Σ + (Q) of the bundle Σ(Q) restricted to M 2m−1 , and we use the Clifford multiplication
Representation of the Dirac equation on manifolds with codimension one foliation
where π + * : Σ + (Q) −→ Σ(M ) is the restriction map and Cl + (Q) is the positive part of Cl(Q).
Representation of the Dirac equation on manifolds with codimension one foliation
Recall that the spin derivatives ∇ η ψ, ∇ g ψ are related, on Q n+1 , by
Representation of the Dirac equation on manifolds with codimension one foliation
In view of the rule of Clifford multiplication described above, we find that, in case of n = 2m, the formula is projected to the slice (M 2m , g) as
Representation of the Dirac equation on manifolds with codimension one foliation
However, in the other case n = 2m − 1, the projection is only possible if ψ = ψ + belongs entirely to the positive part Σ + (Q) of Σ(Q), the projected formula being given by
Representation of the Dirac equation on manifolds with codimension one foliation
Nevertheless, we may regard not only
Representation of the Dirac equation on manifolds with codimension one foliation
, as well-defined spinor fields on Q 2m , not projected to the slice M 2m−1 . Therefore, the following formula makes sense.
Representation of the Dirac equation on manifolds with codimension one foliation
Proof. We compute
Representation of the Dirac equation on manifolds with codimension one foliation
On the other hand,
Representation of the Dirac equation on manifolds with codimension one foliation
Comparing the latter equation with the former, we complete the proof.
Representation of the Dirac equation on manifolds with codimension one foliation
In order to represent the Dirac equation
Representation of the Dirac equation on manifolds with codimension one foliation
with respect to reference metric, we need the following lemma that one verifies straightforwardly using Proposition 3.1. 
Representation of the Dirac equation on manifolds with codimension one foliation
Although the equation in Proposition 4.1 is valid in both cases, n = 2m and n = 2m − 1, it is also very useful, in the latter case n = 2m − 1, to consider the decomposition of spinor fields,
Representation of the Dirac equation on manifolds with codimension one foliation
and rewrite the representation in Proposition 4.1 equivalently as follows.
Representation of the Dirac equation on manifolds with codimension one foliation
We close this section with representing the energy-momentum tensor
Representation of the Dirac equation on manifolds with codimension one foliation
hyperbolically with respect to codimension one foliation. To this end, it is important to notice that, if ψ be a solution of the Dirac equation D η ψ = λ Q ψ, λ Q ∈ R, then the following equation is valid :
Proposition 4.2 For any solution ψ of the Dirac equation
we have
Proposition 4.2 For any solution ψ of the Dirac equation
In case of n = 2m − 1, we consider the decomposition ψ = ψ + + F 2m · ϕ + and can equivalently rewrite the formulas in Proposition 4.2 as follows.
Corollary 4.2 For any solution ψ of the Dirac equation
D η ψ = λ Q ψ on (Q 2m , η) , where ψ = ψ + + F 2m · ϕ + , we have Tr η (T η ) = ǫλ Q 2 (ψ + , ψ + ) + (ϕ + , ϕ + ) , T η (V, W ) = ǫ 4 V · ∇ g W ψ + + W · ∇ g V ψ + , F 2m · ϕ + + ǫ 4 V · ∇ g W ϕ + + W · ∇ g V ϕ + , F 2m · ψ + + ǫ 4 {V · II(W ) + W · II(V )} · ψ + , ϕ + + ǫ 2 II(V, W )(ψ + , ϕ + ), T η (V, F 2m ) = ǫ 4 ∇ g V ψ + − V · ( 2m−1 i=1 F i · ∇ g F i ψ + ), ϕ + − ǫ 4 ∇ g V ϕ + − V · ( 2m−1 i=1 F i · ∇ g F i ϕ + ), ψ + , T η (F 2m , F 2m ) = ǫ 2 F 2m · ( 2m−1 i=1 F i · ∇ g F i ψ + ), ϕ + + ǫ 2 F 2m · ( 2m−1 i=1 F i · ∇ g F i ϕ + ), ψ + + ǫλ Q 2 (ψ + , ψ + ) + (ϕ + , ϕ + ) .
A sufficient condition for the existence of solutions to the weak Killing equation
Let us suppose that (Q n+1 , η) satisfies
A sufficient condition for the existence of solutions to the weak Killing equation
for all V, W ∈ E ⊥ n+1 . Then the weak Killing equation becomes
A sufficient condition for the existence of solutions to the weak Killing equation
Thus we have proved the following proposition.
A sufficient condition for the existence of solutions to the weak Killing equation
Proposition 5.1 Let (Q n+1 , η) satisfy the following conditions:
Under this assumption the weak Killing equation is equivalent to the system of differential equations,
As an application of Proposition 5.1, we are going to prove below that every parallel spinor may evolve to a WK-spinor (Theorem 5.1). For this purpose, we first show that there indeed exist some special metrics satisfying the hypothesis of Proposition 5.1. Let Q n+1 = M n × R be a product manifold, and let the product metric η = g M × g R be the reference metric on Q n+1 , where g M indicates an arbitrary Riemannian metric on M n and g R the standard metric on the real line R. We write g R = dt ⊗ dt, using the standard coordinate t ∈ R. By (E 1 , . . . , E n ) we denote a local orthonormal frame on (M n , g M ) as well as its lift to (Q n+1 , η). Let E n+1 = d dt denote the unit vector field on (R, g R ) as well as the lift to (Q n+1 , η). Then it is clear that Θ(V ) = −∇ η V E n+1 = 0 for all vector fields V ∈ E ⊥ n+1 . For simplicity, we denote by WP(g M ; a) the following class of metrics (the warped products of g M and g R ) :
Under this assumption the weak Killing equation is equivalent to the system of differential equations,
where f : R −→ R is a real-valued function and a ∈ R is a real number.
Under this assumption the weak Killing equation is equivalent to the system of differential equations,
Lemma 5.1 For all η ∈ WP(g M ; a), we have:
Under this assumption the weak Killing equation is equivalent to the system of differential equations,
where we have used the shorthand notation f t := df (E n+1 ) and f tt :=
Under this assumption the weak Killing equation is equivalent to the system of differential equations,
Proof. Since L = e f 2 I (in the notations of Section 3), we have
Under this assumption the weak Killing equation is equivalent to the system of differential equations,
and hence Corollary 3.1 gives
Under this assumption the weak Killing equation is equivalent to the system of differential equations,
Using this, one checks all the equations of the lemma easily.
Under this assumption the weak Killing equation is equivalent to the system of differential equations,
Substituting Lemma 5.1 in Proposition 3.2 as well as in the Gauss-Codazzi equations (Lemma 3.1), we obtain the following lemma immediately.
Lemma 5.2
For all η ∈ WP(g M ; a), we have: Proof. Using Lemma 5.1 and 5.2, we compute
) satisfies the hypothesis of Proposition 5.1 if and only if either
and
) satisfies the hypothesis of Proposition 5.1 if and only if either
the hypothesis of Proposition 5.1 is satisfied if and only if 4f tt + (n − 2a)f t f t = 0, which can be solved completely as given in the lemma.
) satisfies the hypothesis of Proposition 5.1 if and only if either
Then, for any real number λ Q ∈ R, there exists a warped product metric
) satisfies the hypothesis of Proposition 5.1 if and only if either
where h + : R −→ C is a complex-valued function with h + (0) = 1. Let η ∈ WP(g M ; a) be a warped product metric given as in Lemma 5.3. Now recall that, in our realization of Clifford algebra, the volume form µ M of (M 2m , g M ) acts on Σ ± (M ) via
) satisfies the hypothesis of Proposition 5.1 if and only if either
By Proposition 5.1, ψ + is a WK-spinor to WK-number λ Q if and only if the function h + satisfies h
) satisfies the hypothesis of Proposition 5.1 if and only if either
which obviously allows a global solution.
) satisfies the hypothesis of Proposition 5.1 if and only if either
We now extend Proposition 5.2 so as to include the case n = 2m − 1. 
) satisfies the hypothesis of Proposition 5.1 if and only if either
Proof. Because of Proposition 5.2, it suffices to prove the theorem for the case n = 2m−1. Let ϕ + M ∈ Γ(Σ + (Q)) be a parallel spinor on M 2m−1 , and let ϕ = h + ϕ
) satisfies the hypothesis of Proposition 5.1 if and only if either
where h + , k + : R −→ C are complex-valued functions with h + (0) = k + (0) = 1. Let η ∈ WP(g M ; a) be a warped product metric given as in Lemma 5.3. By Proposition 5.1, ϕ is a WK-spinor to WK-number λ Q if and only if (h + , k + ) satisfies the system of differential equations,
) satisfies the hypothesis of Proposition 5.1 if and only if either
f k + and k
) satisfies the hypothesis of Proposition 5.1 if and only if either
This is a linear homogeneous system and hence allows a global solution.
Remark 5.1
The WK-spinors constructed on R 3 at the end of Section 8 in the paper [11] (see p. 171) are a special case of Theorem 5.1 (for the metric η ∈ WP(g M ; a) with f (t) = ct).
The initial-value formulation for the Einstein-Dirac equation
In this section we set up an invariant initial-value formulation for the Einstein-Dirac equation
The initial-value formulation for the Einstein-Dirac equation
where
The initial-value formulation for the Einstein-Dirac equation
The formulation will be applied in the next section to establish a local existence theorem for a specific class of initial data sets(see Theorem 7.1). Following the work [7] as a guideline, we can indeed express the evolution equations as well as the constraints in an invariant form. For simplicity, we write
The initial-value formulation for the Einstein-Dirac equation
The tensor field ∆ decomposes into three parts
The initial-value formulation for the Einstein-Dirac equation
Restricting the equations, ∆ M = 0 and ∆ H = 0, to a fixed slice, we obtain the momentum constraint
The initial-value formulation for the Einstein-Dirac equation
and the Hamiltonian constraint
The initial-value formulation for the Einstein-Dirac equation
where T g denotes the restriction of T η to the slices. The information on the evolution should be contained in
The initial-value formulation for the Einstein-Dirac equation
or any combination of it with the constraints. The evolution equations should be chosen in such a way that, under the evolution, local preservation of the constraints is guaranteed. We consider the evolution equations of the form,
The initial-value formulation for the Einstein-Dirac equation
Note that, under this evolution, the tensor ∆ becomes
The initial-value formulation for the Einstein-Dirac equation
is equivalent to
The initial-value formulation for the Einstein-Dirac equation
Proof. Contracting the equation
The initial-value formulation for the Einstein-Dirac equation
we see that
The initial-value formulation for the Einstein-Dirac equation
from which it follows that
The initial-value formulation for the Einstein-Dirac equation
Let us contract this equation. Then, with the help of the Gauss equation (iii) in Lemma 3.1, we obtain
The initial-value formulation for the Einstein-Dirac equation
which gives the asserted formula immediately. The converse is easy to verify.
The initial-value formulation for the Einstein-Dirac equation
Now we should verify that the constraints are indeed preserved under the evolution
The initial-value formulation for the Einstein-Dirac equation
We note at this point that the divergence of the energy-momentum tensor T η vanishes, div η (T η ) = 0, since T η is defined by eigenspinors of the Dirac operator D η (see [12] ). Then, computing the divergence, div η (∆) = 0, and expressing the covariant derivative ∇ η in terms of ∇ η , we find that
The initial-value formulation for the Einstein-Dirac equation
where we have used the formula established in Proposition 2.1,
The initial-value formulation for the Einstein-Dirac equation
Rewriting the above two equations with respect to η-orthonormal frame (E 1 , . . . , E n , E n+1 ), we arrive at a nonlinear hyperbolic system of first-order differential equations of the form
The initial-value formulation for the Einstein-Dirac equation
and C is a (1,1)-tensor field given by
The initial-value formulation for the Einstein-Dirac equation
We observe that the (1, 1)-tensor fields A(k) and B are symmetric (with respect to reference metric η). Moreover, B is positive definite (B ≥ cI for some positive number c > 0), provided that every slice of Q n+1 is compact. Thus, it is shown that, under our evolution, the constraints are locally preserved. Note that, when we consider the warped product metrics as in Section 5 and Section 7, the local preservation of constraints holds without the assumption that every slice of Q n+1 is compact.
The initial-value formulation for the Einstein-Dirac equation
Next, we state a complete set of evolution equations for the Einstein-Dirac equation. Soon we will also define the corresponding initial data sets precisely. Combining Corollary 3.1, Proposition 3.2, Proposition 4.1, 4.2 and Proposition 6.1 altogether, we easily obtain the evolution system of three differential equations, describing the evolution of metrics L 2 = g , that of symmetric (0,2)-tensor fields II and that of spinor fields ψ , respectively:
The initial-value formulation for the Einstein-Dirac equation
In case of n = 2m − 1, the equation (E3) may be certainly replaced by the one in Corollary 4.1, and the terms for Tr η (T η ) and T η (F n+1 , F n+1 ) in (E2) may be replaced by the ones in Corollary 4.2.
The initial-value formulation for the Einstein-Dirac equation
Let us now define the initial data sets. We derive the constraint equations on initial hypersurfaces in a natural way, by combining Proposition 4.2 (Corollary 4.2) with the relations,
The initial-value formulation for the Einstein-Dirac equation
and
The initial-value formulation for the Einstein-Dirac equation
Definition 6.1 (In case of n = 2m) An initial data set (M 2m , g, II M , ψ M ) for the EinsteinDirac equation on Q 2m+1 consists of a slice M 2m with, defined on it, a metric g, a symmetric (0, 2)-tensor field II M and a spinor field ψ M satisfying the momentum constraint
The initial-value formulation for the Einstein-Dirac equation
as well as the Hamiltonian constraint 
A local existence theorem
For a specific class of initial data sets, we will establish a local existence theorem for the Einstein-Dirac equation. Let us begin with the case n = 2m. Let ψ M = ψ as a set of six unknowns, then the system of evolution equations in Proposition 7.1 reduces to an autonomous eqution d dt Ψ = H(Ψ)
A local existence theorem
for some vector field H defined on the six-dimensional Euclidean space R 6 . This fact implies that, to each initial data, there corresponds a unique smooth local solution to the evolution system in Proposition 7.1. where we can always control ǫ = ±1 and (ψ M , ψ M ) = constant so that
A local existence theorem
is nonnegative. Let II M = − 1 2 f t (0)g M be the symmetric (0,2)-tensor field required to prescribe initial data. Then, with the help of Lemma 7.1, one verifies that the initial data (M 2m , g M , II M , ψ M ) satisfies the constraint equations in Definition 6.1. Moreover, we know that the evolution system in Proposition 7.1 is an autonomous equation and hence allows a local solution satisfying the initial data. This proves the former part of the proposition. Let us now suppose that the spinor ψ M is a parallel spinor (λ M = 0). In this case, we may assume that ψ M = ψ + M ∈ Γ(Σ + (M )) and ψ = h + ψ + M , and hence the evolution system in Proposition 7.1 simplifies to
A local existence theorem
On the other hand, since ψ is (locally) an Einstein spinor, the equation

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