GENERALIZED CYLINDERS IN SEMI-RIEMANNIAN AND SPIN GEOMETRY

CHRISTIAN BÄR, PAUL GAUDUCHON, AND ANDREI MOROIANU

Abstract. We use a construction which we call generalized cylinders to give a new proof of the fundamental theorem of hypersurface theory. It has the advantage of being very simple and the result directly extends to semi-Riemannian manifolds and to embeddings into spaces of constant curvature. We also give a new way to identify spinors for different metrics and to derive the variation formula for the Dirac operator. Moreover, we show that generalized Killing spinors for Codazzi tensors are restrictions of parallel spinors. Finally, we study the space of Lorentzian metrics and give a criterion when two Lorentzian metrics on a manifold can be joined in a natural manner by a 1-parameter family of such metrics.

1. Introduction

� n this paper we give various applications of a construction which we call generalized cylinders. Let M be a manifold and let g_{t} be a smooth 1-parameter family of semi-Riemannian metrics on M, t ∈ I ⊂ ℝ. Then we call the manifold Z = I × M with the metric dt^{2} + g_{t} a generalized cylinder over M. On the

one hand, this ansatz is very flexible. Locally, near a semi-Riemannian hypersurface with spacelike normal bundle every semi-Riemannian manifold is of this form. The restriction to spacelike normal bundle, i. e. to the positive sign in front of dt^{2} in the metric of Z is made for convenience only. Changing the signs of the metrics on M as well as on Z reduces the case of a timelike normal bundle to that of a spacelike normal bundle. On the other hand, this ansatz still allows to closely relate the geometries of M and Z.

In Section 2 we collect basic material on spinors and the Dirac operator on semi-Riemannian manifolds. We do this to fix notation and for the convenience of the reader. Some of the material, such as the spin geometry of submanifolds, is not so easily found in the literature unless one restricts oneself to the Riemannian situation.

In Section 3 we study spinors on a manifold foliated by semi-Riemannian hypersurfaces. In particular, we derive a formula for the commutator of the leafwise Dirac operator and the normal derivative. This formula will be important later.

In Section 4 we collect formulas relating the curvature of a generalized cylinder to geometric data on M.

After these preliminaries we give a first application in Section 5. One technical difficulty when dealing with spinors comes from the fact that the definition of spinors depends on the metric on the manifold. This problem does not arise when one works with tensors. Thus if one wants to compare the Dirac operators for two different metrics, then one first has to identify the spinor bundles in a natural manner. This identification problem can be split into two steps. First, construct an identification for 1-parameter families of metrics and, secondly, given two metrics construct a natural 1-parameter family joining them.

Date: March 10, 2003.

2000 Mathematics Subject Classification. 53C27,53A07,53B30.

Key words and phrases. generalized cylinder, identification of spinors, variation formula for Dirac operator, energy-momentum tensor of a spinor, fundamental theorem of hypersurface theory, generalized Killing spinors, space of Lorentzian metrics.

The second step is trivial for Riemannian metrics; just use linear interpolation. For indefinite semi-Riemannian metrics the situation is much more complicated. In fact, two semi-Riemannian metrics on a manifold cannot always be joined by a continuous path of metrics even if they have the same signature. In Section 9 we study this problem in detail for Lorentzian metrics and we give a criterion when two Lorentzian metrics can be joined in a natural manner.

The first step, identifying spinors for 1-parameter families of semi-Riemannian metrics, is carried out in Section 5. The idea is very simple. Given a 1-parameter family of metrics take the corresponding generalized cylinder and use parallel transport on this cylinder. It turns out that this identification is the same as the one constructed differently by Bourguignon and the second author in [3] for Riemannian metrics. The commutator formula from Section 3 directly translates to the variation formula for Dirac operators.

This variation formula is what one needs to compute the energy-momentum tensor for spinors. To make this precise we briefly summarize Lagrangian field theory in Section 6 and we give a general definition of energy-momentum tensors. Then we compute the example of the Lagrangian for spinors given by the Dirac operator.

In Section 7 we give a new and simple proof of the fundamental theorem of hypersurface theory. A hypersurface of ℝ^{n+1} inherits a Riemannian metric and its Weingarten map must satify the Gauss and Codazzi-Mainardi equations. The fundamental theorem says that, conversely, any Riemannian manifold M with a symmetric endomorphism field of T M satisfying the Gauss and Codazzi-Mainardi equations can, at least locally, be embedded isometrically into ℝ^{n+1} with Weingarten map given by this endomorphism field. Our proof goes like this: We write down an explicit metric on the cylinder Z = I × M and we then check that this metric is flat. Since every flat Riemannian manifold is locally isometric to Euclidean space the theorem follows. This approach directly extends to semi-Riemannian manifolds and to embeddings into spaces of constant sectional curvature not necessarily zero. This kind of approach to the fundamental theorem for hypersurfaces was suggested, but not carried out, by Petersen in [9, p. 95].

In Section 8 we study generalized Killing spinors. They are characterized by the overdetermined equation ∇_{X}^{ΣM}ψ = \frac{1}{2}A(X) · ψ where A is a given symmetric endomorphism field. We show that if A is a Codazzi tensor, then the manifold can be embedded as a hypersurface into a Ricci flat manifold equipped with a parallel spinor which restricts to ψ. This generalizes the case of Killing spinors, A = λ id. The classification of manifolds admitting Killing spinors in [1] was based on the observation that the cone over such a manifold possesses a parallel spinor. This also generalizes the case that A is parallel which was studied in [7].

Acknowledgements. The authors would like to thank W. Ballmann and H. Karcher for valuable suggestions. The authors have been partially supported by the Research and Training Network HPRN-CT-2000-00101 “EDGE” funded by the European Commission. The first author has also been partially supported by the Research and Training Network HPRN-CT-1999-00118 “Geometric Analysis”. The first author would like to thank the Ecole Polytechnique, Palaiseau, and the the Max-Planck-Institut für Mathematik, Bonn, for their hospitality.

2. The Dirac operator on semi-Riemannian manifolds

� n this section we collect the basic facts and conventions concerning spinors and Dirac operators on semi-Riemannian manifolds. For a detailed introduction the reader may consult the book [2]. We start with some algebraic preliminaries. Let r + s = n and consider the nondegenerate symmetric bilinear form of signature (r, s)

⟨v, w⟩ := ∑_{i=1}^{r} v^{i}w^{i} − ∑_{i=r+1}^{n} v^{i}w^{i}

on ℝ^{n}. Define the corresponding orthogonal group by

O(r, s) := {A ∈ GL(n, ℝ) | ⟨Av, Aw⟩ = ⟨v, w⟩ for all v, w ∈ ℝ^{n}}

and the special orthogonal group by

SO(r, s) := {A ∈ O(r, s) | det(A) = 1}.

If r = 0 or s = 0, then SO(r, s) is connected, otherwise it has two connected components.

Now let Cl_{r,s} be the Clifford algebra corresponding to the symmetric bilinear form ⟨·, ·⟩. This is the unital algebra generated by ℝ^{n} subject to the relations

(1)  v · w + w · v + 2 ⟨v, w⟩ · 1 = 0

for all v, w ∈ ℝ^{n}. There is a decomposition into even and odd elements

Cl_{r,s} = Cl_{r,s}^{0} ⊕ Cl_{r,s}^{1}

such that ℝ injects naturally into Cl_{r,s}^{0} and ℝ^{n} into Cl_{r,s}^{1}. The spin group is defined by

Spin(r, s) := {v_{1} ⋯ v_{k} ∈ Cl_{r,s}^{0} | v_{j} ∈ ℝ^{n} such that ⟨v_{j}, v_{j}⟩ = ±1 and k is even}

with multiplication inherited from Cl_{r,s}. Given v ∈ ℝ^{n} such that ⟨v, v⟩ ≠ 0 and arbitrary w ∈ ℝ^{n} we see directly from relation (1) that v^{−1} = −\frac{v}{⟨v,v⟩} and

Ad_{v}(w) := v^{−1} · w · v = −w + 2\frac{⟨v, w⟩}{⟨v, v⟩}v.

Hence −Ad_{v} is the reflection across the hyperplane v^{⊥} and, in particular, leaves ℝ^{n} ⊂ Cl_{r,s} invariant. Thus conjugation gives an action of Spin(r, s) on ℝ^{n} by an even number of reflections across hyperplanes. This yields the exact sequence

1 ⟶ ℤ/2ℤ = {1, −1} ⟶ Spin(r, s) ⟶^{Ad} SO(r, s) ⟶ 1.

If n = r + s is even the Clifford algebra possesses an irreducible complex module Σ_{r,s} of complex dimension dimension 2^{n/2}, the complex spinor module. When restricted to Cl_{r,s}^{0} the spinor module decomposes into

Σ_{r,s} = Σ_{r,s}^{+} ⊕ Σ_{r,s}^{−},

the submodules of spinors of positive resp. negative chirality. In particular, the spin group Spin(r, s) ⊂ Cl_{r,s}^{0} acts on Σ_{r,s}^{+} and on Σ_{r,s}^{−}. This action

ρ = ρ^{+} ⊕ ρ^{−} : Spin(r, s) → Aut(Σ_{r,s}^{+}) × Aut(Σ_{r,s}^{−}) ⊂ Aut(Σ_{r,s})

is called the spinor representation of Spin(r, s). Given an orientation on ℝ^{n} the Cl_{r,s}^{0}- submodules Σ_{r,s}^{+} and Σ_{r,s}^{−} can be characterized by the action of the volume element vol := e_{1} ⋯ e_{n} ∈ Cl_{r,s}^{0} which acts on Σ_{r,s}^{+} as +i^{s+n(n+1)/2}id and on Σ_{r,s}^{−} as −i^{s+n(n+1)/2}id where e_{1}, …, e_{n} is a positively oriented orthonormal basis of ℝ^{n}.

If n is odd, then Cl_{r,s} has two inequivalent irreducible modules Σ_{r,s}^{0} and Σ_{r,s}^{1}, both of complex dimension 2^{(n−1)/2}. These two modules are again distinguished by the action of the volume element vol = e_{1} ⋯ e_{n} ∈ Cl_{r,s}^{1}, namely vol acts as +i^{s+n(n+1)/2}id on Σ_{r,s}^{0} and as −i^{s+n(n+1)/2}id on Σ_{r,s}^{1}. When restricted to Cl_{r,s}^{0} the two modules become equivalent and we simply write Σ_{r,s} := Σ_{r,s}^{0}. This time the spinor representation

ρ : Spin(r, s) → Aut(Σ_{r,s})

is irreducible. All spinor modules carry nondegenerate symmetric sesquilinear forms ⟨·, ·⟩ (in general not definite) which are invariant under the action of Spin(r, s). The action of a vector v ∈ ℝ^{n} ⊂ Cl_{r,s} on Σ_{r,s} is skewsymmetric with respect to ⟨·, ·⟩, i. e. ⟨v · σ_{1}, σ_{2}⟩ = − ⟨σ_{1}, v · σ_{2}⟩.

To prepare for the study of submanifolds later on we now look at an embedding of ℝ^{n} into ℝ^{n+1} such that (ℝ^{n})^{⊥} is spacelike. Let (ℝ^{n})^{⊥} be spanned by a spacelike unit vector e_{0}. The map ℝ^{n} → Cl_{r+1,s}, v ↦ e_{0} · v, induces an algebra isomorphism Cl_{r,s} → Cl_{r+1,s}^{0} under which the volume element of Cl_{r,s} is mapped to the volume element of Cl_{r+1,s} in case n is odd.

If n is even, then Σ_{r+1,s} pulls back to Σ_{r,s} under this algebra isomorphism. In other words, we can regard Σ_{r+1,s} as the spinor representation of Cl_{r,s} provided we define the action of Cl_{r,s} on Σ_{r+1,s} by

v ⊗ σ ↦ e_{0} · v · σ

where v ∈ ℝ^{n} and · denotes the action of Cl_{r+1,s}.

Similarly, if n is odd, then the action of the volume forms shows that Σ_{r+1,s}^{+} pulls back to Σ_{r,s}^{0} while Σ_{r+1,s}^{−} pulls back to Σ_{r,s}^{1}.

Now we turn to geometry. Let X denote an oriented n-dimensional differentiable manifold. The bundle P_{GL}^{+}(X) of positively oriented tangent frames forms a GL^{+}(n, ℝ)-principal bundle over X. Here and henceforth GL^{+}(n, ℝ) denotes the group of real n × n-matrices with positive determinante and A : G̃L^{+}(n, ℝ) → GL^{+}(n, ℝ) its connected twofold covering group. A spin structure of X is a G̃L^{+}(n, ℝ)-principal bundle PG̃L(X) over X together with a twofold covering map Θ : PG̃L^{+}(X) → P_{GL}^{+}(X) such that the following diagram commutes

(2)  \xymatrix{PG̃L^{+}(X) × G̃L^{+}(n, ℝ) \ar[dd]^{Θ×A} \ar[r] & PG̃L^{+}(X) \ar[dd]_{Θ} \ar[dr] \\  &  & X \\ P_{GL}^{+}(X) × GL^{+}(n, ℝ) \ar[r] & P_{GL}^{+}(X) \ar@{-->}[ur]}

where the horizontal arrows denote the group actions on the principal bundles. This definition of a spin structure has the advantage of being independent of the choice of any semi-Riemannian metric on X. An oriented manifold together with a spin structure will be called a spin manifold.

Let X now in addition carry a semi-Riemannian metric of signature (r, s), r + s = n. The bundle P_{SO}(X) ⊂ P_{GL}^{+}(X) of positively oriented orthonormal tangent frames forms an SO(r, s)-principal bundle over X. Restricting A : G̃L^{+}(n, ℝ) → GL^{+}(n, ℝ) to the preimage of SO(r, s) ⊂ GL^{+}(n, ℝ) we recover Ad : Spin(r, s) → SO(r, s). Putting P_{Spin}(X) := Θ^{−1}(P_{SO}(X)) we get a Spin(r, s)-principal bundle and and the maps in diagram (2) restrict to the following commutative diagram

\xymatrix{P_{Spin}(X) × Spin(r, s) \ar[dd]^{Θ×Ad} \ar[r] & P_{Spin}(X) \ar[dd]_{Θ} \ar[dr] \\  &  & X \\ P_{SO}(X) × SO(r, s) \ar[r] & P_{SO}(X) \ar[ur]}

Very often in the literature P_{Spin}(X) is called a spin structure of X and we will call X together with P_{Spin}(X) a semi-Riemannian spin manifold.

On a semi-Riemannian spin manifold we define the spinor bundle of X as the complex vector bundle associated to the spinor representation, i. e.

ΣX := P_{Spin}(X) ×_{ρ} Σ_{r,s}.

In other words, for p ∈ X the fiber of Σ_{p}X of ΣX over p consists of equivalence classes of pairs [b, σ] where b ∈ P_{Spin}(X)_{p} and σ ∈ Σ_{r,s} subject to the relation

[b, σ] = [bg^{−1}, gσ]

for all g ∈ Spin(r, s). Unfortunately, the spinor bundle cannot be defined independently of the metric using PG̃L^{+}(X) instead of P_{Spin}(X) because the spinor representation ρ of

Spin(r, s) on Σ_{r,s} does not extend to a representation of G̃L^{+}(n, ℝ) on Σ_{r,s}. We will come back to this problem in Section 5.

Note that the tangent bundle can also be written in a similar manner, T X = P_{SO}(X)×_{τ} ℝ^{n} where τ is the standard representation of SO(r, s) on ℝ^{n}. One defines Clifford multiplication T_{p}X ⊗ Σ_{p}X → Σ_{p}X by

[Θ(b), v] · [b, σ] := [b, v · σ]

where b ∈ P_{Spin}(X)_{p}, v ∈ ℝ^{n}, and σ ∈ Σ_{r,s}. For g ∈ Spin(r, s) we see from

[Θ(bg), v] · [bg, σ] = [Θ(b)Ad_{g}, v] · [bg, σ] = [Θ(b), Ad_{g}v] · [b, gσ]
= [b, gvg^{−1}gσ] = [b, gvσ] = [bg, vσ]

that this is well-defined. It is this point that goes wrong when one tries to work with nonoriented manifolds and pin structures. Had we defined Σ_{r,s} = Σ_{r,s}^{1} instead of Σ_{r,s} = Σ_{r,s}^{0} in odd dimensions, then we would have obtained the Clifford multiplication with the opposite sign.

Clifford multiplication inherits the relations of the Clifford algebra, i. e. for X, Y ∈ T_{p}X and φ ∈ Σ_{p}X we have

X · Y · φ + Y · X · φ + 2 ⟨X, Y ⟩ φ = 0.

In even dimensions the spinor bundle splits into the positive and the negative half-spinor bundles,

(3)  ΣX = Σ^{+}X ⊕ Σ^{−}X

where Σ^{±}X = P_{Spin}(X)×_{ρ^{±}}Σ_{r,s}^{±}. Clifford multiplication by a tangent vector interchanges Σ^{+}X and Σ^{−}X.

The Spin(r, s)-invariant nondegenerate symmetric sesquilinear forms on Σ_{r,s} and Σ_{r,s}^{±} induce (in general indefinite) inner products on ΣX and Σ^{±}X which we again denote by ⟨·, ·⟩.

The connection 1-form ω^{X} on P_{SO}(X) for the Levi-Civita connection ∇^{X} can be lifted via Θ to P_{Spin}(X), i. e. ω^{ΣX} := Ad_{∗}^{−1} ∘Θ^{∗}(ω^{X}). Composing with Ad_{∗}^{−1} is necessary because the connection 1-form on P_{Spin}(X) must take values in the Lie algebra of Spin(r, s) rather than in that of SO(r, s). Now ω^{ΣX} induces a covariant derivative ∇^{ΣX} on ΣX.

An equivalent, but less invariant, way of describing ∇^{ΣX} is as follows: If b is a local section in P_{Spin}(X), then Θ(b) = (e_{1}, …, e_{n}) is a local oriented orthonormal tangent frame, ⟨e_{i}, e_{j}⟩ ≡ ε_{i}δ_{ij} where ε_{i} = ±1. The Christoffel symbols of ∇^{X} with respect to this frame are given by

∇_{e_{i}}^{X}e_{j} = ∑_{k=1}^{n} Γ_{ij}^{k} e_{k}.

Now the covariant derivative of a locally defined spinor field φ = [b, σ], σ a function with values in Σ_{r,s}, is given by

(4)  ∇_{e_{i}}^{ΣX}φ = [b, d_{e_{i}}σ + \frac{1}{2} ∑_{j<k} Γ_{ij}^{k} ε_{j} e_{j} · e_{k} · σ] .

One checks that ∇^{ΣX} is a metric connection and that it leaves the splitting (3) in even dimensions invariant. Moreover, it satisfies the following Leibniz rule:

∇_{Z}^{ΣX}(Y · φ) = (∇_{Z}^{X}Y) · φ + Y · ∇_{Z}^{ΣX}φ

for all vector fields Z and Y and all spinor fields φ.

The curvature tensor R^{ΣX} of ∇^{ΣX} can be computed in terms of the curvature tensor R^{X} of the Levi-Civita connection,

R^{ΣX}(Y, Z)φ = \frac{1}{2} ∑_{i<j} ε_{i}ε_{j} ⟨R^{X}(Y, Z)e_{i}, e_{j}⟩ e_{i} · e_{j} · φ.

Using the first Bianchi identity one easily computes

(5)  ∑_{i=1}^{n} ε_{i} e_{i} · R^{ΣX}(e_{i}, Y)φ = \frac{1}{2}Ric^{X}(Y) · φ.

Here Ric^{X} denotes the Ricci curvature considered as an endomorphism field on T M. The Ricci curvature considered as a symmetric bilinear form will be written ric^{X}(Y, Z) = ⟨Ric^{X}(Y), Z⟩.

The Dirac operator maps spinor fields to spinor fields and is defined by

D^{X}φ = ∑_{i=1}^{n} ε_{i}e_{i} · ∇_{e_{i}}^{ΣX}φ.

Given two spinor fields φ and ψ one can define a vector field Y by the requirement ⟨Y, Z⟩ = ⟨Z · φ, ψ⟩ for all vector fields Z and one easily computes

div(Y) = ⟨D^{X}φ, ψ⟩ − ⟨φ, D^{X}ψ⟩ .

Hence the Dirac operator is formally selfadjoint, i. e. if the intersection of the supports of φ and ψ is compact, then

(D^{X}φ, ψ) = (φ, D^{X}ψ)

where (φ, ψ) = ∫_{M} ⟨φ, ψ⟩ dV.

3. The Dirac operator on manifolds foliated by hypersurfaces

� et Z be an oriented (n + 1)-dimensional semi-Riemannian spin manifold. Let Θ : P_{Spin}(Z) → P_{SO}(Z) be a spin structure on Z. Let M ⊂ Z be a semi-Riemannian hypersurface with trivial spacelike normal bundle. This means there is a vector field ν on Z along M satisfying ⟨ν, ν⟩ = +1 and ⟨ν, T M⟩ = 0. If

the signature of M is (r, s), then the signature of Z is (r + 1, s).

In this situation M inherits a spin structure as follows: The bundle of oriented orthonormal frames of M, P_{SO}(M), can be embedded into the bundle of oriented orthonormal frames of Z restricted to M, P_{SO}(Z)|_{M}, by the map ι : (e_{1}, …, e_{n}) ↦ (ν, e_{1}, …, e_{n}). Then P_{Spin}(M) := Θ^{−1}(ι(P_{SO}(M))) defines a spin structure on M. We will always implicitly assume that this spin structure be taken on M. The same discussion is possible on the level of G̃L^{+}(n, ℝ)-bundles.

The algebraic remarks in the previous section show that if n is even, then

ΣZ|_{M} = ΣM

where the Clifford multiplication with respect to M is given by X ⊗ φ ↦ ν · X · φ and “·” always denotes the Clifford multiplication with respect to Z. If n is odd, then

Σ^{+}Z|_{M} = ΣM

and again Clifford multiplication with respect to M is given by X ⊗ φ ↦ ν · X · φ while

Σ^{−}Z|_{M} = ΣM

with Clifford multiplication with respect to M given by X ⊗ φ ↦ −ν · X · φ. The minus sign comes from the fact that in odd dimensions we defined Σ_{r,s} = Σ_{r,s}^{0} while Σ_{r,s}^{1} leads to the opposite sign for the Clifford multiplication. The identifications preserve the natural inner products ⟨·, ·⟩.

Let W denote the Weingarten map with respect to ν, i. e.

(6)  ∇_{X}^{Z}Y = ∇_{X}^{M}Y + ⟨W (X), Y ⟩ ν

for all vector fields X and Y on M. The Weingarten map is symmetric with respect to the semi-Riemannian metric, ⟨W (X), Y ⟩ = ⟨X, W (Y)⟩ and is also given by W (X) = −∇_{X}^{Z}ν. If we denote the Christoffel symbols of M with respect to a local orthogonal tangent frame (e_{1}, …, e_{n}) by Γ_{ij}^{M,k} and the Christoffel symbols of Z with respect to (e_{0}, e_{1}, …, e_{n}), e_{0} = ν, by Γ_{ij}^{Z,k}, then (6) implies for 1 ≤ i, j, k ≤ n

(7)  Γ_{ij}^{Z,k} = Γ_{ij}^{M,k},
(8)  Γ_{ij}^{Z,0} = ⟨W (e_{i}), e_{j}⟩ ,
(9)  Γ_{i0}^{Z,k} = −ε_{0}ε_{k}Γ_{ik}^{Z,0} = −ε_{k} ⟨W (e_{i}), e_{k}⟩ .

Plugging this into (4) we get for a section φ = [b, σ] of ΣZ|_{M} and 1 ≤ i ≤ n

∇_{e_{i}}^{ΣZ}φ = [b, d_{e_{i}}σ + \frac{1}{2} (− ∑_{k=1}^{n} ε_{k} ⟨W (e_{i}), e_{k}⟩ ε_{0}e_{0} · e_{k} + ∑_{1≤j<k≤n} Γ_{ij}^{M,k}ε_{j}e_{j} · e_{k}) · σ]

= [b, d_{e_{i}}σ + \frac{1}{2} (−e_{0} · W (e_{i}) + ∑_{1≤j<k≤n} Γ_{ij}^{M,k}ε_{j}e_{0} · e_{j} · e_{0} · e_{k}) · σ]

= ∇_{e_{i}}^{ΣM}φ − \frac{1}{2}ν · W (e_{i}) · φ.

Hence for each X ∈ T M and each section φ of ΣZ|_{M} we have

(10)  ∇_{X}^{ΣZ}φ = ∇_{X}^{ΣM}φ − \frac{1}{2}ν · W (X) · φ.

Now let φ be a section of ΣZ defined in a neighborhood of M. On the one hand,

D^{Z}φ = ∑_{i=1}^{n} ε_{i}e_{i} · ∇_{e_{i}}^{ΣZ}φ + ν · ∇_{ν}^{ΣZ}φ.

On the other hand by (10),

∑_{i=1}^{n} ε_{i}e_{i} · ∇_{e_{i}}^{ΣZ}φ = ∑_{i=1}^{n} ε_{i} e_{i} · ∇_{e_{i}}^{ΣM}φ − \frac{1}{2} ∑_{i=1}^{n} ε_{i} e_{i} · ν · W (e_{i}) · φ

= −ν · ∑_{i=1}^{n} ε_{i} ν · e_{i} · ∇_{e_{i}}^{ΣM}φ + \frac{1}{2} ∑_{i=1}^{n} ε_{i} ν · e_{i} · W (e_{i}) · φ

= −ν · D̃^{M} − \frac{1}{2} tr(W)ν · φ

where D̃^{M} = D^{M} if n is even and D̃^{M} = (\begin{matrix}D^{M} & 0 \\ 0 & −D^{M} \end{matrix}) if n is odd. Thus the Dirac

operators on M and on Z are related by

(11)  ν · D^{Z} = D̃^{M} + \frac{n}{2}H − ∇_{ν}^{ΣZ}

where H = \frac{1}{n} tr(W) denotes the mean curvature.

Next we consider the situation that Z carries a semi-Riemannian foliation by hypersurfaces. The commutator of the leafwise Dirac operator and the normal derivative will be of central importance later.

Proposition 3.1. Let Z be an (n + 1)-dimensional semi-Riemannian spin manifold. Let Z carry a semi-Riemannian foliation by hypersurfaces with trivial spacelike normal bundle, i. e. the leaves M are semi-Riemannian hypersurfaces and there exists a vector field ν on Z perpendicular to the leaves such that ⟨ν, ν⟩ = 1 and ∇_{ν}^{Z}ν = 0. Let W denote the Weingarten map of the leaves with respect to ν and let H = \frac{1}{n} tr(W) be the mean curvature.

Then the commutator of the leafwise Dirac operator and the normal derivative is given by

[∇_{ν}^{ΣZ}, D̃^{M}] φ = 𝔇^{W} φ − \frac{n}{2} ν · grad^{M}(H) · φ + \frac{1}{2} ν · div^{M}(W) · φ.

Here grad^{M} denotes the leafwise gradient, div^{M}(W) = ∑_{i=1}^{n} ε_{i} (∇_{e_{i}}^{M}W)(e_{i}) denotes the leafwise divergence of the endomorphism field W, 𝔇^{W} φ = ∑_{i=1}^{n} ε_{i} ν · e_{i} · ∇_{W (e_{i})}^{ΣM}φ, and “·” denotes Clifford multiplication on Z.

Proof. We choose a local oriented orthonormal tangent frame (e_{1}, …, e_{n}) for the leaves and we may assume for simplicity that ∇_{ν}^{Z}e_{i} = 0. We compute

[∇_{ν}^{ΣZ}, D̃^{M}] φ = ∑_{i=1}^{n} ε_{i} (∇_{ν}^{ΣZ}(ν · e_{i} · ∇_{e_{i}}^{ΣM}φ) − ν · e_{i} · ∇_{e_{i}}^{ΣM}∇_{ν}^{ΣZ}φ)

= ∑_{i=1}^{n} ε_{i} ν · e_{i} · (∇_{ν}^{ΣZ}∇_{e_{i}}^{ΣM}φ − ∇_{e_{i}}^{ΣM}∇_{ν}^{ΣZ}φ)

=^{(10)} ∑_{i=1}^{n} ε_{i} ν · e_{i} · (∇_{ν}^{ΣZ}(∇_{e_{i}}^{ΣZ} + \frac{1}{2}ν · W (e_{i}))

−(∇_{e_{i}}^{ΣZ} + \frac{1}{2}ν · W (e_{i}))∇_{ν}^{ΣZ})φ

= ∑_{i=1}^{n} ε_{i} ν · e_{i} · (R^{ΣZ}(ν, e_{i}) + ∇_{[ν,e_{i}]}^{ΣZ} + \frac{1}{2}ν · (∇_{ν}^{Z}W)(e_{i}))φ

=^{(5)} −\frac{1}{2}ν · Ric^{Z}(ν) · φ + ∑_{i=1}^{n} ε_{i} ν · e_{i} · (∇_{W (e_{i})}^{ΣZ} + \frac{1}{2}ν · (∇_{ν}^{Z}W)(e_{i}))φ

=^{(10)} −\frac{1}{2}ν · Ric^{Z}(ν) · φ

+ ∑_{i=1}^{n} ε_{i} ν · e_{i} · (∇_{W (e_{i})}^{ΣM} − \frac{1}{2}ν · W^{2}(e_{i}) + \frac{1}{2}ν · (∇_{ν}^{Z}W)(e_{i}))φ

= −\frac{1}{2}ν · Ric^{Z}(ν) · φ + 𝔇^{W} φ

+\frac{1}{2} ∑_{i=1}^{n} ε_{i} e_{i} · ( − W^{2}(e_{i}) + (∇_{ν}^{Z}W)(e_{i}))φ. (12)

The Riccati equation for the Weingarten map (∇_{ν}^{Z}W)(X) = R^{Z}(X, ν)ν + W^{2}(X) yields

[∇_{ν}^{ΣZ}, D̃^{M}] φ = −\frac{1}{2}ν · Ric^{Z}(ν) · φ + 𝔇^{W} φ + \frac{1}{2} ∑_{i=1}^{n} ε_{i} e_{i} · (R^{Z}(e_{i}, ν)ν) · φ

= −\frac{1}{2}ν · Ric^{Z}(ν) · φ + 𝔇^{W} φ + \frac{1}{2}ric^{Z}(ν, ν)φ
(13)  = 𝔇^{W} φ − \frac{1}{2} ∑_{i=1}^{n} ε_{i} ric^{Z}(ν, e_{i}) ν · e_{i} · φ.

The Codazzi-Mainardi equation [8, p. 115] gives for X, Y, V ∈ T_{p}M

⟨R^{Z}(X, Y)V, ν⟩ = ⟨(∇_{X}^{M}W)(Y), V ⟩ − ⟨(∇_{Y}^{M}W)(X), V ⟩ .

Thus

ric^{Z}(ν, X) = ∑_{i=1}^{n} ε_{i} ⟨R^{Z}(X, e_{i})e_{i}, ν⟩

= ∑_{i=1}^{n} ε_{i} (⟨(∇_{X}^{M}W)(e_{i}), e_{i}⟩ − ⟨(∇_{e_{i}}^{M}W)(X), e_{i}⟩)
= tr(∇_{X}^{M}W) − ⟨div^{M}(W), X⟩ .

Plugging this into (13) we get

[∇_{ν}^{ΣZ}, D̃^{M}] φ = 𝔇^{W} φ − \frac{1}{2} ∑_{i=1}^{n} ε_{i} (tr(∇_{e_{i}}^{M}W) − ⟨div^{M}(W), e_{i}⟩) ν · e_{i} · φ

= 𝔇^{W} φ − \frac{1}{2} ∑_{i=1}^{n} ε_{i} d_{e_{i}} tr(W)ν · e_{i} · φ + \frac{1}{2}ν · div^{M}(W) · φ

= 𝔇^{W} φ − \frac{n}{2}ν · grad^{M}(H) · φ + \frac{1}{2}ν · div^{M}(W) · φ.
□

4. The generalized cylinder

� et M be an n-dimensional differentiable manifold, let g_{t} be a smooth 1parameter family of semi-Riemannian metrics on M, t ∈ I where I ⊂ ℝ is an interval. We define the generalized cylinder by

Z := I × M

with semi-Riemannian metric

g_{Z} := dt^{2} + g_{t}.

The generalized cylinder is an (n + 1)-dimensional semi-Riemannian manifold (with boundary if I has boundary) of signature (r + 1, s) if the signature of g_{t} is (r, s). The vector field ν := \frac{∂}{∂t} is spacelike of unit length and orthogonal to the hypersurfaces M_{t} := {t} × M. Let W denote the Weingarten map of M_{t} with respect to ν and let H be the mean curvature.

If X is a local coordinate field on M, then ⟨X, ν⟩ = 0 and [X, ν] = 0. Thus

0 = d_{ν} ⟨X, ν⟩ = ⟨∇_{ν}^{Z}X, ν⟩ + ⟨X, ∇_{ν}^{Z}ν⟩ = ⟨∇_{X}^{Z}ν, ν⟩ + ⟨X, ∇_{ν}^{Z}ν⟩
= − ⟨W (X), ν⟩ + ⟨X, ∇_{ν}^{Z}ν⟩ = ⟨X, ∇_{ν}^{Z}ν⟩

and differentiating ⟨ν, ν⟩ = 1 yields ⟨ν, ∇_{ν}^{Z}ν⟩ = 0. Hence

∇_{ν}^{Z}ν = 0,

i. e. for p ∈ M the curves t ↦ (t, p) are geodesics parametrized by arclength. So the assumptions of Proposition 3.1 are satisfied for the foliation (M_{t})_{t∈I}.

Now fix p ∈ M and X, Y ∈ T_{p}M. We define the first and second derivative of g_{t} by

ġ_{t}(X, Y) := \frac{d}{dt}(g_{t}(X, Y)),
g̈_{t}(X, Y) := \frac{d^{2}}{dt^{2}} (g_{t}(X, Y)).

Then ġ_{t} and g̈_{t} are smooth 1-parameter families of symmetric (2, 0)-tensors on M.

Proposition 4.1. On a generalized cylinder Z = I × M with semi-Riemannian metric g^{Z} = ⟨·, ·⟩ = dt^{2} + g_{t} the following formulas hold:

(14)  ⟨W (X), Y ⟩ = −\frac{1}{2}ġ_{t}(X, Y),
(15)  ⟨R^{Z}(U, V)X, Y ⟩ = ⟨R^{M_{t}}(U, V)X, Y ⟩

+\frac{1}{4} (ġ_{t}(U, X)ġ_{t}(V, Y) − ġ_{t}(U, Y)ġ_{t}(V, X)) ,

(16)  ⟨R^{Z}(X, Y)U, ν⟩ = \frac{1}{2} ((∇_{Y}^{M_{t}} ġ_{t})(X, U) − (∇_{X}^{M_{t}} ġ_{t})(Y, U)) ,
(17)  ⟨R^{Z}(X, ν)ν, Y ⟩ = −\frac{1}{2} (g̈_{t}(X, Y) + ġ_{t}(W (X), Y)) ,
(18)  ric^{Z}(ν, ν) = tr(W^{2}) − \frac{1}{2} tr_{g_{t}}(g̈_{t}),
(19)  ric^{Z}(X, ν) = d_{X} tr(W) − ⟨div^{M}(W), X⟩ ,
(20)  ric^{Z}(X, Y) = ric^{M_{t}}(X, Y) + 2 ⟨W (X), W (Y)⟩

− tr(W) ⟨W (X), Y ⟩ − \frac{1}{2}g̈_{t}(X, Y),

(21)  Scal^{Z} = Scal^{M_{t}} + 3 tr(W^{2}) − tr(W)^{2} − tr_{g_{t}}(g̈_{t}),

where X, Y, U, V ∈ T_{p}M, p ∈ M.

Proof. To show (14) we extend X and Y to local coordinate fields on M so that all Lie brackets vanish. Then the Koszul formula [8, p. 61] for the Levi-Civita connection of Z yields

⟨W (X), Y ⟩ = − ⟨∇_{X}^{Z}ν, Y ⟩ = −\frac{1}{2} (d_{X} ⟨ν, Y ⟩ + d_{ν} ⟨Y, X⟩ − d_{Y} ⟨X, ν⟩)
= −\frac{1}{2}d_{ν} ⟨Y, X⟩ = −\frac{1}{2} \frac{∂}{∂t}g_{t}(X, Y) = −\frac{1}{2}ġ_{t}(X, Y).

Equation (15) follows directly from (14) and the Gauss equation [8, p. 100]

⟨R^{Z}(U, V)X, Y ⟩ = ⟨R^{M_{t}}(U, V)X, Y ⟩ + ⟨W (U), X⟩ ⟨W (V), Y ⟩
− ⟨W (U), Y ⟩ ⟨W (V), X⟩ .

Equation (16) follows directly from (14) and the Codazzi-Mainardi equation [8, p. 115]

⟨R^{Z}(X, Y)U, ν⟩ = ⟨(∇_{X}^{M_{t}}W)(Y), U⟩ − ⟨(∇_{Y}^{M_{t}}W)(X), U⟩ .

The Riccati equation for W

(∇_{ν}^{Z}W)(X) = R^{Z}(X, ν)ν + W^{2}(X)

gives

⟨R^{Z}(X, ν)ν, Y ⟩ = ⟨(∇_{ν}^{Z}W)(X), Y ⟩ − ⟨W^{2}(X), Y ⟩
= \frac{∂}{∂t} ⟨W (X), Y ⟩ − ⟨W (∇_{ν}^{Z}X), Y ⟩ − ⟨W (X), ∇_{ν}^{Z}Y ⟩

+\frac{1}{2}ġ_{t}(W (X), Y)

= −\frac{1}{2} \frac{∂}{∂t}ġ_{t}(X, Y) − ⟨W (∇_{X}^{Z}ν), Y ⟩ − ⟨W (X), ∇_{Y}^{Z}ν⟩

+\frac{1}{2}ġ_{t}(W (X), Y)

= −\frac{1}{2}g̈_{t}(X, Y) + ⟨W (W (X)), Y ⟩ + ⟨W (X), W (Y)⟩

+\frac{1}{2}ġ_{t}(W (X), Y)

= −\frac{1}{2}g̈_{t}(X, Y) − \frac{1}{2}ġ_{t}(W (X), Y)

which is (17). The Ricci curvature is now easily computed.

ric^{Z}(ν, ν) = ∑_{i=1}^{n} ε_{i} ⟨R^{Z}(e_{i}, ν)ν, e_{i}⟩
=^{(17)} −\frac{1}{2} ∑_{i=1}^{n} ε_{i} (g̈_{t}(e_{i}, e_{i}) + ġ_{t}(W (e_{i}), e_{i}))
=^{(14)} −\frac{1}{2} tr_{g_{t}}(g̈_{t}) + tr(W^{2})

which is (18). Moreover,

ric^{Z}(X, ν) = ∑_{i=1}^{n} ε_{i} ⟨R^{Z}(X, e_{i})e_{i}, ν⟩
=^{(16)} \frac{1}{2} ∑_{i=1}^{n} ε_{i} ((∇_{e_{i}}^{M_{t}} ġ_{t})(X, e_{i}) − (∇_{X}^{M_{t}} ġ_{t})(e_{i}, e_{i}))
=^{(14)} − ∑_{i=1}^{n} ε_{i} (⟨(∇_{e_{i}}^{M_{t}}W)(X), e_{i}⟩ − ⟨(∇_{X}^{M_{t}}W)(e_{i}), e_{i}⟩)
= − ⟨div^{M_{t}}, X⟩ + tr(∇_{X}^{M_{t}}W)
= − ⟨div^{M_{t}}, X⟩ + d_{X} tr(W)

thus showing (19). Furthermore,

ric^{Z}(X, Y)  =  ∑_{i=1}^{n} ε_{i} ⟨R^{Z}(e_{i}, X)Y, e_{i}⟩ + ⟨R^{Z}(ν, X)Y, ν⟩

=^{(15),(17)} ∑_{i=1}^{n} ε_{i}( ⟨R^{M_{t}}(e_{i}, X)Y, e_{i}⟩ + \frac{1}{4}ġ_{t}(e_{i}, Y)ġ_{t}(X, e_{i})

−\frac{1}{4}ġ_{t}(e_{i}, e_{i})ġ_{t}(X, Y)) − \frac{1}{2} (g̈_{t}(X, Y) + ġ_{t}(W (X), Y))

=  ric^{M_{t}}(X, Y) + ∑_{i=1}^{n} ε_{i}(⟨W (e_{i}), Y ⟩ ⟨W (X), e_{i}⟩

− ⟨W (e_{i}), e_{i}⟩ ⟨W (X), Y ⟩) − \frac{1}{2}g̈_{t}(X, Y) + ⟨W^{2}(X), Y ⟩

=  ric^{M_{t}}(X, Y) + 2 ⟨W (X), W (Y)⟩ − tr(W) ⟨W (X), Y ⟩

−\frac{1}{2}g̈_{t}(X, Y)

shows (20). Formula (21) for the scalar curvature follows from (18) and (20). □

Example 4.2. A simple special case of a generalized cylinder is that of a warped product, i. e. g_{t} = f(t)^{2}g where f : I → ℝ is a smooth positive function. Then ġ_{t} = 2 f ḟ g = \frac{2ḟ}{f}g_{t}

and g̈_{t} = 2(ḟ^{2} + ff̈)g = 2\frac{ḟ^{2}+ff̈}{f^{2}}g_{t} and the formulas in Proposition 4.1 reduce to

W = −\frac{ḟ}{f}id,
R^{Z}(X, Y)U = R^{M_{t}}(X, Y)U + \frac{ḟ^{2}}{f^{2}} (⟨X, U⟩ Y − ⟨Y, U⟩ X) ,
R^{Z}(X, ν)ν = −\frac{f̈}{f}X,
ric^{Z}(X, Y) = ric^{M_{t}}(X, Y) − \frac{(n − 1)ḟ^{2} + ff̈}{f^{2}} ⟨X, Y ⟩ ,
ric^{Z}(X, ν) = 0,
ric^{Z}(ν, ν) = −n\frac{f̈}{f},
Scal^{Z} = Scal^{M_{t}} − n \frac{(n − 1)ḟ^{2} + 2ff̈}{f^{2}},

compare [8, Ch. 7].

5. Identifying spinors and the variation formula for the Dirac operator

� t is an annoying problem that the definition of spinors, in contrast to that of differential forms and tensors, depends on the semi-Riemannian metric of the manifold. Hence if one wants to compare the Dirac operators for two different metrics one first has to identify the underlying spinor bundles.

The problem of constructing such identifications can be split into two steps: First construct identifications for any two metrics in a 1-parameter family of metrics. The identification of spinors for two metrics will in general depend on the 1-parameter family of metrics joining them. Secondly, given two metrics construct a natural curve of metrics joining them.

Both steps have been carried out very satisfactorily for the case of Riemannian metrics in [3]. In the present section we will deal only with the first step. The second step cannot always be carried out. In Section 9 we will discuss this problem for the case of Lorentz metrics in great detail.

Now let g_{t}, t ∈ I, be a smooth 1-parameter family of semi-Riemannian metrics of signature (r, s) on a manifold M. We form the generalized cylinder Z := I × M with metric g = dt^{2} + g_{t}. For t ∈ I we abbreviate the semi-Riemannian manifold (M, g_{t}) by M_{t}.

Spin structures on M and on Z are in 1-1-correspondence. As explained in Section 3 spin structures on Z can be restricted to spin structures on M_{t} = M. Conversely, given a spin structure on M it can be pulled back to I × M yielding a G̃L^{+}(n, ℝ)-principal bundle on Z. Enlarging the structure group via the embedding G̃L^{+}(n, ℝ) ↪ G̃L^{+}(n + 1, ℝ)

covering the standard embedding GL^{+}(n, ℝ) ↪ GL^{+}(n + 1, ℝ), a ↦ (\begin{matrix}1 & 0 \\ 0 & a \end{matrix}), yields

the spin structure on Z which restricts to the given spin structure on M.

Let us write “·” for the Clifford multiplication on Z and “•_{t}” for the Clifford multiplication on M_{t}. Recall from Section 3 that ΣZ|_{M_{t}} = ΣM_{t} as Hermitian vector bundles if n = r+s is even and Σ^{+}Z|_{M_{t}} = ΣM_{t} if n is odd. In both cases the Clifford multiplications are related by X •_{t} φ = ν · X · φ. For given x ∈ M and t_{0}, t_{1} ∈ I parallel translation on Z along the curve t ↦ (t, x) is a linear isometry τ_{t_{0}}^{t_{1}} : Σ_{x}M_{t_{0}} → Σ_{x}M_{t_{1}}. Since “·” and ν are parallel along the curve t ↦ (t, x) so is the family of Clifford multiplications “•_{t}” and τ_{t_{0}}^{t_{1}} preserves Clifford multiplication in the following sense:

τ_{t_{0}}^{t_{1}}(X •_{t_{0}} φ) = (τ_{t_{0}}^{t_{1}}X) •_{t_{1}} (τ_{t_{0}}^{t_{1}}φ).

In general, the covariant derivative and hence parallel transport depends on the semi-Riemannian metric and its first derivatives. We note here that for fixed x ∈ M the parallel transport τ_{t_{0}}^{t_{1}} : T_{x}M_{t_{0}} → T_{x}M_{t_{1}} or τ_{t_{0}}^{t_{1}} : Σ_{x}M_{t_{0}} → Σ_{x}M_{t_{1}} is determined by g_{t}(x) and ġ_{t}(x), no x-derivatives of g_{t} enter. Namely, if x^{1}, …, x^{n} are local coordinates on M and X(t, x) = ∑_{j=1}^{n} ξ^{j}(x, t)\frac{∂}{∂x^{j}} is a parallel vector field along t ↦ (t, x), then this means by (9) and (14)

0 = \frac{∇}{dt}X = ∑_{j=1}^{n} (ξ̇^{j} + ∑_{k=1}^{n} Γ_{k,0}^{Z,j}ξ^{k}) \frac{∂}{∂x^{j}}

= ∑_{j=1}^{n} (ξ̇^{j} + \frac{1}{2} ∑_{k,ℓ=1}^{n} g_{t}^{jℓ}ġ_{t,kℓ}ξ^{k}) \frac{∂}{∂x^{j}} .

Thus τ_{t_{0}}^{t_{1}} : T_{x}M_{t_{0}} → T_{x}M_{t_{1}} is given by solving the system of ordinary differential equations

ξ̇^{j}(t, x) = −\frac{1}{2} ∑_{k,ℓ=1}^{n} g_{t}^{jℓ}(x)ġ_{t,kℓ}(x)ξ^{k}(t, x).

For spinors the situation is similar. By [3, Prop. 2] this shows that our identification τ_{t_{0}}^{t_{1}} of spinors for different metrics coincides with the one in [3].

Now we rewrite the commutator formula of Proposition 3.1. For a section φ of ΣZ (or Σ^{+}Z if n is odd) we have

(22)  [∇_{ν}^{ΣZ}, D^{M_{t}}] φ = 𝔇^{W_{t}}φ − \frac{n}{2} grad^{M_{t}}(H_{t}) •_{t} φ + \frac{1}{2} div^{M_{t}}(W_{t}) •_{t} φ

where D^{M_{t}} is the Dirac operator of M_{t}, grad^{M_{t}} is the gradient and div^{M_{t}} the divergence (of endomorphisms) on M_{t}, W_{t} is the Weingarten map of M_{t} in Z and H_{t} = \frac{1}{n} tr(W_{t}) the mean curvature and finally 𝔇^{W_{t}}φ = ∑_{i=1}^{n} ε_{i} e_{i} •_{t} ∇_{W_{t}(e_{i})}^{ΣM_{t}}φ for any orthonormal basis e_{1}, …, e_{n}. From (14) we have div^{M_{t}}(W_{t}) = −\frac{1}{2} div^{M_{t}}(ġ_{t}), H_{t} = −\frac{1}{2n} tr_{g_{t}}(ġ_{t}) and 𝔇^{W_{t}} = −\frac{1}{2}𝔇ġ_{t} where 𝔇ġ_{t}φ = ∑_{i,j=1}^{n} ε_{i}ε_{j}ġ_{t}(e_{i}, e_{j})e_{i} •_{t} ∇_{e_{j}}^{ΣM_{t}}φ. Thus (22) can be rewritten as

(23) [∇_{ν}^{ΣZ}, D^{M_{t}}] φ = −\frac{1}{2}𝔇ġ_{t}φ + \frac{1}{4} grad^{M_{t}}(tr_{g_{t}}(ġ_{t})) •_{t} φ − \frac{1}{4} div^{M_{t}}(ġ_{t}) •_{t} φ.

Now if φ is parallel along the curves t ↦ (t, x), i. e. it is of the form φ(t, x) = τ_{t_{0}}^{t}ψ(x) for some spinor field ψ on M_{t_{0}}, then the left hand side of (23) is at t = t_{0}

[∇_{ν}^{ΣZ}, D^{M_{t}}] φ = ∇_{ν}^{ΣZ}D^{M_{t}} φ = \frac{d}{dt}|_{t=t_{0}} τ_{t}^{t_{0}}D^{M_{t}} φ
= \frac{d}{dt}|_{t=t_{0}} τ_{t}^{t_{0}}D^{M_{t}}τ_{t_{0}}^{t}ψ.

We have shown the variation formula for the Dirac operator: Theorem 5.1. Let g_{t} be a smooth 1-parameter family of semi-Riemannian metrics on a spin manifold M. We write briefly M_{t} for the semi-Riemannian spin manifold (M, g_{t}). Let τ_{t_{0}}^{t_{1}} be the identification of spinor spaces for M_{t_{0}} and M_{t_{1}} constructed above, let D^{M_{t}} be the Dirac operator of M_{t}, let “•_{t}” be Clifford multiplication on M_{t} and let 𝔇ġ_{t}φ = ∑_{i,j=1}^{n} ε_{i}ε_{j}ġ_{t}(e_{i}, e_{j})e_{i} •_{t} ∇_{e_{j}}^{ΣM_{t}}φ.

Then for any smooth spinor field ψ on M_{t_{0}} we have

\frac{d}{dt}|_{t=t_{0}} τ_{t}^{t_{0}}D^{M_{t}}τ_{t_{0}}^{t}ψ = −\frac{1}{2}𝔇ġ_{t_{0}} ψ + \frac{1}{4} grad^{M_{t_{0}}} (tr_{g_{t_{0}}} (ġ_{t_{0}})) •_{t_{0}} ψ − \frac{1}{4} div^{M_{t_{0}}} (ġ_{t_{0}}) •_{t_{0}} ψ.

This is exactly the formula given in [3, Thm. 21] for Riemannian manifolds.

6. Energy-momentum tensors

� heorem 5.1 can be used to compute the energy-momentum tensor for spinors. In order to explain what this means we briefly sketch Lagrangian field theory, see [4, p. 153 ff] for a more detailed introduction. Let M denote a differentiable manifold and let G be a set of (smooth) semi-Riemannian metrics on M, open

in the C^{∞}-topology. Let π : E → G × M be a fiber bundle with finite dimensional fibers. For example, if M carries a spin structure the fiber over (g, x) ∈ G × M could be the spinor space at x with respect to the metric g, E_{(g,x)} = Σ_{x}^{g}M. For each fixed g ∈ G the restriction π^{−1}({g} × M) → M is a fiber bundle over M and we can form the space of smooth sections S_{g} of this bundle. These Fréchet manifolds S_{g} give rise to a Fréchet fiber bundle S := ⋃_{g∈G} S_{g} → G. Let F ⊂ S be a Fréchet submanifold such that the restriction π : F → G is again a Fréchet fiber bundle.

Now let L : F → Ω^{|n|}(M) be a smooth map where Ω^{|n|}(M) denotes the space of smooth densities on M, i. e. smooth sections of Λ^{n}T^{∗}M ⊗ 𝔬_{M} where 𝔬_{M} is the orientation line bundle. We assume that L is local in the sense that for φ ∈ F the density L(φ) evaluated at x ∈ M depends only on φ(x) and the M-derivatives of φ at x. In other words, L(φ)(x) is a function of the jet j_{M}^{∞}φ(x). We call L the Lagrangian density. In physics it is customary to integrate over M and call ∫_{M} L(φ) the Lagrangian or the action. We avoid this integration since in general the integral ∫_{M} L(φ) need not exist.

We call a smooth 1-parameter family φ_{t} ∈ F_{g} with φ_{0} = φ compactly supported if it is constant outside a compact subset K ⊂ M, i. e. φ_{t}(x) = φ(x) for all x ∈ M \ K and all t. Since L is local L(φ_{t}) is constant outside K as well so that ∫_{M}(L(φ_{t}) − L(φ)) exists and

\frac{d}{dt}|_{t=0} ∫_{M}(L(φ_{t}) − L(φ)) = ∫_{M} \frac{d}{dt}|_{t=0} L(φ_{t}).

The section φ ∈ F_{g} is called critical for L if for each compactly supported deformation φ_{t}

∫_{M} \frac{d}{dt}|_{t=0} L(φ_{t}) = 0.

To explain the concept of energy-momentum tensors we need one more piece of structure. Let H ⊂ T F be a connection. This means that for any φ ∈ F we have T_{φ}F = T_{φ}(F_{π(φ)}) ⊕ H_{φ} and the restriction dπ|_{H_{φ}} : H_{φ} → T_{π(φ)}G is an isomorphism. For fixed φ ∈ F and g := π(φ) we have the linear map dL ∘ (dπ|_{H_{φ}})^{−1} : T_{g}G → Ω^{|n|}(M). Recall that T_{g}G is nothing but the space of smooth (2, 0)-tensors. A smooth symmetric (2, 0)- tensor Q_{φ} will be called the energy-momentum tensor for φ with respect to the Lagrangian L if

dL ∘ (dπ|_{H_{φ}})^{−1}(k) = ⟨Q_{φ}, k⟩_{g} dV_{g}

for all k ∈ T_{g}G. Here ⟨·, ·⟩_{g} denotes the (pointwise) metric on symmetric (2, 0)-tensors induced by g and dV_{g} is the Riemannian volume measure for g. If it exists Q_{φ} is obviously unique. By its definition the energy-momentum tensor describes the behavior of the Lagrangian under variations of the metric. Example 6.1. Let M carry a spin structure, let G be the set of all semi-Riemannian metrics on M and let E be the universal spinor bundle, E_{(g,x)} = Σ_{x}^{g}M. Then S is the universal bundle of spinor fields and we put F := S. We fix λ ∈ ℝ and we define the Lagrangian L by

L(φ) := Re ⟨φ, (D^{g} − λ)φ⟩_{g} dV_{g}

where D^{g} is the Dirac operator with respect to the metric g = π(φ). If φ_{t} is a compactly supported deformation of φ we write \frac{d}{dt}|_{t=0}φ_{t} = φ̇ and we compute

∫_{M} \frac{d}{dt}|_{t=0} L(φ_{t}) = ∫_{M} Re(⟨φ̇, (D^{g} − λ)φ⟩_{g} + ⟨φ, (D^{g} − λ)φ̇⟩_{g}) dV_{g}
= 2 Re ∫_{M} ⟨φ̇, (D^{g} − λ)φ⟩_{g} dV_{g}.

Thus φ is critical if and only if (D^{g} − λ)φ = 0, i. e. if φ is a Dirac-eigenspinor for the eigenvalue λ.

The connection H is determined by the parallel translation τ_{t_{0}}^{t_{1}} used in the previous section to identify spinors for different metrics. More precisely, H_{φ} is the set of all \frac{d}{dt} |_{t=0} τ_{0}^{t}φ for all smooth curves g_{t} of metrics with g_{0} = π(φ).

Now let g_{t} be such a 1-parameter family of metrics and write k := ġ_{0}. We compute

dL ∘ (dπ|_{H_{φ}})^{−1}(k)

= \frac{d}{dt}|_{t=0} L(τ_{0}^{t}φ)
= \frac{d}{dt}|_{t=0} Re ⟨τ_{0}^{t}φ, (D^{g_{t}} − λ)(τ_{0}^{t}φ)⟩_{g_{t}} dV_{g_{t}}
= \frac{d}{dt}|_{t=0} Re ⟨φ, (τ_{t}^{0}D^{g_{t}}τ_{0}^{t} − λ)φ⟩_{g_{0}} \frac{dV_{g_{t}}}{dV_{g_{0}}} dV_{g_{0}}

= Re (⟨φ, \frac{d}{dt}|_{t=0} (τ_{t}^{0}D^{g_{t}}τ_{0}^{t}φ)⟩_{g_{0}} + ⟨φ, (D^{g_{0}} − λ)φ⟩_{g_{0}} \frac{d}{dt}|_{t=0} \frac{dV_{g_{t}}}{dV_{g_{0}}} ) dV_{g_{0}}.

The first term is given by the variation formula for the Dirac operator. Since Clifford multiplication with tangent vectors is skewadjoint all terms of the form Re ⟨φ, X •_{g_{0}} φ⟩ vanishes. Thus Theorem 5.1 yields

Re ⟨φ, \frac{d}{dt}|_{t=0} (τ_{t}^{0}D^{g_{t}}τ_{0}^{t}φ)⟩_{g_{0}} = −\frac{1}{2} Re ⟨φ, 𝔇^{k}φ⟩_{g_{0}} .

For the second term we use

\frac{d}{dt}|_{t=0} \frac{dV_{g_{t}}}{dV_{g_{0}}} = \frac{1}{2} tr_{g_{0}}(k).

Thus

dL ∘ (dπ|_{H_{φ}})^{−1}(k) = \frac{1}{2} Re (− ⟨φ, 𝔇^{k}φ⟩_{g_{0}} + ⟨φ, (D^{g_{0}} − λ)φ⟩_{g_{0}} tr_{g_{0}}(k)) dV_{g_{0}}
= ⟨Q_{φ}, k⟩_{g_{0}} dV_{g_{0}}

for the symmetric (2, 0)-tensor

Q_{φ}(X, Y) = −\frac{1}{4} Re (⟨φ, X •_{g_{0}} ∇_{Y}^{ΣM}φ⟩ + ⟨φ, Y •_{g_{0}} ∇_{X}^{ΣM}φ⟩)

+\frac{1}{2} Re ⟨φ, (D^{g_{0}} − λ)φ⟩ g_{0}(X, Y).

If φ is critical, i. e. if D^{g_{0}}φ = λφ, then the energy-momentum tensor simplifies to

(24)  Q_{φ}(X, Y) = −\frac{1}{4} Re (⟨φ, X •_{g_{0}} ∇_{Y}^{ΣM}φ⟩ + ⟨φ, Y •_{g_{0}} ∇_{X}^{ΣM}φ⟩) .

Example 6.2. Again, let M carry a spin structure, let G be the set of all semi-Riemannian metrics on M and let E be the universal spinor bundle, E_{(g,x)} = Σ_{x}^{g}M. Then again S is the universal bundle of spinor fields and we this time we put F_{g} := {φ ∈ S_{g} | ∫_{M} ⟨φ, φ⟩_{g} dV_{g} = ±1}. We define the Lagrangian L by

L(φ) := Re ⟨φ, D^{g}φ⟩_{g} dV_{g}.

Now φ is critical if and only if

∫_{M} \frac{d}{dt}|_{t=0} L(φ_{t}) = 2 Re ∫_{M} ⟨φ̇, D^{g}φ⟩_{g} dV_{g} = 0

for all φ̇ perpendicular to φ, i. e. if and only if D^{g}φ is a multiple of φ. This way we obtain all nonnull eigenspinors for all eigenvalues simultaneously as critical φ’s.

This time the connection has to be chosen differently because τ_{t_{0}}^{t_{1}} is a pointwise isometry but the volume element dV_{g} also depends on the semi-Riemannian metric. Therefore τ_{t_{0}}^{t_{1}} does not give an isometry for the L^{2}-product used to define F. This can be corrected by defining the connection H̄ as the set of all \frac{d}{dt} |_{t=0} \sqrt{\frac{dV_{g_{t}}}{dV_{g_{0}}}} τ_{0}^{t}φ for all smooth curves g_{t} of metrics with g_{0} = π(φ).

Then we have for such a 1-parameter family of metrics g_{t} with k := ġ_{0}

dL ∘ (dπ|H̄_{φ})^{−1}(k) = Re ⟨φ, \frac{d}{dt}|_{t=0} (τ_{t}^{0}D^{g_{t}}τ_{0}^{t}φ)⟩_{g_{0}} dV_{g_{0}}

and therefore

Q_{φ}(X, Y) = −\frac{1}{4} Re (⟨φ, X •_{g_{0}} ∇_{Y}^{ΣM}φ⟩ + ⟨φ, Y •_{g_{0}} ∇_{X}^{ΣM}φ⟩)

for all φ, critical or not.

These two examples show that for noncritical φ the energy-momentum tensor also depends on the choice of connection H. In contrast, for critical φ the differential dL descends to a map dL : T_{φ}F/T_{φ}(F_{π(φ)}) → Ω^{|n|}(M). Thus the map dL ∘ dπ^{−1} : T_{π(φ)}G → Ω^{|n|}(M) is well defined without any reference to H.

7. Embeddings of hypersurfaces

� e will now apply the cylinder construction described in Section 4 to study the question whether a given manifold can be isometrically immersed as a hypersurface into a manifold of constant curvature. The classical example for such a result is the fundamental theorem for hypersurfaces which can be stated as

follows:

Theorem 7.1. Let (M^{n}, g) be a Riemannian manifold and let A be a field of symmetric endomorphisms of T M satisfying the equations of Gauss and Codazzi-Mainardi:

(25)  (∇_{X}^{M}A)Y = (∇_{Y}^{M}A)X,
(26)  R^{M}(X, Y)Z = ⟨A(Y), Z⟩ A(X) − ⟨A(X), Z⟩ A(Y)

for all X, Y, Z ∈ T_{p}M, p ∈ M.

Then every point of M has a neighborhood which can be isometrically embedded into Euclidean (n + 1)-space ℝ^{n+1}, with Weingarten map A. If M is simply connected, then there exists a global isometric immersion of M into ℝ^{n+1} with the above property.

A proof can be found in [6, Ch. VII.7], but here we will give a more geometrical argument based on the cylinder construction. This will allow us to extend the result without effort to the semi-Riemannian case and to embeddings into model spaces of constant sectional curvature not necessarily zero. We will construct an explicit metric of constant curvature on the cylinder I × M, whose restriction to the leaf {0} × M is g.

For a constant κ ∈ ℝ define the generalized sine and cosine functions

𝔰_{κ}(t) := \{ \begin{matrix}\frac{1}{\sqrt{κ}} sin(\sqrt{κ} · t) , κ & > 0 \\ t  , κ & = 0 \\ \frac{1}{\sqrt{|κ|}} sinh(\sqrt{|κ|} · t), κ & < 0 \end{matrix}  and 𝔠_{κ}(t) := \{ \begin{matrix}cos(\sqrt{κ} · t) , κ & > 0 \\ 1  , κ & = 0 \\ cosh(\sqrt{|κ|} · t), κ & < 0 \end{matrix}

One easily checks 𝔰_{κ}(0) = 0, 𝔠_{κ}(0) = 1, κ𝔰_{κ}^{2} + 𝔠_{κ}^{2} = 1, 𝔰′_{κ} = 𝔠_{κ}, and 𝔠′_{κ} = −κ𝔰_{κ}. Theorem 7.2. Let (M^{n}, g) be a semi-Riemannian manifold and let κ ∈ ℝ. Let A be a field of symmetric endomorphisms of T M satisfying

(27)  (∇_{X}^{M}A)Y = (∇_{Y}^{M}A)X,
R^{M}(X, Y)Z = ⟨A(Y), Z⟩ A(X) − ⟨A(X), Z⟩ A(Y)

+ κ(⟨Y, Z⟩ X − ⟨X, Z⟩ Y) (28)

for all X, Y, Z ∈ T_{p}M, p ∈ M. Define a family of metrics on M by

g_{t}(X, Y) := g((𝔠_{κ}(t) id − 𝔰_{κ}(t)A)^{2}X, Y).

Then the metric dt^{2} + g_{t} on Z = I × M has constant sectional curvature κ on its domain of definition (i. e. for |t| sufficiently small).

Proof. Put R_{κ}^{Z}(X, Y)Z := R^{Z}(X, Y)Z − κ(⟨Y, Z⟩ X − ⟨X, Z⟩ Y). Having constant sectional curvature κ is equivalent to R_{κ}^{Z} ≡ 0. The proof is based on the following lemma:

Lemma 7.3. Let Z = I × M be a generalized cylinder and let κ ∈ ℝ. Assume that g(R_{κ}^{Z}(X, ν)ν, Y) = 0 for all vector fields X and Y on Z, where ν denotes the vector \frac{∂}{∂t}.

(i) If the Weingarten map A of the hypersurface {0} × M of Z satisfies (27), then g(R_{κ}^{Z}(X, Y)Z, ν) = 0 for all vector fields X, Y and Z on Z.

(ii) If, moreover, A also satisfies (28), then R_{κ}^{Z} ≡ 0, i. e. Z has constant sectional curvature κ.

Assume this lemma for a moment. We will check that the metric dt^{2} + g_{t} satisfies the hypothesis of the lemma for g_{t}(X, Y) = g((𝔠_{κ}(t) id − 𝔰_{κ}(t)A)^{2}X, Y). Let W_{t} denote the Weingarten tensor of the hypersurface {t} × M of Z. This gives rise to a tensor field W on Z, vanishing in the direction of ν. From the definition of g_{t} we compute

ġ_{t}(X, Y) = −2g((𝔠_{κ}(t) id − 𝔰_{κ}(t)A))(κ𝔰_{κ}(t) id + 𝔠_{κ}(t)A)X, Y)
= −2g_{t}((𝔠_{κ}(t) id − 𝔰_{κ}(t)A))^{−1}(κ𝔰_{κ}(t) id + 𝔠_{κ}(t)A)X, Y)

hence by (14)

W = (𝔠_{κ}(t) id − 𝔰_{κ}(t)A))^{−1}(κ𝔰_{κ}(t) id + 𝔠_{κ}(t)A).

Moreover,

g̈_{t}(X, Y) = −2g ([κ(𝔠_{κ}(t) id − 𝔰_{κ}(t)A)^{2} − (κ𝔰_{κ}(t) id + 𝔠_{κ}(t)A)^{2}]X, Y ) .

Equation (17) yields

g_{t}(R^{Z}(X, ν)ν, Y) = −\frac{1}{2}g̈_{t}(X, Y) − \frac{1}{2}ġ_{t}(W (X), Y)
= g(κ(𝔠_{κ}(t) id − 𝔰_{κ}(t)A)^{2}X, Y)
= κ g_{t}(X, Y),

thus R^{Z}(X, ν)ν = κ X and hence R_{κ}^{Z}(X, ν)ν = 0. All conditions of the lemma are satisfied and the theorem follows. □

Proof of the lemma. The modified curvature tensor R_{κ}^{Z} has all the symmetries of a curvature tensor including the Bianchi identities.

i) Consider the family of tensors on M defined by K_{t}(X, Y, Z)_{x} := ⟨R_{κ}^{Z}(X, Y)Z, ν⟩_{(t,x)}. Using the second Bianchi identity on Z, together with the fact that ν commutes with vectors on M and the formula W (X) = −∇_{X}^{Z}ν = −∇_{ν}^{Z}X + [ν, X] = −∇_{ν}^{Z}X we see

K̇_{t}(X, Y, Z) = d_{ν} ⟨R_{κ}^{Z}(X, Y)Z, ν⟩
= ⟨(∇_{ν}^{Z}R_{κ}^{Z})(X, Y)Z, ν⟩
− ⟨R_{κ}^{Z}(W (X), Y)Z + R_{κ}^{Z}(X, W (Y))Z + R_{κ}^{Z}(X, Y)W (Z), ν⟩
= ⟨(∇_{X}^{Z}R_{κ}^{Z})(ν, Y)Z, ν⟩ + ⟨(∇_{Y}^{Z}R_{κ}^{Z})(X, ν)Z, ν⟩

+(W^{∗}K_{t})(X, Y, Z) (29)

where W^{∗} denotes the induced action of W as a derivation on tensors. From the assumption in the lemma we conclude

0 = d_{X} ⟨R_{κ}^{Z}(ν, Y)Z, ν⟩
= ⟨(∇_{X}^{Z}R_{κ}^{Z})(ν, Y)Z, ν⟩ + ⟨R_{κ}^{Z}(∇_{X}^{Z}ν, Y)Z, ν⟩ + ⟨R_{κ}^{Z}(ν, ∇_{X}^{Z}Y)Z, ν⟩
+ ⟨R_{κ}^{Z}(ν, Y)∇_{X}^{Z}Z, ν⟩ + ⟨R_{κ}^{Z}(ν, Y)Z, ∇_{X}^{Z}ν⟩
= ⟨(∇_{X}^{Z}R_{κ}^{Z})(ν, Y)Z, ν⟩ − ⟨R_{κ}^{Z}(W (X), Y)Z, ν⟩ + 0
+ 0 − ⟨R_{κ}^{Z}(ν, Y)Z, W (X)⟩

thus

⟨(∇_{X}^{Z}R_{κ}^{Z})(ν, Y)Z, ν⟩ = ⟨R_{κ}^{Z}(W (X), Y)Z, ν⟩ + ⟨R_{κ}^{Z}(ν, Y)Z, W (X)⟩

and similarly

⟨(∇_{Y}^{Z}R_{κ}^{Z})(X, ν)Z, ν⟩ = ⟨R_{κ}^{Z}(X, W (Y))Z, ν⟩ + ⟨R_{κ}^{Z}(X, ν)Z, W (Y)⟩ .

Plugging this into (29) yields

K̇_{t}(X, Y, Z) = ⟨R_{κ}^{Z}(W (X), Y)Z, ν⟩ + ⟨R_{κ}^{Z}(ν, Y)Z, W (X)⟩
+ ⟨R_{κ}^{Z}(X, W (Y))Z, ν⟩ + ⟨R_{κ}^{Z}(X, ν)Z, W (Y)⟩
+(W^{∗}K_{t})(X, Y, Z).

Hence K̇_{t} = F (t)(K_{t}) for some linear endomorphism F of the space of 3-tensors. This is a linear first order ODE for K_{t}. The initial condition K_{0} = 0 follows from (16) because W_{0} = A is a Codazzi tensor. This shows that K_{t} ≡ 0.

ii) Similarly, using the identity ⟨R_{κ}^{Z}(X, Y)Z, ν⟩ ≡ 0 that we just obtained, we see that the family of tensors on M defined by R_{t}(X, Y, Z, V)_{x} := ⟨R_{κ}^{Z}(X, Y)Z, V ⟩_{(t,x)} satisfies a linear ODE. Moreover, (15) implies R_{0} ≡ 0 because W_{0} = A satisfies the Gauss equation. Thus R_{t} ≡ 0 for all t. This proves the lemma. □

Now recall that any semi-Riemannian manifold of constant sectional curvature κ is locally isometric to 𝕄_{κ}^{r,s}. Here 𝕄_{κ}^{r,s} is the model space of constant sectional curvature κ and signature (r, s). If κ = 0, then 𝕄_{0}^{r,s} is semi-Euclidean space ℝ^{n} with the metric g_{r,s} = (dx^{1})^{2} + ⋯ + (dx^{r})^{2} − (dx^{r+1})^{2} − ⋯ − (dx^{n})^{2}. If κ > 0, then 𝕄_{κ}^{r,s} is a pseudosphere, more precisely, it is the semi-Riemannian hypersurface of (ℝ^{n+1}, g_{r+1,s}) defined by ⟨x, x⟩_{r+1,s} = 1/κ and x^{1} > 0 if r = 0. If κ < 0, then 𝕄_{κ}^{r,s} is a pseudohyperbolic space, more precisely, it is the semi-Riemannian hypersurface of (ℝ^{n+1}, g_{r,s+1}) defined by ⟨x, x⟩_{r,s+1} = 1/κ and x^{n+1} > 0 if r = 0. In all cases 𝕄_{κ}^{r,s} is connected and homogeneous. Moreover, 𝕄_{κ}^{r,s} is simpy connected except for 𝕄_{κ}^{1,n−1} if κ > 0 and 𝕄_{κ}^{n−1,1} if κ < 0, compare [8, p. 108 ff].

The local isometry is essentially given by the Riemannian exponential map, see [10, Cor. 2.3.8], and it is uniquely determined by its differential at a point. Applying this to the cylinder constructed in Theorem 7.2 yields the local statement in the fundamental theorem for hypersurfaces for semi-Riemannian manifolds. Corollary 7.4. Let (M^{n}, g) be a semi-Riemannian manifold of signature (r, s) and let κ ∈ ℝ. Let A be a field of symmetric endomorphisms of T M satisfying the equations of Gauss and Codazzi-Mainardi:

(∇_{X}^{M}A)Y = (∇_{Y}^{M}A)X,
R^{M}(X, Y)Z = ⟨A(Y), Z⟩ A(X) − ⟨A(X), Z⟩ A(Y)
+ κ(⟨Y, Z⟩ X − ⟨X, Z⟩ Y)

for all X, Y, Z ∈ T_{p}M, p ∈ M.

Then for every point p ∈ M, for every q ∈ 𝕄_{κ}^{r+1,s}, and for every linear isometric embedding F : T_{p}M → T_{q}𝕄_{κ}^{r+1,s} there exists a neighborhood U of p in M and an isometric embedding f : U → 𝕄_{κ}^{r+1,s} as a semi-Riemannian hypersurface with Weingarten map A, such that f(p) = q and df(p) = F.

Moreover, any two such local embeddings f_{1} and f_{2} must agree in a neighborhood of p if f_{1}(p) = f_{2}(p) =: q and df_{1}(p) = df_{2}(p) : T_{p}M → T_{q}𝕄_{κ}^{r+1,s}.

Now that this local result is established exactly the same proof as in [6, Ch. VII, Thm. 7.2] can be used to show the corresponding global immersion statement in the simply connected case. Corollary 7.5. Let (M^{n}, g) be a simply connected semi-Riemannian manifold of signature (r, s), let κ ∈ ℝ and let A be a field of symmetric endomorphisms of T M satisfying the two equations (27) and (28) above.

Then M can be isometrically immersed as a semi-Riemannian hypersurface into the model space 𝕄_{κ}^{r+1,s} with Weingarten map A. Any two such immersions differ by an isometry of 𝕄_{κ}^{r+1,s}.

8. Generalized Killing spinors

� e now turn our attention to restrictions of spinors to hypersurfaces. Let M^{n} ⊂ Z^{n+1} be a hypersurface of a spin manifold Z admitting a parallel spinor Ψ. If n + 1 is even, we will assume that Ψ lies in Σ^{+}Z. From the discussion in Section 3 we see that the restriction ψ of Ψ to M is actually a spinor on M and

(10) reads

(30)  0 = ∇_{X}^{ΣZ}Ψ = ∇_{X}^{ΣM}ψ − \frac{1}{2}A(X) • ψ

for all X ∈ T M where A is the Weingarten tensor of the submanifold M and “•” denotes Clifford multiplication on M. If ψ is an eigenspinor of the Dirac operator, then A is closely related to the energy-momentum tensor of ψ. More precisely, using (24) one computes

Q_{ψ}(X, Y) = \frac{1}{4} ⟨X, A(Y)⟩ ⟨ψ, ψ⟩

where ⟨ψ, ψ⟩ is constant since ψ is parallel on Z. Spinors satisfying (30) will be called generalized Killing spinors. They are closely related to the so–called T –Killing spinors studied by Friedrich and Kim in [5].

Conversely, given a generalized Killing spinor ψ on a manifold M^{n} with ∇_{X}^{ΣM}ψ− \frac{1}{2}A(X)• ψ, it is natural to ask whether the tensor A can be realized as the Weingarten tensor of some isometric embedding of M in a manifold Z^{n+1} carrying parallel spinors. Morel studied this problem in the case where the tensor A is parallel, see [7].

The next result provides an affirmative answer to the above question, for the case where the energy-momentum tensor of ψ is a Codazzi tensor. Theorem 8.1. Let (M^{n}, g) be a semi-Riemannian spin manifold and let A be a field of symmetric endomorphisms of T M satisfying equation (25) on M. Let ψ be a spinor on (M^{n}, g) satisfying for all X ∈ T M

(31)  ∇_{X}^{ΣM}ψ = \frac{1}{2}A(X) • ψ.

Then the generalized cylinder Z = I × M with the metric dt^{2} + g_{t}, where g_{t}(X, Y) = g((id − tA)^{2}X, Y), and with the spin structure inducing the given one on {0} × M by restriction has a parallel spinor, whose restriction to the leaf {0} × M is just ψ.

Proof. The spinor ψ defines a spinor Ψ on Z by parallel transport along the geodesics ℝ × {x}. More precisely, we define Ψ_{(0,x)} := ψ_{x} via the identification Σ_{x}M ≅ Σ_{(0,x)}Z (resp. Σ_{(0,x)}^{+}Z for n odd) and Ψ_{(t,x)} = τ_{0}^{t}Ψ_{(0,x)}. By construction we have

(32)  ∇_{ν}^{ΣZ}Ψ ≡ 0 and ∇_{X}^{ΣZ}Ψ|_{{0}×M} = 0

for all X ∈ T M.

The explicit form of the metrics g_{t} yields ⟨R^{Z}(X, ν)ν, Y ⟩ = 0 on Z for all X and Y tangent to M as in the proof of Theorem 7.2. Since the Codazzi equation (25) holds Lemma 7.3 (i) yields ⟨R^{Z}(ν, X)Y, Z⟩ = 0 on all of Z. Hence R^{Z}(ν, X) = 0 for all X ∈ T M.

Let X be a fixed arbitrary vector field on M, identified as usual with the vector field (0, X) on Z. Using (32) we get 0 = \frac{1}{2}R^{Z}(ν, X) · Ψ = ∇_{ν}^{ΣZ}∇_{X}^{ΣZ}Ψ, thus showing that the spinor field ∇_{X}^{ΣZ}Ψ is parallel along the geodesics ℝ × {x}. Now (32) shows that this spinor vanishes for t = 0, hence it is zero everywhere on Z. Since X was arbitrary, this shows that Ψ is parallel on Z. □

This theorem generalizes the result from [1] where the case A = λ · id is treated, λ ∈ ℝ, and it is shown that the cone over a manifold with Killing spinors admits parallel spinors, as well as a more recent result by Morel [7] for the case when A is parallel. Nevertheless, the question whether a manifold with a spinor satisfying (31) can be isometrically embedded in a manifold with parallel spinors such that A becomes the Weingarten tensor of the embedding without assuming that A is a Codazzi tensor is left open in the present article.

9. The space of Lorentzian metrics

� n the final section we address the problem of connecting any two semi-Riemannian metrics of signature (r, s) on some manifold M of dimension n = r + s, by a curve g_{t} of semi-Riemannian metrics of the same signature

in a unique and universal manner. The latter requirement reduces this problem

to the purely algebraic issue of finding a universal way of relating any two inner products of signature (r, s) on some real vector space E ≅ ℝ^{n} in the manifold M_{r,s} of all inner products of signature (r, s) on E.

In the positive or negative definite case an obvious candidate is the linear interpolation g_{t} = tg_{1} + (1 − t)g_{0} which, however, cannot be used for other signatures. An alternative solution, which has been considered in the definite case, see e.g. [3], but holds in a formally identical way for all signatures, relies on the geometry of M_{r,s}, as a (semi-Riemannian) symmetric space that we now recall briefly.

For any signature (r, s) the identity component of the general linear group GL^{+}(E) ≅ GL^{+}(n, ℝ) acts transitively on M_{r,s} by

(γ · g)(u, v) = g(γ^{−1}u, γ^{−1}v)

for γ ∈ GL^{+}(E), g ∈ M_{r,s}, and u, v ∈ E. For any chosen g_{0} in M_{r,s}, the isotropy group of g_{0} in GL^{+}(E) is the special orthogonal group SO(g_{0}) relative to g_{0}. Recall that, except in the definite case where SO(g_{0}) is connected, SO(g_{0}) has two connected components. We thus get the identification M_{r,s} = GL^{+}(E)/SO(g_{0}) or, equivalently, M_{r,s} = ℝ^{+} × SL(E)/SO(g_{0}), where ℝ^{+} acts by homotheties, and SL(E) ≅ SL(n, ℝ) denotes the special linear group of elements of determinant 1 in GL^{+}(E). Hence M_{r,s}^{0} := SL(E)/SO(g_{0}) can be regarded as the space of inner products on E of signature (r, s) and with a fixed volume element. Concerning the problem addressed in this section, it is clearly sufficient to restrict our attention to M_{r,s}^{0}.

The homogeneous geometry of M_{r,s}^{0} = SL(E)/SO(g_{0}) can be described as follows. For simplicity, write G := SL(E), H := SO(g_{0}), let 𝔤 be the Lie algebra of G, identified with the Lie algebra of trace-free endomorphisms of E, and let 𝔥 be the Lie algebra of H, identified with the Lie algebra of g_{0}-skewsymmetric endomorphisms. Denote by 𝔪 the orthogonal complement of 𝔥 in 𝔤 with respect to the Killing form of 𝔤, so that 𝔤 = 𝔥 ⊕ 𝔪. Recall that the Killing form of 𝔤 equals the bilinear form a, b ↦ tr(ab), up to a positive universal constant, so that 𝔪 is the space of g_{0}-symmetric elements of 𝔤. Since the Killing form is G-invariant, 𝔪 is stable under the adjoint action of H, making M_{r,s}^{0} a reductive homogeneous space. Moreover, we clearly have the Lie bracket relations [𝔥, 𝔥] ⊂ 𝔥, [𝔥, 𝔪] ⊂ 𝔪, and [𝔪, 𝔪] ⊂ 𝔥 showing that M_{r,s}^{0} is actually a symmetric homogeneous space.

In the positive definite case, M_{n,0}^{0} is a Riemannian symmetric space of noncompact type, hence a Hadamard space. It follows that any two points of M_{n,0}^{0} can be joined by a unique geodesic. If g and g_{0} are any two points of M_{n,0}, then g = g_{0}(A·, ·), for a uniquely defined automorphism A of E, where A is symmetric and positive definite for both g_{0} and g. Then A = exp(a) for a uniquely defined symmetric endomorphism a of E and the unique geodesic connecting g_{0} to g is the curve g_{t} := g_{0}(exp(ta)·, ·) = g_{0}(A^{t}·, ·), for t ∈ [0, 1] where exp : 𝔤 → G denotes the exponential mapping.

In the general case, the restriction of the Killing form to 𝔪 is an H-invariant inner product of signature (\frac{r(r+1)}{2} + \frac{s(s+1)}{2} − 1, rs), making M_{r,s}^{0} a semi-Riemannian symmetric space of this signature.

The fact that M_{r,s}^{0} is symmetric, as a semi-Riemannian homogeneous space, implies that the Levi-Civita connection of the semi-Riemannian metric coincides with the canonical homogeneous connection. In particular, all (semi-Riemannian) geodesics emanating from g_{0} are of the form

exp(tX) · g_{0}

for X ∈ 𝔪 = T_{g_{0}}M_{r,s}.

As a homogeneous semi-Riemannian manifold M_{r,s}^{0} is certainly geodesically complete in the sense that geodesics are defined on all of ℝ, but for (r, s) ≠ (n, 0), (0, n), it is not longer true that any two points can be joined by a geodesic and, if so, there is no guarantee that the geodesic be unique. This will be illustrated firstly in the case that (r, s) = (1, 1), then in the general Lorentzian case when (r, s) = (n − 1, 1).

9.1. The space of Lorentzian inner products in dimension 2. Let E denote an oriented real vector space of dimension 2. We fix a positive generator ω of the real line Λ^{2}E^{∗}, which can be viewed as a symplectic form on E. Now G ≅ SL(2, ℝ), 𝔤 ≅ 𝔰𝔩(2, ℝ) is the Lie algebra of trace-free endomorphisms of E, and M_{1,1}^{0} is the space of all Lorentzian inner products on E, whose volume form with respect to the given orientation is ω. For any chosen point g_{0} ∈ M_{1,1}^{0} we then have M_{1,1}^{0} = SL(2, ℝ)/SO(1, 1). Note that SO(1, 1) has two connected components. The connected component of the identity SO_{0}(1, 1) is isomor-

phic the the additive group ℝ of real numbers via the isomorphism t ↦ (\begin{matrix}cosh t & sinh t \\ sinh t & cosh t \end{matrix}).

The other connected component equals −SO_{0}(1, 1). Differentiation with respect to t shows that the corresponding isotropy Lie algebra 𝔥 is the Lie algebra of 2 × 2-matrices of the

form (\begin{matrix}0 & b \\ b & 0 \end{matrix}), for b ∈ ℝ.

An endomorphism α of E is tracefree if and only if it is “antisymmetric” with respect to ω, i. e. if and only if it satisfies: ω(α·, ·) + ω(·, α·) = 0.

For any g ∈ M_{1,1}^{0} there is one and only one automorphism I_{g} of E such that

(35)  g = ω(·, I_{g}·).

Since g is symmetric I_{g} is trace-free. Its determinant equals −1 because g is Lorentzian, with volume form equal to ω. In particular, I_{g}^{2} = 1. The light cone of g is the union of the two eigenspaces of I_{g}, for the eigenvalues ±1. The latter are generated by v ± I_{g}v respectively, for any nonzero v ∈ E.

Conversely, for any automorphism I of E of trace equal to 0 and of determinant equal to −1, the bilinear form g defined by g = ω(·, I·) is a Lorentzian inner product, with volume form equal to ω and I = I_{g}.

The automorphism I_{g} belongs to the Lie algebra 𝔤, on which G acts by the adjoint representation, and the map g ↦ I_{g} is G-equivariant. Indeed, by definition of G, we have that ω(γ·, γ·) = ω(·, ·) for each γ ∈ G, so that

γ · g = g(γ^{−1}·, γ^{−1}·) = ω(γ^{−1}·, I_{g} γ^{−1}·) = ω(·, γ I_{g}γ^{−1}·).

The map g ↦ I_{g} is then a G-equivariant identification of M_{1,1}^{0} with the adjoint orbit of all elements of 𝔤 of determinant equal to −1.

As a function defined on 𝔤 ≅ ℝ^{3}, the opposite of the determinant is a nondegenerate quadratic form of signature (2, 1), equal to the (suitably normalized) Killing form. We denote the symmetric bilinear form corresponding to − det by ⟨·, ·⟩, i. e. ⟨u, u⟩ = − det(u) = \frac{1}{2} tr(u^{2}). The adjoint orbit is then the pseudosphere 𝕄_{1}^{1,1} of elements u such that ⟨u, u⟩ = 1 in the 3-dimensional Minkowski space (𝔤, ⟨·, ·⟩). The restriction of ⟨·, ·⟩ to 𝕄_{1}^{1,1} makes the latter a G-homogeneous Lorentzian manifold, known as the 2-dimensional de Sitter universe.

The map M_{1,1}^{0} → 𝕄_{1}^{1,1}, g ↦ I_{g}, is a G-equivariant isometry.

Reflection with respect to ⟨·, ·⟩ about a vector subspace is an isometry of (𝔤, ⟨·, ·⟩) and it preserves 𝕄_{1}^{1,1}. Since the fixed point set of an isometry is a totally geodesic submanifold the geodesics of 𝕄_{1}^{1,1} are precisely the intersections of 𝕄_{1}^{1,1} with 2-dimensional vector subspaces E ⊂ 𝔤. There are three types of geodesics: timelike geodesics (hyperbolas) corresponding to Minkowski planes, spacelike geodesics (ellipses) corresponding to spacelike planes, and null geodesics (straight lines) corresponding to degenerate planes (tangent to the light cone).

\begin{aligned}& E \\ 𝕄_{1}^{1,1} \end{aligned} \begin{aligned}& E \\ 𝕄_{1}^{1,1} \end{aligned} \begin{aligned}& E \\ 𝕄_{1}^{1,1} \end{aligned}

Fig. 1

Now let I, I′ be two different points in 𝕄_{1}^{1,1}. If I′ = −I, then each plane E containing I also contains I′. In the timelike or in the null case I′ lies on the other connected component of E ∩ 𝕄_{1}^{1,1}. Thus all spacelike geodesics emanating from I hit I′ = −I, but the timelike and null geodesics emanating from I miss I′ = −I.

If I′ ≠ −I, then I and I′ are linearly independent, so the plane E containing I and I′ is uniquely determined. Thus I′ is hit by the geodesic emanating from I if and only if it does not lie on the “wrong” connected component of E ∩ 𝕄_{1}^{1,1} (in the timelike or null case). In other words, the points on 𝕄_{1}^{1,1} which cannot be reached by a geodesic emanating from I are precisely the ones lying on timelike or null geodesics emanating from −I.

−I I 𝕄_{1}^{1,1}  unreachable points

Fig. 2

The two null geodesics emanating from −I are cut out of 𝕄_{1}^{1,1} by the affine plane {⟨I, I′⟩ = −1}. Thus the points I′ ∈ 𝕄_{1}^{1,1} with ⟨I, I′⟩ < −1 cannot be attained by a geodesic from I.

Similarly, by looking at the affine plane {⟨I, I′⟩ = +1} we see that the points I′ with ⟨I, I′⟩ > 1 are the ones that lie on timelike geodesics emanating from I, the ones with ⟨I, I′⟩ = 1 are the ones that lie on null geodesics emanating from I, and the ones with −1 < ⟨I, I′⟩ < 1 lie on spacelike geodesics emanating from I.

We now retranslate this information back to M_{1,1}^{0}. If g, g′ ∈ M_{1,1}^{0}, then

g′ = g(A·, ·),

with

A = I_{g}^{−1}I_{g′} = I_{g}I_{g′}.

We then have

⟨I_{g}, I_{g′}⟩ = \frac{1}{2}tr A.

Note that A is g- and g′-symmetric and of determinant equal to +1.

By choosing g as a base-point, we conclude that M_{1,1}^{0} can also be identified with the space of all g-symmetric automorphisms of determinant 1 of E. We summarize: Proposition 9.1. The space M_{1,1}^{0} of Lorentzian inner products on a 2-dimensional real vector space that have a fixed volume element carries a natural Lorentzian metric making it SL(2, ℝ)-equivariantly isometric to the 2-dimensional de Sitter universe. For g, g′ ∈ M_{1,1}^{0} there is a unique endomorphism A such that g′ = g(A·, ·). Moreover, the following holds:

• If tr(A) > 2, then there is a unique geodesic in M_{1,1}^{0} joining g and g′. This geodesic is timelike.

• If tr(A) = 2, then there is a unique geodesic in M_{1,1}^{0} joining g and g′. This geodesic is null.

• If −2 < tr(A) < 2, then there is a unique geodesic in M_{1,1}^{0} joining g and g′. This geodesic is spacelike.

• If tr(A) < −2, then there is no geodesic in M_{1,1}^{0} joining g and g′.

• If tr(A) = −2 and g ≠ −g′, then there is no geodesic in M_{1,1}^{0} joining g and g′.

• If tr(A) = −2 and g = −g′, then all spacelike geodesics in M_{1,1}^{0} emanating from g pass through g′ while the timelike and null geodesics in M_{1,1}^{0} emanating from g miss g′.

This proposition shows that given two Lorentzian metrics on a 2-dimensional manifold we can construct a canonical 1-parameter family of Lorentzian metrics joining them only if the endomorphism field A relating the two metrics satisfies tr(A) > −2. A restriction like this does not come as a surprise because there are pairs of Lorentzian metrics e. g. on the 2-torus which cannot even be joined by any continuous curve of Lorentzian metrics.

9.2. The space of Lorentzian inner products in higher dimensions. We now consider the manifold M_{n−1,1} = ℝ^{+} × M_{n−1,1}^{0} of all Lorentzian inner products of signature (n − 1, 1) on some n-dimensional real vector space E.

As observed before the manifold M_{n−1,1}^{0} is a symmetric semi-Riemannian space of sig-

nature (\frac{n(n−1)}{2}, n − 1) and the geodesics emanating from any chosen base-point g_{0} are of the form exp(tX) · g_{0}, where X belongs to the space 𝔪 of trace-free g_{0}-symmetric endomorphisms of E, 𝔪 being naturally identified with the tangent space T_{g_{0}}M_{n−1,1}^{0}.

The goal of this section is to determine the set of elements g ∈ M_{n−1,1} which can be joined from g_{0} by a geodesic in M_{n−1,1}, and whether or not this geodesic is unique. This has just been done in detail in the case that n = 2 and, as we shall see, the general case can essentially be reduced to the 2-dimensional case. More precisely, we have Proposition 9.2. Let g_{0} and g be two distinct points in M_{n−1,1}. Then there is the following alternative: Either

(i) E splits as

E = E_{1,1} ⊕ E_{n−2,0},

where the sum is orthogonal, E_{1,1} is of signature (1, 1), E_{n−2,0} is of signature (n − 2, 0) for g_{0} and g. Both g_{0} and g belong to the corresponding totally geodesic submanifold M_{1,1} × M_{n−2,0} ⊂ M_{n−1,1}. Thus the issue of the existence and uniqueness of geodesics connecting g_{0} to g is reduced to the same issue for the 2-dimensional Lorentzian metrics g_{0}_{|E_{1,1}} and g_{|E_{1,1}} in M_{1,1} as described in Proposition 9.1, or

(ii) E splits as

E = E_{2,1} ⊕ E_{n−3,0},

where the sum is orthogonal, E_{2,1} is of signature (2, 1), E_{n−3,0} is of signature (n − 3, 0) for g_{0} and g. Both g_{0} and g belong to the corresponding totally geodesic submanifold M_{2,1} × M_{n−3,0} ⊂ M_{n−1,1}. The 3-dimensional Lorentzian metrics g_{0}_{|E_{2,1}} and g_{|E_{2,1}} are related by g_{|E_{2,1}} = g_{0}_{|E_{2,1}}(B·, ·), where B is an automorphism of E_{2,1} of the form k(id + x), where k is a positive real number and x is an endomorphism of E_{2,1} satisfying x^{3} = 0 but x^{2} ≠ 0. Thus g_{0} and g are connected by a unique geodesic whose E_{2,1}-part is of the form

g_{t}_{|E_{2,1}} = g_{0}_{|E_{2,1}}(B_{t}·, ·),

with B_{t} = k^{t} exp(t(x − \frac{1}{2}x^{2})) = k^{t} (1 + tx + \frac{t(t−1)}{2}x^{2}).

The rest of the paper is devoted to the proof of Proposition 9.2.

Recall that for any g and g_{0} in M_{n−1,1}, there exists a uniquely defined automorphism A of E — with detA > 0 — such that g = g_{0}(A·, ·): A = (γ^{−1})^{∗}γ^{−1}, for any γ ∈ GL(E) such that g = γ · g_{0} and A is symmetric relative to both g and g_{0}. Then g_{0} can be joined with g by a geodesic in M_{n−1,1} if and only if A is of the form A = exp(a), for some g_{0}-symmetric endomorphism a of E, and the corresponding geodesic is then the curve g_{t} := g_{0}(exp(ta)·, ·) for t ∈ [0, 1].

The proof of Proposition 9.2 requires the spectral analysis of A. For this purpose it is convenient to introduce a positive definite Euclidean inner product (·, ·) on E such that g_{0} = (I·, ·) where I is of the form

(43)  I = id − 2(u, ·)u,

for some element u ∈ E such that |u|^{2} = 1. Here, and henceforth, | · | denotes the norm with respect to (·, ·). For g_{0} the vector u is timelike with g_{0}(u, u) = −1. Conversely, any such u determines a Euclidean inner product as above.

Then g = g_{0}(A·, ·) can be written as g = (S·, ·) for a uniquely defined (·, ·)-symmetric automorphism S of E with exactly n − 1 positive and 1 negative eigenvalues.

Conversely, for any such automorphism S, the inner product g = (S·, ·) belongs to M_{n−1,1} with

A = I^{−1}S = IS.

The spectral decomposition of S reads

S = λ_{0}Π_{0} + ⨁_{r=1}^{ℓ} λ_{j}Π_{r},

with λ_{0} < 0 < λ_{1} < … λ_{ℓ}, where Π_{j} denotes the (·, ·)-orthogonal projection onto the d_{j}-dimensional eigenspace E_{j} of S corresponding to the eigenvalue λ_{j}, j = 0, 1, …, ℓ. Note that d_{0} = 1.

Via the decomposition E = E_{0} ⊕ ⨁_{r=1}^{ℓ} E_{r} the unit vector u appearing in (43) splits as

u = u_{0} + u_{1} + … + u_{ℓ}.

We denote by Δ the subset of j ∈ {0, 1, …, ℓ} such that u_{j} ≠ 0, and by m the cardinality of Δ. For each j ∈ Δ such that d_{j} > 1 we denote by Ẽ_{j} the (·, ·)-orthogonal complement of u_{j} in E_{j}. Let Ẽ be the subspace of E defined by

(47)  Ẽ := ⨁_{j∈Δ,d_{j}>1} Ẽ_{j} ⊕ ⨁_{j∉Δ} E_{j},

and W the m-dimensional subspace of E defined by

(48)  W = ⨁_{j∈Δ} ℝ u_{j}

so that

E = Ẽ ⊕ W.

Both Ẽ and W are left invariant by A, I, and S. The sum is orthogonal with respect to (·, ·), g_{0}, and g.

Note that if 0 ∉ Δ, i. e. if u_{0} = 0, then Ẽ is of signature (n − m − 1, 1) and W is of signature (m, 0), whereas, if 0 ∈ Δ, i. e. if u_{0} ≠ 0, W is of signature (m − 1, 1) and Ẽ is of signature (n − m, 0) for g (but W is always of signature (m − 1, 1) for g_{0}, as Ẽ is orthogonal to u).

Since Ẽ is orthogonal to u, I_{|Ẽ} = id and A_{|Ẽ} = S_{|Ẽ}. In particular, A_{|Ẽ} is symmetric for g_{0}, g and (·, ·) and its spectral decomposition coincides with the one of S_{|Ẽ}, given by (47), with eigenvalues λ_{j} for each j ∉ Δ and each j ∈ Δ with d_{j} > 1.

The spectral study of A is then reduced to the spectral study of A_{|W} and the latter is summarized by the following lemma. Lemma 9.3. (i) The characteristic polynomial P of A_{|W} defined by P (t) = det(t id − A_{|W}) is given by

(50)  P (t) = ∏_{j∈Δ}(t − λ_{j}) + 2 ∑_{j∈Δ} λ_{j}|u_{j}|^{2} ∏_{k∈Δ\{j}}(t − λ_{k}).

In particular, the roots of P are all distinct from the λ_{j}, j ∈ Δ.

(ii) For each real root μ of P the corresponding eigenspace is the one-dimensional vector space generated by the element v_{μ} ∈ W defined by

(51)  v_{μ} = ∑_{j∈Δ} \frac{u_{j}}{μ − λ_{j}} .

Moreover,

(52)  g(v_{μ}, v_{μ}) = μ g_{0}(v_{μ}, v_{μ}) = −\frac{1}{2} \frac{P′(μ)}{Q(μ)}

where Q denotes the polynomial defined by Q(t) = ∏_{j∈Δ}(t − λ_{j}). In particular, v_{μ} is a null-vector — for both g and g_{0} — if and only if μ is a multiple root of P.

Proof. By definition, any v ∈ W is of the form v = ∑_{j∈Δ} y_{j}u_{j}, for real numbers y_{1}, …, y_{m}, so that

Av = ISv = ∑_{j∈Δ}(λ_{j}y_{j} − 2(Su, v)) u_{j}.

Note that v is an eigenvector of A_{|W} for some eigenvalue μ if and only if

(54)  (μ − λ_{j}) y_{j} = −2(Su, v),

for each j ∈ Δ. It is easily checked that (Su, v) cannot be equal to 0 if v ≠ 0. Indeed, suppose for a contradiction that v satisfies (54) with (Su, v) = 0 and v ≠ 0. Since v ≠ 0, one of the y_{j}, say y_{1}, is nonzero, so that μ = λ_{1}. This implies μ ≠ λ_{j}, for j ≠ 1, as the λ_{j} are pairwise distinct. It follows that y_{j} = 0 for all j ≠ 1, so that v = y_{1}u_{1}. Then (Su, v) = λ_{1}y_{1}|u_{1}|^{2} ≠ 0 as y_{1} ≠ 0, a contradiction.

In particular, this shows μ ≠ λ_{j} for each j ∈ Δ so that we can write

(55)  v = −2(Su, v) ∑_{j∈Δ} \frac{u_{j}}{μ − λ_{j}} .

Moreover, by computing (Su, v) = (Sv, u) from (55), we get

(56)  ∑_{j∈Δ} \frac{λ_{j}|u_{j}|^{2}}{μ − λ_{j}} = −\frac{1}{2}.

It follows that each eigenvalue of A_{|W} is a root of the polynomial P defined by (50). Since P is monic and of degree m, it must coincide with the characteristic polynomial of A_{|W}. We readily see from (50) that the roots of P are distinct from the λ_{j} (recall that the latter are pairwise distinct). From (55) we immediately see that the eigenspace corresponding to μ is generated by the vector v_{μ} defined by (51).

Conversely, for each root μ of P the vector v_{μ} defined by (51) is certainly an eigenvector of A_{|W} for the eigenvalue μ.

Since the roots of P are distinct from the λ_{j}, P can also be expressed by

(57)  \frac{P (t)}{Q(t)} = 1 + 2 ∑_{j∈Δ} \frac{λ_{j}|u_{j}|^{2}}{t − λ_{j}} ,

where we put Q(t) := ∏_{j∈Δ}(t − λ_{j}). Differentiating (57) at t = μ, we get (52). It follows that v_{μ} is a null vector if and only if P′(μ) = 0, meaning that μ is a multiple root. □

For further use, we need more information about the sign of the characteristic polynomial P at t = λ_{j}, j ∈ Δ, and at t = 0. In the sequel, we use the notation P (t_{0}) ≡ (−1)^{r}, for some integer r, to mean that P has the sign of (−1)^{r} — in particular is not zero — at t = t_{0}. Lemma 9.4. (i) If 0 ∉ Δ, we re-label the λ_{j} so that Δ = {1, …, m}, and 0 < λ_{1} < … < λ_{m}. We then have:

\begin{matrix}P (−∞) ≡ P (λ_{0}) ≡ (−1)^{m}, \\ P (0) ≡ (−1)^{m−1}, \\ P (λ_{j}) ≡ (−1)^{m−j}, j = 1, …, m. \end{matrix} ; (58)

In particular, P has then exactly m distinct real roots μ_{0} < 0 < μ_{1} < … < μ_{m−1}, with μ_{0} ∈ (λ_{0}, 0) and μ_{i} ∈ (λ_{i}, λ_{i+1}), for i = 1, …, m − 1.

(ii) If 0 ∈ Δ, we re-label the λ_{j} so that Δ = {0, 1, …, m − 1} and λ_{0} < 0 < λ_{1} < … < λ_{m−1}.We then have

\begin{matrix}P (−∞) ≡ P (λ_{0}) ≡ P (0) ≡ (−1)^{m}, \\ P (λ_{j}) ≡ (−1)^{m−j−1}, j = 1, …, m − 1. \end{matrix} ; (59)

In particular, P has then at least (m − 2) distinct real roots 0 < μ_{1} < … < μ_{m−2}, with μ_{i} ∈ (λ_{i}, λ_{i+1}), for i = 1, …, m − 2.

Proof. Easy consequence of (50). □

We now consider the two cases when 0 does or does not belong to Δ.

Case 1: 0 ∉ Δ.

According to Lemma 9.4 (i), A_{|W} is diagonalizable (over ℝ) with one negative eigenvalue μ_{0} and m − 1 distinct positive eigenvalues. Moreover, we easily see from (52) that the m corresponding eigenvectors v_{μ}, defined by (51), are all spacelike. On the other hand, A_{|Ẽ} is also diagonalizable with one negative eigenvalue, namely λ_{0} — whose eigenspace is E_{0} — and n − m − 1 positive eigenvalues. Denote by E_{1,1} the direct sum of E_{0} and the (one-dimensional) eigenspace of μ_{0}, and by E_{n−2,0} the orthogonal complement of E_{1,1} for g or g_{0}. Then, both g and g_{0} are of signature (1, 1) on E_{1,1} and positive definite on E_{n−2,0}. Accordingly, A splits as the sum of two operators A = A_{1,1} ⊕A_{n−2,0}, where A_{1,1} acts trivially on E_{n−2,0} and is diagonalizable, with negative eigenvalues on E_{1,1}, whereas A_{n−2,0} acts trivially on E_{1,1} and is positive definite, as well as g_{0}- and g-symmetric on E_{n−2,0}. This can be interpreted as follows. Denote by M_{1,1} the space of Lorentzian inner products of E_{1,1}, by M_{n−2,0} the space of positive definite inner products of E_{n−2,0}. Then the product M_{1,1} × M_{n−2,0} is naturally embedded as a totally geodesic submanifold of M_{n−1,1} and both g = g_{|E_{1,1}} ⊕ g_{|E_{n−2,0}} and g_{0} = g_{0}_{|E_{1,1}} ⊕ g_{0}_{|E_{n−2,0}} belong to it. In M_{n−2,0} any two elements, in particular g_{|E_{n−2,0}} and g_{0}_{|E_{n−2,0}}, are joined by a unique geodesic. The situation concerning M_{1,1} has been explored in detail in the first part of this section. In the present case, g_{|E_{1,1}} and g_{0}_{|E_{1,1}} are related by the automorphism A_{|E_{1,1}} which is diagonalizable with distinct negative eigenvalues, so that g_{|E_{1,1}} and g_{0}_{|E_{1,1}} cannot be linked by a geodesic.

Case 2: 0 ∈ Δ.

According to Lemma 9.4 (ii), there exist at least m−2 distinct positive eigenvalues of A_{|W}, namely 0 < μ_{1} < … < μ_{m−2}. Then, either these eigenvalues are all simple roots of P, or one of them — and only one — is a triple root. The case that two of them are double roots is impossible since, according to Lemma 9.3 (ii), the corresponding eigenvectors defined by (51) would then form an orthogonal pair of nonzero null vectors in the Lorentzian space (E, g).

In the case when all μ_{i} are simple roots, we easily check by using (52) that the corresponding eigenvectors are all spacelike. Denote by E_{n−2,0} the direct sum of the corresponding eigenspaces and Ẽ, and by E_{1,1} ⊂ W the orthogonal complement of E_{n−2,0} for g or g_{0}. Then, both g and g_{0} are positive definite on E_{n−2,0} and of signature (1, 1) on E_{1,1}. The situation is then quite similar to the previous one, except that all cases considered in Section 9.1 for M_{1,1} may now happen, depending on whether the missing two roots of P are complex conjugate, both positive (equal or distinct) or both negative (equal or distinct).

It remains to consider the case that one of the μ_{i}, say μ_{j} := k > 0, is a triple root of P. Then, according to Lemma 9.3 (iii), the corresponding eigenvector v_{μ_{j}} is a null vector. Again, it is easily checked that the v_{μ_{i}}, for i ≠ j, are all spacelike. Denote by E_{n−3,0} the direct sum of the eigenspaces corresponding to the μ_{i}, i ≠ j, and E^{0}, and by E_{2,1} ⊂ W the orthogonal complement of E_{n−3,0} for g or g_{0}. Then, both g and g_{0} are positive definite on E_{n−3,0} and of signature (2, 1) on E_{2,1}. It follows that g and g_{0} both belong to a same totally geodesic subspace M_{2,1} × M_{n−3,0}. Moreover, the restriction of A to E_{2,1}, which relates g|_{E_{2,1}} and g_{0}_{|E_{2,1}}, is of the form k(id + x), where x is nilpotent and regular (this is because μ_{j} has no other eigenvector than v_{μ_{j}}). Now, id+x is the exponential of id+x− \frac{x^{2}}{2} , which is certainly symmetric for both g_{0} and g (since x = (id + x) − id is symmetric) and is the only symmetric “logarithm” of id + x. We thus get a unique (null) geodesic between g_{0}_{|E_{2,1}} and g_{|E_{2,1}} in M_{2,1}, hence also between g_{0} and g in M_{n,1}.

This completes the proof of Proposition 9.2. □

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GENERALIZED CYLINDERS IN SEMI-RIEMANNIAN AND SPIN GEOMETRY 29

Universität Hamburg, FB Mathematik, Bundesstr. 55, 20146 Hamburg, Germany

École Polytechnique, Centre de Mathématiques, 91128 Palaiseau Cedex, France

École Polytechnique, Centre de Mathématiques, 91128 Palaiseau Cedex, France

E-mail address: baer@math.uni-hamburg.de
E-mail address: pg@math.polytechnique.fr
E-mail address: am@math.polytechnique.fr