PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 103(117) (2018), 77–89 DOI: https://doi.org/10.2298/PIM1817077H

SECTIONAL CURVATURE IN 4-DIMENSIONAL MANIFOLDS OF NEUTRAL SIGNATURE

Graham Hall

Abstract. We consider the sectional curvature function on a 4-dimensional manifold admitting a metric g of neutral signature, (+, +, −, −) together with a review of the situation for the other two signatures. The main results of the paper are: first, that if the sectional curvature function is not a constant function at any m ∈ M (actually a slightly weaker assumption is made), the conformal class of g is always uniquely determined and in almost all cases g is uniquely determined on M, second, a study of the special cases when this latter uniqueness does not hold, third, the construction of the possible metrics in this latter case, fourth, some remarks on sectional curvature preserving vector fields and finally the complete solution when (M, g) is Ricci flat.

1. Introduction

The sectional curvature function has been studied for metrics of positive definite and Lorentz signature and a brief summary of this work which is relevant to the present article will be given at the beginning of Section 2. The purpose of this paper is to consider the case when g has neutral signature.

To settle notation, g, unless otherwise stated, is a smooth metric of neutral signature (+, +, −, −) on a 4-dimensional, smooth, connected manifold M with Levi-Civita connection ∇, sometimes written, collectively, as (M, g). For m ∈ M, T_{m}M denotes the tangent space to M at m, B_{m} denotes the 6-dimensional vector space of 2-forms (here referred to as bivectors) at m and ∗ denotes the Hodge duality (linear) operator on B_{m}. Let ∼ denote the equivalence relation on nonzero bivectors at m given by B_{1} ∼ B_{2} ⇔ B_{1} = kB_{2} for k ∈ ℝ and let the resulting equivalence classes of projective bivectors be denoted by P B_{m}. Topologically, B_{m} and P B_{m} are ℝ^{6} and P ℝ^{5}. The matrix rank of any nonzero member F ∈ B_{m} is an even number and if it is 2, F is called simple and may be written in components as F^{ab} = p^{a}q^{b} − q^{a}p^{b} for p, q ∈ T_{m}M (sometimes abbreviated to p ∧ q). The 2dimensional subspace (2-space) of T_{m}M spanned by p and q is uniquely determined by F and called the blade of F. The set of all simple members of B_{m} is denoted by SB_{m} and one easily passes to the subset (projective simple bivectors) of P B_{m} denoted by P SB_{m} which is in bijective correspondence with the collection of blades of members of SB_{m}. If u, v ∈ T_{m}M the inner product g(m)(u, v) is denoted by u · v and 0 ≠ u ∈ T_{m} is called spacelike, respectively timelike or null, if u · u > 0, (respectively < 0 or = 0). A 2-space W of T_{m}M is called spacelike if each nonzero member of W is spacelike, or each nonzero member of W is timelike, timelike if W contains exactly two, null 1-dimensional subspaces (null directions), null if V contains exactly one null direction and totally null if each nonzero member of V is null. Thus a totally null 2-space consists, apart from the zero vector, of null vectors any two of which are orthogonal. This classification of 2-spaces is mutually exclusive and exhaustive.

2010 Mathematics Subject Classification: Primary 53A07; Secondary 53A35.

Key words and phrases: sectional curvature, 4-dimensions, neutral signature.

The set of all spacelike (respectively, timelike) 2-spaces at m is denoted by S_{m} (respectively, T_{m}) and the combined set of all null and totally null 2-spaces at m is denoted by N_{m}. A member F ∈ SB_{m} is called spacelike, (respectively timelike, null or totally null) if its blade is of that type and F is then spacelike if F^{ab}F_{ab} > 0, timelike if F^{ab}F_{ab} < 0 and null or totally null if F^{ab}F_{ab} = 0. If one defines an orthonormal basis x, y, s, t of T_{m}M by x · x = y · y = −s · s = −t · t = 1 and its associated null basis (of null vectors) l, n, L, N where \sqrt{2}l = x + t, \sqrt{2}n = x − t, \sqrt{2}L = y + s and \sqrt{2}N = y − s so that l.n = L.N = 1 (and all other such inner products are zero) then, for example, x ∧ y and s ∧ t are spacelike, x ∧ t (= −l ∧ n) is a timelike, l ∧ y and l ∧ s are null and l ∧ N and n ∧ L are totally null 2-spaces at m. One may always extend an orthonormal basis at m to a smooth orthonormal basis on some open neighbourhood of m and hence, as above, extend a null basis at m to a smooth null basis in this neighbourhood. The manifold theory associated with the above bivector (and related) sets is discussed in [12] and one may identify P SB_{m} with the 4-dimensional, compact, connected Grassmann manifold G_{m} of all (real) 2-spaces at m and define its open submanifold \overline{G}_{m} ≡ G_{m} ∖ N_{m} which is the disjoint union of the two open submanifolds S_{m} and T_{m}. In the usual manifold topology on G_{m} each member of N_{m} is a limit point of S_{m} and of T_{m}.

Now define the Lie algebras (under matrix commutation) S^{+}_{m} ≡ {A ∈ B_{m} :

A^{∗} = A} and S^{−}_{m} ≡ {A ∈ B_{m} : A^{∗} = −A} which are Lie-isomorphic to each other and to o(1, 2). It is known that the Lie algebra B_{m} is the Lie algebra product

B_{m} ≡ S^{+}_{m} ⊕ S^{−}_{m} and that the only simple members of S^{+}_{m} and S^{−}_{m} are totally null

[22] and conversely any totally null bivector lies in S^{+}_{m} or S^{−}_{m}. Further, the blades

of two totally null bivectors, one of which is in S^{+}_{m} and the other in S^{−}_{m}, intersect

in a unique null direction whereas if they are both in S^{+}_{m} or both in S^{−}_{m} their intersection is trivial. It is also useful to note that if F ∈ B_{m} is null its unique null direction is orthogonal to all other members of its blade and that it may be written

as F = F^{+} + F^{−} for unique, nonzero, totally null members F^{+} ∈ S^{+}_{m} and F^{−} ∈ S^{−}_{m} and

where the blades of F^{+} and F^{−} intersect in the unique null direction in the blade of

F. If P ∈ S^{+}_{m} and Q ∈ S^{−}_{m}, P^{ab}Q_{ab} = 0.

The Riemann tensor Riem with components R^{a}_{bcd}, its associated Ricci tensor Ricc with components R_{ab} ≡ R^{c}_{acb}, the Ricci scalar R ≡ R_{ab}g^{ab} and the corresponding Weyl conformal tensor C with components C^{a}_{bcd} are related at each m ∈ M by

(1.1)  R_{abcd} = C_{abcd} + E_{abcd} + \frac{R}{6}G_{abcd}

where G is the tensor with components G_{abcd} = \frac{1}{2}[g_{ac}g_{bd} − g_{ad}g_{bc}],

(1.2)  E_{abcd} = \frac{1}{2}[R̃_{ac} g_{bd} − R̃_{ad}g_{bc} + R̃_{bd}g_{ac} − R̃_{bc}g_{ad}]

and where the tracefree Ricci tensor R̃icc has been introduced with components R̃_{ab} = R_{ab} − \frac{R}{4} g_{ab}. Then E(m) = 0 if and only if R̃icc(m) = 0, that is, Ricc = \frac{R}{4} g at m, which is equivalent to the Einstein space condition at m.

The Weyl conformal tensor at m may be classified algebraically [8] in a manner similar to that given by Petrov in the Lorentz case [16] (the Petrov classification). The details of this are not needed except for the concept of a principal null direction of C at m. A null direction spanned by a null l ∈ T_{m}M is called a principal null direction for C(m) ≠ 0 if C_{abcd}l^{a}l^{c} = l_{b}p_{d} + p_{b}l_{d} for some 1-form p at m. Such null directions are important in this classification and have the property (which is useful later) that if k_{0} ∈ T_{m}M spans a principal null direction of C(m) then the null directions spanned by those null vectors in some open neighbourhood of k_{0} in the topology of T_{m}M cannot all be principal null directions of C(m) since they force the contradiction C(m) = 0. This result is trivially true in the Lorentz case since, for C(m) ≠ 0, only finitely many such directions may exist, but less obviously true in the present case since it is possible for infinitely many such directions to exist in neutral signature even if C(m) ≠ 0 [8].

2. The Sectional Curvature Function on (M,g)

The sectional curvature function at m is a real-valued function defined on the open submanifold \overline{G}_{m} of G_{m}, σ_{m} : \overline{G}_{m} → ℝ and given, for a representative member F of SB_{m}, by

(2.1)  σ_{m}(F) ≡ \frac{R_{abcd}F^{ab}F^{cd}}{2G_{abcd}F^{ab}F^{cd}} = \frac{R_{abcd}F^{ab}F^{cd}}{2F^{ab}F_{ab}}

It is clear that this definition is independent of the representative member F used to fix the member of \overline{G}_{m}. (For the function σ_{m} the identification of P SB_{m} with G_{m}, through a representative member of SB_{m}, will always be made.) Now a simple bivector F is null or totally null if and only if G_{abcd}F^{ab}F^{cd}(= F^{ab}F_{ab}) = 0 and so such members of G_{m} must be excluded from the domain of σ_{m}. The interpretation of σ_{m}(F) for F ∈ \overline{G}_{m} is as follows; there exists a neighbourhood U of m such that the geodesics of ∇ starting from m with initial directions in the blade of F generate a 2-dimensional submanifold M′ of U (and hence of M) such that M′ has an induced metric h from g and σ_{m}(F) is then the Gauss curvature of M′ with respect to h at m. It seems that sectional curvature is essentially what Riemann had in mind in his famous address on curvature [18]. If σ_{m} is a constant function on \overline{G}_{m}, (1.1) holds at m with C = E = 0 and σ_{m} is, in this case, trivially continuously extendible to a (constant) function on G_{m}.

If g is positive definite σ_{m} is defined on the whole of G_{m} for each m and provided it is not a constant function on G_{m} for any m in some open dense subset of M, the collection of functions σ_{m} at each m ∈ M uniquely determines the metric on M from which it came (and this applies for all dimensions ⩾ 4) [15]. For Lorentz signature, all 2-spaces are either spacelike, timelike or null, as defined above, (with totally null ones impossible) and thus σ_{m} is only immediately defined on the set of non-null 2-spaces \overline{G_{m}}. If, however, σ_{m} is continuously extendible to a single null 2-space (in the usual manifold topology on G_{m}), it is a constant function on \overline{G_{m}} and hence (trivially) continuously extendible to a constant function on the whole of G_{m} [6, 7, 13] and the components R_{abcd} are proportional to G_{abcd} at m. [In fact, to achieve the constancy of σ_{m} on \overline{G}_{m} it is sufficient to achieve it on S_{m} or on T_{m}, as is easily checked (cf [5]).] Again in this case if σ_{m} is not a constant function on \overline{G_{m}} for any m ∈ M the collection of functions σ_{m} at each m ∈ M uniquely determines each set N_{m} = G_{m} ∖ \overline{G_{m}} and hence, as is easily shown, the null cone at each m. In fact it uniquely determines the metric on M from which it came except in certain very special circumstances [6, 19, 7]. Further, if g is Ricci-flat and nonflat (in the sense that Riem does not vanish over any nonempty open subset of M) the functions σ_{m} for each m ∈ M uniquely determine the metric on M which gave rise to them in all cases [6, 7].

From now on, unless specified to the contrary, g has neutral signature.

Lemma 1. (i) If σ_{m} is a constant function on S_{m} or on T_{m} it is a constant function on \overline{G_{m}} and hence continuously extendible to a constant function on G_{m}.

(ii) If k_{0} ∈ T_{m}M is null and R_{ab}k^{a}k^{b} = 0 for each null k in some neighbourhood of k_{0} ∈ T_{m}M, (M, g) satisfies the Einstein space condition R_{ab} = \frac{R}{4} g_{ab} at m.

(iii) Let F_{0} ∈ N_{m} and suppose R_{abcd}F^{ab}F^{cd} = 0 for each F ∈ U ∩ N_{m} for some open neighbourhood U of F_{0} in G_{m}. Then the constant curvature condition R_{abcd} = \frac{R}{6} G_{abcd} holds at m.

Proof. (i) The proof is straightforward. Suppose σ_{m} is a constant function on S_{m} with value d. In the usual basis let Q = x ∧ t be timelike and construct the

spacelike members P^{α} ≡ x ∧ (t + αy) for α > 1. Then define the tensor H_{abcd} =

R_{abcd} −2dG_{abcd} so that H_{abcd} P^{ab}^{α}P^{cd}^{α} = 0 for each α. This is a polynomial in α and vanishes for all α > 1. It follows that H_{abcd}Q^{ab}Q^{cd} = 0 and hence that σ_{m}(Q) = d. The proof when σ_{m} is a constant function on T_{m} is similar.

(ii) The set of null directions at m is a 2-dimensional, smooth submanifold of P ℝ^{3}. Using the null basis l, n, L, N for T_{m}M given in Section 1 and choosing projective coordinates y, z in this submanifold about the null direction spanned by k_{0}(= l) = (1, 0, 0, 0), so that k = (1, −yz, y, z) for suitable y, z ∈ ℝ, the condition R_{ab}k^{a}k^{b} = 0 becomes a polynomial relation P (y, z) = 0 on an open subset of ℝ^{2} and hence all polynomial coefficients are zero. So R_{01} = R_{23} with all other components of Ricc vanishing at m and the result follows from the completeness relation g_{ab} = l_{a}n_{b} + n_{a}l_{b} + L_{a}N_{b} + N_{a}L_{b}.

(iii) Suppose F_{0} is null in the given condition for (iii). So in a null basis at m chosen so that, say, F_{0} = l ∧ y, one has R_{abcd}l^{a}y^{b}l^{c}y^{d} = 0. Defining a symmetric tensor T at m by T_{ab} = R_{cadb}l^{c}l^{d} and considering the members l ∧ (y + μL) and l ∧ (y + νN) with μ, ν ∈ ℝ sufficiently small that these members are in U ∩ N_{m}, one gets T (y, y)(≡ T_{ab}y^{a}y^{b}) = T (y, L) = T (L, L) = T (y, N) = T (N, N) = 0 and, since T_{ab}l^{b} = 0, also T (l, l) = T (l, n) = T (l, L) = T (l, N) = 0, which imply T (L, N) = 0. It follows that T_{ab} = al_{a}l_{b} + b(l_{a}L_{b} + L_{a}l_{b}) + c(l_{a}N_{b} + N_{a}l_{b}) and so g^{ab}T_{ab} = 0, that is, R_{ab}l^{a}l^{b} = 0 at m from the definition of T. Since this last equation then holds for each null vector in some neighbourhood of l in T_{m}M it follows from part (ii) that the Einstein space condition holds at m. Then the remarks following (1.2) show that E(m) = 0 and so from (1.1) and the above form for T each null direction in some open neighbourhood of that spanned by l is a principal null direction for C(m) as given at the end of Section 1 [8] with, allowing for an abuse of notation, the 1-form p a linear combination of l, L and N. It follows that C(m) = 0. If F_{0} is totally null, say F_{0} = l ∧ N in the null basis, a similarly argument and result arise by considering the bivectors l ∧ (N + λL) for λ ∈ ℝ sufficiently small. The result now follows and, for dimension 4, strengthens a result in [2]. □

Lemma 2. For (M, g), if F and F′ are bivector representatives of two distinct members of N_{m}, the bivector F + λF′, for some 0 ≠ λ ∈ ℝ, (is simple and) also represents a member of N_{m} if and only if the 2-spaces represented by F and F′ intersect in a null direction (and then the bivector F + λF′ represents a member of N_{m} for all λ ∈ ℝ). Thus the subset N_{m} ⊂ G_{m} may be used to fix the collection of null vectors of g at m.

Proof. Let F, F′ represent distinct members of N_{m}. Then F and F′ are independent and simple and if their blades intersect in a null direction, say k, F = k ∧ u and G = k ∧ v for u, v ∈ T_{m}M. Then whether F and F′ are null or totally null, k.u = k.v = 0 and it easily follows that F + λF′ represents a member of N_{m} for all λ ∈ ℝ. Conversely, write F = p ∧ q, F′ = r ∧ s and F + λF′ = e ∧ f for p, q, r, s, e, f ∈ T_{m}M, for some 0 ≠ λ ∈ ℝ and with F, F′ and F + λF′ representing members of N_{m}. One may always choose p.p = p.q = r.r = r.s = e.e = e.f = 0 and then a contraction of F + λF′ = e ∧ f with e gives (e.q)p − (e.p)q + λ(s.e)r − λ(e.r)s = 0 . If p, q, r and s are independent in T_{m}M one achieves the contradiction that e is orthogonal to each member of the basis p, q, r, s for T_{m}M and so it follows that p, q, r and s are not independent and hence that the blades of F and F′ intersect in a direction spanned, say, by k ∈ T_{m}M. Now write F = k∧r′and F′ = k∧s′and suppose that k is not null. Then one may choose r′and s′ null with each orthogonal to k. Then since F, F′ and H ≡ F + λF′ represent members of N_{m}, F_{ab}F^{ab} = F′_{ab}F^{′ab} = H_{ab}H^{ab} = 0 from which it follows that F_{ab}F^{′ab} = 0 and so (k.k)(r′.s′) = 0. Since k.k ≠ 0 one finds r′.s′ = 0 and so r′∧ s′ is totally null. But k.r′ = k.s′ = 0 and so k ∈ r′∧ s′ which contradicts the fact that k.k ≠ 0 (or, alternatively, that F and F′ are independent). So k is null (and hence F + λF′ is null or totally null for all λ ∈ ℝ). [It is remarked that a consideration of the null 2-spaces l ∧y and n∧y in a null basis shows that it is insufficient for simple bivectors F, F′ representing 2-spaces in N_{m} to have intersecting blades in order that F + λF′ represent a 2-space in N_{m} for any 0 ≠ λ ∈ ℝ.] It is then clear that N_{m}, in particular those “pencils” of the form F + λF′ which lie in N_{m}, reproduce the set of null vectors of g at m. □

It is remarked first that the 2-dimensional manifold of null directions which emerges in this lemma serves to distinguish the signature of g, being topologically 𝕊^{1} × 𝕊^{1} for neutral signature and topologically 𝕊^{2} for Lorentz signature. Second, it is pointed out that in the Lorentz case the set Z of all members of all 2-spaces in N_{m} consists, apart from the zero vector, of all spacelike and all null vectors at m and, as an alternative to lemma 2, the null cone is then the boundary of this set (in the natural topology on T_{m}M) and is hence determined by it (cf [17]). This fails for neutral signature since Z is then equal to T_{m}M.

The next lemma considers the continuous extendibility of the sectional curvature function to N_{m}

Lemma 3. (i) Consider (M, g) with m ∈ M, let A ≡ S_{m} ∪ T_{m} = \overline{G_{m}} and let F be a representative bivector in N_{m}. Suppose that R_{abcd}F^{ab}F^{cd} ≠ 0. If U is any open neighbourhood of F in G_{m}, σ_{m} is unbounded on A ∩ U.

(ii) If σ_{m} is continuously extendible to a single member of N_{m} it is a constant function on \overline{G_{m}} and hence may be extended continuously to G_{m}. Thus the constant curvature condition holds at m.

Proof. Since F is necessarily a limit point of A, the proof of part (i) is essentially the same as that given in [13] (see [17] and cf [1]). The idea is to consider the restriction f to S^{5} of the natural projection ℝ^{6} → P ℝ^{5} (that is, B_{m} → P B_{m}) and to show that the map h : S^{5} → ℝ given (using an identification mentioned earlier) by h(Q) = R_{abcd}Q^{ab}Q^{cd} is bounded away from zero on some compact neighbourhood of Q_{0} ∈ f^{−1} (F) whilst noting that G_{abcd}F^{ab}F^{cd} = 0.

For part (ii), if σ_{m} is continuously extendible to F_{0} ∈ N_{m} then σ_{m} is continuous as a map {F_{0}}∪A → ℝ. Suppose σ_{m}(F_{0}) = a and let V = (a−ϵ, a+ϵ) for 0 < ϵ ∈ ℝ. Then if U = σ_{m}^{−1} V, F_{0} ∈ U, U is open in {F_{0}} ∪ A with U = ({F_{0}} ∪ A) ∩ W for W open in G_{m} and U ∩A ≠ ∅ since F_{0} is a limit point of A. Then σ_{m} is bounded on U and since A ∩ W ⊂ U, σ_{m} is bounded on A ∩ W. Also W is an open neighbourhood of F_{0} in G_{m} and then it follows from part (i) that R_{abcd}F_{0}^{ab}F_{0}^{cd} = 0 and, in fact, this latter result is true for each member of W ∩ N_{m}. Then Lemma 1(iii) shows that the constant curvature condition holds at m. Thus σ_{m} is a constant function on \overline{G_{m}} and trivially continuously extendible to G_{m}. □

Theorem 1. Suppose that g and g′ are smooth metrics on a 4-dimensional, smooth, connected manifold M with g of neutral signature and that the (closed) subset B ⊂ M of precisely those points where Riem (for g) vanishes has empty interior in the topology of M (so that (M, g) is nonflat.) Suppose also that g and g′ determine the same sectional curvature function at each m ∈ M and that for no m ∈ M ∖ B is it a constant function. Then, using a prime to denote corresponding quantities with respect to g′ one has, on M, for some real-valued, nowhere-zero function ϕ : M → ℝ

{(i) g′ = ϕg, \atop (ii) R′_{abcd} = ϕ^{2}R_{abcd} ,} {(iii) R^{′a}_{bcd} = ϕR^{a}_{bcd} , \atop (iv) R′_{ab} = ϕR_{ab},}  {(v) R′ = R, \atop (vi) C′ = ϕC.}

Proof. It is perhaps unrealistic to demand that Riem is nowhere zero on M and this is why the subset B is introduced (but with the reasonable nonflat assumption, intB = ∅, where int denotes the interior operator in the manifold topology of M). The condition that the sectional curvature function σ_{m} for g is not constant at any m ∈ M ∖ B and equals the sectional curvature function σ′_{m} for g′ on this subset means (Lemma 3(ii)) that the subset of G_{m} on which σ_{m} and σ′_{m} are not defined is the same subset for both g and g′ and, in fact, equals the set N_{m} as defined for g and for g′. Thus from lemma 2 the set of null vectors for these metrics agree everywhere on M ∖ B and hence g′ is conformally related to g on this subset. It follows that g′ and g are conformally related on M (and that g′ has neutral signature) and so part (i) of the theorem holds and for ϕ : M → ℝ smooth since g and g′ are. It is remarked that since Riem vanishes on B the sectional curvature functions σ′_{m} and σ_{m} are zero maps for each m ∈ B. It then follows that Riem′ vanishes on precisely the subset B. To see this note that (2.1) and part (i) of the theorem show that R′_{abcd}F^{ab}F^{cd} = 0 at each m ∈ B and for each (simple) F representing a spacelike or timelike 2-space at m and hence, by continuity, for each simple F at each such m. Applying this for the simple bivectors x ∧ y, x ∧ s,...,s ∧ t in \overline{G_{m}} formed from the orthonormal basis x, y, s, t for T_{m}, together with some other judiciously chosen simple bivectors and the symmetries of R′_{abcd} one sees that Riem′ vanishes on B and, by reversing the argument, nowhere else. Now the equality of the sectional curvatures together with part (i) gives (R′_{abcd} − ϕ^{2}R_{abcd})F^{ab}F^{cd} = 0 for each simple F and then, by a similar argument to that given in the last sentence, one sees that (ii) holds on M. Then (iii), (iv) and (v) immediately follow. For (vi) use (1.1) and (1.2) to compute the Weyl conformal tensor directly using the results previously found (including R̃′_{ab} = ϕR̃_{ab} and recalling that C has components C^{a}_{bcd}). [It is remarked that, although (i) above implies the equality of the Weyl tensors, they may each be zero. In fact, if one assumes that the Weyl tensor of g (and hence of g′) is nowhere zero on M one immediately has ϕ = 1 on M (from (vi)) and so g′ = g on M but this will not be assumed here. The result (vi) applies in all cases. □

Continuing with the assumptions of theorem 1 the argument now follows that given in [6, 7]. Let U ⊂ M be the open subset of M on which the Weyl tensor of g (and hence of g′) is not zero (and so U ⊂ M ∖ B) and let V ⊂ M be the open set on which dϕ does not vanish. Then ϕ = 1 on U, U ∩ V = ∅ and ϕ is a (nonzero) constant on each component of int(M ∖V). It follows that Riem′ = Riem on int(M ∖ V) since their metrics differ only by a constant conformal factor. Now Riem′ and Riem cannot vanish over any nonempty open subset of int(M ∖ V) (by the definition of B) and so part (iii) above shows that ϕ = 1 on int(M ∖ V). Define the closed subset W ⊂ M comprising precisely those points of M at which ϕ = 1. Then U ⊂ W and int(M ∖ V) ⊂ W. Then disjointly decompose M as M = V ∪ intW ∪ K where the closed set K is defined by the disjointness of the decomposition and has empty interior in M since any nonempty open subset U′ contained in K would necessarily satisfy U′∩ V = ∅ and so ϕ would be constant on each of its components. Thus, since intB = ∅, U′ is not contained in B (and so Riem is nonzero and ϕ = 1 over the nonempty, open subset U′∩ (M ∖ B) of U′). It follows that U′∩ (intW) ≠ ∅ contradicting the disjointness of the decomposition. Thus M has been disjointly decomposed into the open subset intW where ϕ = 1 (and hence g′ = g) together with the open subset V where dϕ does not vanish and which is conformally flat for g and g′ (since U ∩ V = ∅) and a closed subset K with empty interior. So leaving aside the subset K one is left with the open set intW, where g′ = g, together with (perhaps more interestingly) the conformally flat open subset V upon which dϕ is nowhere zero.

3. Analysis of the Subset V

On V one has a nowhere-zero closed 1-form dϕ with components ϕ_{a} and V is conformally flat for g and g′, C′ = C = 0. One could now argue as in [7] but the following is a little easier and gives the same results. From C = 0 and (1.1) and (1.2) one has at each m ∈ M and for the metric g

(3.1) R_{abcd}(≡ g_{ae} R^{e}_{bcd}) = \frac{1}{2}[R̃_{ac} g_{bd} − R̃_{ad}g_{bc} + R̃_{bd}g_{ac} − R̃_{bc}g_{ad}] + \frac{R}{6}G_{abcd}

and also the conformally flat Bianchi identities, one for each of the connections ∇ of g and ∇′ of g′,

(3.2) R_{ca;b} − R_{cb;a} = \frac{1}{6}[g_{ac}R_{,b} − g_{cb}R_{,a}]  R′_{ca|b} − R′_{cb|a} = \frac{1}{6}[g′_{ac}R′_{,b} − g′_{cb}R′_{,a}]

where a comma denotes a partial derivative and a semi-colon and the symbol | denote covariant derivatives with respect to ∇ and ∇′, respectively. Now evaluate the second in (3.2), using Theorem 1 parts (iv) and (v), and subtract from it ϕ times the first in (3.2). The partial derivatives disappear and terms in P_{bc}^{a} ≡ Γ_{bc}^{′a} − Γ_{bc}^{a} arise where Γ and Γ′ are the Christoffel symbols from ∇ and ∇′, respectively, and where, since g′ = ϕg (and ϕ^{a} ≡ g^{ab}ϕ_{b})

(3.3)  P_{bc}^{a} = \frac{1}{2}ϕ^{−1} [ϕ_{c}δ_{b}^{a} + ϕ_{b}δ_{c}^{a} − ϕ^{a}g_{bc}]

One finds after a short calculation

(3.4)  ϕ_{b}R_{ac} − ϕ_{a}R_{bc} = ϕR_{ae} P_{bc}^{e} − ϕR_{be}P_{ac}^{e} = R_{be}ϕ^{e}g_{ac} − R_{ae} ϕ^{e}g_{bc}

A contraction with g^{ac} then shows that R̃_{ab}ϕ^{b} = 0 and a back substitution into (3.4) reveals that R̃_{ac} ϕ_{b}−R̃_{bc}ϕ_{a} = 0 and hence that R̃_{ab} = ψϕ_{a}ϕ_{b} for some function ψ : V → ℝ. Thus ψ(ϕ^{a}ϕ_{a}) = 0 on V. If, at any m ∈ V, ψ(m) = 0 then R̃icc(m) = 0 and so E(m) = 0 from (1.2). Then (1.1) shows that R_{abcd} = \frac{R}{6} G_{abcd} at m. Now if R(m) ≠ 0, σ_{m} is a constant function at m ∈ M ∖B and a contradiction is obtained and if R(m) = 0, Riem(m) = 0. It follows, since intB = ∅, that ψ cannot vanish over any nonempty open subset of V. So ϕ^{a}ϕ_{a} = 0 on V and hence ϕ^{a} is null on V (and, from Theorem 1(i), with respect to both metrics g and g′). [Thus in the positive definite case and from the definition of V, V = ∅ and the result in [15] is recovered.] A use of (3.3) now gives ϕ_{a;b} = ϕ_{a|b} + ϕ^{−1} ϕ_{a}ϕ_{b} and this enables further ∇ and ∇′ covariant derivatives to be taken as follows.

(3.5)  ϕ_{a;bc} = (ϕ_{a|b})_{;c} − ϕ^{−2} ϕ_{a}ϕ_{b}ϕ_{c} + ϕ^{−1} ϕ_{a}ϕ_{b;c} + ϕ^{−1} ϕ_{a;c} ϕ_{b}

and so

(3.6) (ϕ_{a|b})_{;c} = ϕ_{a|bc} = ϕ_{a|e} P_{bc}^{e} + ϕ_{e|b}P_{ac}^{e} = (2ϕ)^{−1} [ϕ_{a|c} ϕ_{b} + ϕ_{c|b}ϕ_{a} + 2ϕ_{a;b}ϕ_{c}]

Putting (3.5) and (3.6) together gives

(3.7)  ϕ_{a;bc} − ϕ_{a;cb} = ϕ_{a|bc} − ϕ_{a|cb} + (2ϕ)^{−1} [ϕ_{a;c} ϕ_{b} − ϕ_{a;b}ϕ_{c}]

(which corrects a minus sign error in equation (9.48) in [7] but which ultimately cancels out without problem in that reference). Using the Ricci identities for ∇ and ∇′ in (3.7) followed by a contraction with ϕ^{b} (and noting that since ϕ_{a;b} = ϕ_{b;a}, ϕ_{a;b}ϕ^{b} = 0) one gets

(3.8)  ϕ^{b}[ϕ_{d}R^{d}_{abc}] = ϕ^{b}[ϕ_{d}R^{′d}_{abc}] = ϕ[ϕ^{b}ϕ_{d}R^{d}_{abc}]

Then (3.1), the condition ϕ^{a}ϕ_{a} = 0 on V and the above expression for R̃icc give R_{abcd}ϕ^{b}ϕ^{d} = −\frac{R}{12} ϕ_{a}ϕ_{c}. Thus if R(m) ≠ 0 for some m ∈ V, R does not vanish in some open neighbourhood of m and hence ϕ = 1 and dϕ vanishes on this neighbourhood, contradicting the definition of V. It follows that R = 0 (that is, R̃icc = Ricc) and R_{ab} = ψϕ_{a}ϕ_{b} on V. Thus

(3.9)  R_{abcd}ϕ^{d} = 0

holds on V whilst (3.7) and the Ricci identities give ϕ_{a;c} ϕ_{b} = ϕ_{a;b}ϕ_{c} on V. So ϕ_{a;b} = αϕ_{a}ϕ_{b} for some function α on V and another application of the Ricci identity together with (3.9) shows that αϕ_{a} is locally a gradient, αϕ_{a} = ρ_{,a} on some suitable open neighbourhood U′′ of any m ∈ V and for some function ρ on U′′. Thus χ ≡ e^{−ρ} dϕ is a parallel 1-form satisfying χ =du on U′′ for u : U′′ → ℝ and so ϕ = ϕ(u) and ρ = ρ(u). Finally it follows from (3.2) that R_{ab} = γχ_{a}χ_{b} for γ : U′′→ ℝ with γ = γ(u).

Since Riem cannot vanish over any nonempty open subset of M there exists m ∈ V and a neighbourhood of m such that Riem does not vanish at any point of this neighbourhood. Construct a null basis l, n, L, N at m with l^{a} chosen equal to g^{ab}χ_{b}(m) ≡ χ^{a}(m) and then a basis l∧N, l∧n, n∧L, l∧L, L∧N and n∧N for B_{m} in terms of symmetrized products of which Riem(m) may be written. Now since the Weyl tensor C and R vanish on V, (1.1) gives Riem = E and so (see, e.g.[4])

∗ Riem = − Riem^{∗} on V. Also, from (3.9) Riem(m) may be reduced to a sum of symmetrized products of A ≡ l∧N, B ≡ l∧L and C ≡ L∧N and the expression for Ricc shows that the terms in C vanish. Then the relation^{∗} Riem = − Riem^{∗}, noting

that A ∈ S^{+}_{m} and B ∈ S^{−}_{m} (see Section 1) shows that in some open neighbourhood V′ of m (assumed to satisfy all the conditions placed on the above set U′′) where Riem does not vanish and chosen to allow a smooth extension of the null basis to V′ consistent with l^{a} = χ^{a} one has

(3.10)  R_{abcd} = β(A_{ab}B_{cd} + B_{ab}A_{cd})

for a smooth nowhere-zero function β : V′ → ℝ (and from which one now has R_{ab} = 2βl_{a}l_{b}). A consequence of this is that for any totally null 2-space represented

by a totally null bivector F at m ∈ V′ (and which is necessarily in S^{+}_{m} or S^{−}_{m}),

R_{abcd}F^{ab}F^{cd} = 0, whereas if F is null, one may write F = F^{+} + F^{−} with F^{+} ∈

S^{+}_{m}, F^{−} ∈ S^{−}_{m} and with each totally null (see Section 1) and so R_{abcd}F^{ab}F^{cd} =

2β(F^{ab}^{+}A_{ab})(F^{ab}^{−}B_{ab}) which is not zero unless F^{+} is a multiple of A or F^{−} is a multiple of B (cf Lemma 3(i)). Also if X is a spacelike (respectively timelike) 2-space at m and Y its spacelike (respectively timelike) orthogonal complement and which

are represented, respectively, by the (simple) bivectors F and F^{∗} whose blades are

X and Y (and noting that F + F^{∗} ∈ S^{+}_{m}), the conditions ^{∗} Riem = − Riem^{∗} and

F^{ab}F_{ab} = F^{ab}^{∗}F_{ab}^{∗} show that the sectional curvatures of X and Y differ only in sign.

Returning to (3.10) one has A_{ab}l^{b} = 0 and hence (A_{ab}l^{b})_{;c} = 0 on V′. Thus, since l is parallel on V′, (A_{ab;c}k^{c})l^{b} = 0 for any k ∈ T_{m}M (m ∈ V′) and so, since

A_{ab;c}k^{c} is in S^{+}_{m} for each k, A is a recurrent 2-form on V′, A_{ab;c} = A_{ab}p_{c} and similarly for B, B_{ab;c} = B_{ab}q_{c} for 1-forms p and q on V′. One can then compute from the Ricci identity and (3.10)

(3.11)  A_{ab;cd} − A_{ab;dc} [= A_{ab}(p_{c;d} − p_{d;c})] = A_{eb}R^{e}_{acd} + A_{ae} R^{e}_{bcd} = 0

to see that p, (and similarly q) are locally gradients of, say, the functions a and b, respectively and which are defined on the, possibly reduced, subset V′. Thus A and B may be locally scaled to parallel bivectors A′ ≡ e^{−a} A and B′ = e^{−b} B, respectively. Thus Riem is recurrent on V′, R_{abcd;e} = R_{abcd}r_{e}, on V′where r ≡ p+q is a gradient (as it must be by Walker’s theorem [21]). Clearly Ricc (R_{ab} = 2βl_{a}l_{b}) is also recurrent on V′, R_{ab;c} = R_{ab}r_{c} with r_{a} = β^{−1} β_{a} and (3.2) with R = 0 reveals that β_{,a} is proportional to l_{a} and is hence null. Thus the infinitesimal holonomy group of (M, g) has constant dimension (= 2) on the open subset V′ of V and so equals the holonomy group on any component of V′. This holonomy group is represented by the (2-dimensional) holonomy algebra labelled 2g in [22]. [It is remarked that the above calculation shows that, on V′, Ricc is of Segre type {(211)} and recurrent on V′ (but not properly recurrent since it is (locally) proportional to a parallel, symmetric second order tensor on V′) and again this holonomy result follows from [10].] One thus has

Theorem 2. Let M be a 4-dimensional manifold with nonflat metric g of neutral signature. Let g′ be any other metric on M whose sectional curvature function σ′_{m} at m ∈ M equals that of g at each m ∈ M. Suppose that for no m ∈ M ∖ B is σ_{m} a constant function. Then g and g′ are conformally related on M (and hence g′ has neutral signature). Also one may disjointly decompose M = V ∪ intW ∪ K such that g′ = g on intW, K is a closed subset of M with intK = ∅ and V is an open subset of M on which g and g′ are conformally flat and the above general analysis of this section holds.

One can add more here. From [21] one sees that on the open dense subset of V where Riem does not vanish the conditions of Walker’s nonsimple K_{n}^{∗} spaces are satisfied and so in some connected, open neighbourhood of any point of this neighbourhood one may choose coordinates u, v, x, y with u as above such that the metric takes the form

(3.12)  ds^{2} = H(u, x, y)du^{2} + 2du dv + dx^{2} − dy^{2}

where l_{a} = u_{,a} and, from the conformally flat condition, H(u, x, y) = δ(u)(x^{2} − y^{2}) for some smooth function δ. This is the analogue, for this signature, of the conformally flat plane waves of general relativity [20], the latter appearing in a similar way in the study of sectional curvature in Lorentz signature [6, 19]. To see that nontrivially related pairs g and g′ of solutions exist with identical sectional curvature functions suppose that M = ℝ^{4} with metric g as in (3.12). Then since the function ϕ above is a function only of u and may be chosen positive, let g′ = e^{2ρ(u)} g for some smooth function ρ on M. A computation of Riem′ and use of Theorem 1(ii) shows that the sectional curvatures of g and g′ are equal if and only if

ρ̈− ρ̇^{2} = δ(u)(e^{2ρ} − 1)

(where a dot denotes d/du) solutions of which are known (see [19]).

4. Sectional Curvature Preserving Vector Fields

Let f be a smooth map on some open neighbourhood W of M into M, let u, v be independent members of T_{m}M for m ∈ W and let m′ ∈ M with f(m) = m′. With f_{∗} the usual differential of f and assumed to be an isomorphism at each m ∈ W, suppose the sectional curvatures of u ∧ v at m and f_{∗}u ∧ f_{∗}v at m′ are equal for each such choice of m, u and v (in the sense that neither is defined or each is defined and equality holds). Then f is called sectional curvature preserving on W. Similarly a smooth vector field X on M is called sectional curvature preserving on M if each of its local flows is sectional curvature preserving in the above sense [6]. It is now easily checked from theorem 1 that the collection of all sectional preserving vector fields SCP(M) on M is a subalgebra of the (finite-dimensional) Lie algebra of conformal vector fields on M (each member satisfying, from theorem 1, L_{X} g = wg and L_{X} Ricc = w Ricc where L denotes the Lie derivative and w is some function on M). Clearly SCP(M) contains the Killing algebra of M but contains no proper homothetic vector fields. Now consider a (connected) coordinate neighbourhood W′ in the open subset V (as described in the last section) of the type in which (3.12) is written. Then R_{ab} = −2δ(u)l_{a}l_{b} and X ∈ SCP(W′) if and only if either (i) L_{X} g = wg and L_{X} Ricc = w Ricc for some function w(u) (and necessarily ẅ = 2wδ), or (ii) L_{X} g = wg where w(u) satisfies ẅ = 2wδ (and necessarily L_{X} Ricc = w Ricc). The details of the Lie algebra SCP(W′) may then be explored in a similar way to that in the Lorentz case [9, 14]. This should be compared with, for example, a study of such vector fields on any component of the open subset intW of Section 2 where SCP(intW) coincides with the Killing algebra on intW.

5. Final Remarks Including the Ricci-Flat Case

The restriction regarding the nonconstancy of σ_{m} in Theorem 2 can, in a special case, be removed in the following way. Suppose, with the above notation, that (M, g) is nonflat and Ricci flat. Then C = Riem on M and so C and Riem are each nowhere zero on an open dense subset U ⊂ M and each vanish on M ∖ U where the latter subset, because of the nonflat condition, (is closed and) has empty interior in M. Then σ_{m} is necessarily nowhere a constant function on U (otherwise the Ricci-flat condition would force the contradiction Riem = 0 at such points). Also ϕ = 1 on U and hence on M. Thus one has

Theorem 3. Let M be a 4-dimensional manifold with a nonflat, Ricci flat metric g of neutral signature. Let g′ be any other metric on M whose sectional function σ_{m} equals that of g′ at each m ∈ M. Then g′ = g on M.

It is noted that, quite generally for this dimension and signature, ^{∗}G^{∗} = G, ∗C^{∗} = C and ^{∗}E^{∗} = −E (see e.g. [4]) and hence, from (1.1), Riem−^{∗}Riem^{∗} = 2E. So, for any spacelike or timelike 2-space represented by the simple bivector F and

with orthogonal complement represented by F^{∗}, F^{ab}F_{ab} = F^{ab}^{∗}F_{ab}^{∗} and the following holds

σ_{m}(F) − σ_{m}(F^{∗}) = \frac{R_{abcd}F^{ab}F^{cd}}{2G_{abcd}F^{ab}F^{cd}} − \frac{^{∗}R_{abcd}^{∗}F^{ab}F^{cd}}{2G_{abcd}F^{ab}F^{cd}} = \frac{E_{abcd}F^{ab}F^{cd}}{G_{abcd}F^{ab}F^{cd}}

It follows that for each such 2-space the difference σ_{m}(F) − σ_{m}(F^{∗}) is controlled by the tensor E (that is, by R̃icc(m)) (see e.g. [11]) and that, just as in the Lorentz

case [3], σ_{m}(F) = σ_{m}(F^{∗}) for each such 2-space if and only if E(m) = 0, that is, if and only if the Einstein space condition holds at m.

Acknowledgements. The author wishes to thank the organisers of the XIX Geometrical Seminar in Zlatibor, Serbia, for their friendship and hospitality during his visit there. He also thanks Dr Bahar Kırık for many helpful discussions.

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Institute of Mathematics University of Aberdeen Aberdeen Scotland, UK

    g.hall@abdn.ac.uk