The local structure of conformally symmetric manifolds
THE LOCAL STRUCTURE OF CONFORMALLY
SYMMETRIC MANIFOLDS
ANDRZEJ DERDZINSKI AND WITOLD ROTER
Abstract. This is a final step in a local classification of pseudo-Riemannian man-
ifolds with parallel Weyl tensor that are not conformally flat or locally symmetric.
Introduction
The present paper provides a finishing touch in a local classification of essentially
conformally symmetric pseudo-Riemannian metrics.
A pseudo-Riemannian manifold of dimension n ≥ 4 is called essentially conformal-
ly symmetric if it is conformally symmetric [2] (in the sense that its Weyl conformal
tensor is parallel) without being conformally flat or locally symmetric.
The metric of an essentially conformally symmetric manifold is always indefinite
[4, Theorem 2]. Compact essentially conformally symmetric manifolds are known
to exist in all dimensions n ≥ 5 with n ≡ 5 (mod 3), where they represent all in-
definite metric signatures [8], while examples of essentially conformally symmetric
pseudo-Riemannian metrics on open manifolds of all dimensions n ≥ 4 were first
constructed in [16].
On every conformally symmetric manifold there is a naturally distinguished parallel
distribution D, of some dimension d, which we call the Olszak distribution. As shown
by Olszak [13], for an essentially conformally symmetric manifold d ∈ {1, 2}.
In [7] we described the local structure of all conformally symmetric manifolds with
d = 2. See also Section 3. This paper establishes an analogous result (Theorem 4.1)
for the case d = 1.
In both cases, some of the metrics in question are locally symmetric. In Remark 4.2
we explain why a similar classification result cannot be valid just for essentially con-
formally symmetric manifolds.
Essentially conformally symmetric manifolds with d = 1 are all Ricci-recurrent, in
the sense that, for every tangent vector field v, the Ricci tensor ρ and the covariant
derivative ∇vρ are linearly dependent at each point. The local structure of essentially
conformally symmetric Ricci-recurrent manifolds at points with ρ ⊗∇ρ 6= 0 has
already been determined by the second author [16]. Our new contribution settles the
2000 Mathematics Subject Classification. 53B30.
Key words and phrases. Parallel Weyl tensor, conformally symmetric manifold.
http://arxiv.org/abs/0704.0596v1
2 A. DERDZINSKI AND W. ROTER
one case still left open in the local classification problem, namely, that of essentially
conformally symmetric manifolds with d = 1 at points where ρ⊗∇ρ = 0.
The literature dealing with conformally symmetric manifolds includes, among oth-
ers, [9, 10, 12, 15, 17, 18] and the papers cited above. A local classification of homo-
geneous essentially conformally symmetric manifolds can be found in [3].
1. Preliminaries
Throughout this paper, all manifolds and bundles, along with sections and connec-
tions, are assumed to be of class C∞. A manifold is, by definition, connected. Unless
stated otherwise, a mapping is always a C∞ mapping betweeen manifolds.
Given a connection ∇ in a vector bundle E over a manifold M , a section ψ of
E , and vector fields u, v tangent to M , we use the sign convention
(1) R(u, v)ψ = ∇v∇uψ − ∇u∇vψ + ∇[u,v]ψ
for the curvature tensor R = R∇.
The Levi-Civita connection of a given pseudo-Riemannian manifold (M, g) is al-
ways denoted by ∇. We also use the symbol ∇ for connections induced by ∇, in
various ∇-parallel subbundles of TM and their quotients.
The Schouten tensor σ and Weyl conformal tensor W of a pseudo-Riemannian
manifold (M, g) of dimension n ≥ 4 are given by σ = ρ − (2n − 2)−1 sg, with ρ
denoting the Ricci tensor, s = trgρ standing for the scalar curvature, and
(2) W = R − (n− 2)−1g ∧ σ.
Here ∧ is the exterior multiplication of 1-forms valued in 1-forms, which uses the
ordinary ∧ as the valuewise multiplication; thus, g∧σ is a 2-form valued in 2-forms.
Let (t, s) 7→ x(s, t) be a fixed variation of curves in a pseudo-Riemannian manifold
(M, g), that is, an M-valued C∞ mapping from a rectangle (product of intervals) in
the ts-plane. By a vector field w along the variation we mean, as usual, a section of
the pullback of TM to the rectangle (so that w(t, s) ∈ Tx(t,s)M). Examples are xs
and xt, which assign to (t, s) the velocity of the curve t 7→ x(t, s) or s 7→ x(t, s) at
s or t. Further examples are provided by restrictions to the variation of vector fields
on M . The partial covariant derivatives of a vector field w along the variation are
the vector fields wt, ws along the variation, obtained by differentiating w covariantly
along the curves t 7→ x(t, s) or s 7→ x(t, s). Skipping parentheses, we write wts, wstt,
etc., rather than (wt)s, ((ws)t)t for higher-order derivatives, as well as xss, xst instead
of (xs)s, (xs)t. One always has wts = wst + R(xt, xs)w, cf. [11, formula (5.29) on
p. 460], and, since the Levi-Civita connection ∇ is torsionfree, xst = xts. Thus,
whenever (t, s) 7→ x(s, t) is a variation of curves in M ,
(3) xtss = xsst + R(xt, xs)xs .
CONFORMALLY SYMMETRIC MANIFOLDS 3
2. The Olszak distribution
The Olszak distribution of a conformally symmetric manifold (M, g) is the parallel
subbundle D of TM , the sections of which are the vector fields u with the property
that ξ∧Ω = 0 for all vector fields v, v ′ and for the differential forms ξ = g(u, · ) and
Ω =W (v, v ′, · , · ). The distribution D was introduced, in a more general situation,
by Olszak [13], who also proved the following lemma.
Lemma 2.1. The following conclusions hold for the dimension d of the Olszak
distribution D in any conformally symmetric manifold (M, g) with dimM = n ≥ 4.
(i) d ∈ {0, 1, 2, n}, and d = n if and only if (M, g) is conformally flat.
(ii) d ∈ {1, 2} if (M, g) is essentially conformally symmetric.
(iii) d = 2 if and only if rankW=1, in the sense that W, as an operator acting
on exterior 2-forms, has rank 1 at each point.
(iv) If d = 2, the distribution D is spanned by all vector fields of the form
W (u, v)v′ for arbitrary vector fields u, v, v′ on M .
Proof. See Appendix I. �
In the next lemma, parts (a) and (d) are due to Olszak [13, 2o and 3o on p. 214].
Lemma 2.2. If d ∈ {1, 2}, where d is the dimension of the Olszak distribution D
of a given conformally symmetric manifold (M, g) with dimM = n ≥ 4, then
(a) D is a null parallel distribution,
(b) at any x ∈M the space Dx contains the image of the Ricci tensor ρx treated,
with the aid of gx, as an endomorphism of TxM,
(c) the scalar curvature is identically zero and R = W + (n− 2)−1g ∧ ρ,
(d) W (u, · , · , · ) = 0 whenever u is a section of D,
(e) R(v, v ′, · , · ) = W (v, v ′, · , · ) = 0 for any sections v and v ′ of D⊥,
(f) of the connections in D and E = D⊥/D, induced by the Levi-Civita connec-
tion of g, the latter is always flat, and the former is flat if d = 1.
Proof. Assertion (e) for W is immediate from the definition of D. Namely, at any
point x ∈M , every 2-form Ωx in the image of Wx (for Wx acting on 2-forms at x) is
∧-divisible by ξ = gx(u, · ) for each u ∈ Dxr{0}, and so Ωx(v, v
′) = 0 if v, v ′ ∈ Dx
We now proceed to prove (a), (b), (c) and (d).
First, let d = 2. By Lemma 2.1(iii), this amounts to the condition rankW=1, so
that (a), (b) and (c) follow from Lemma 2.1(iv) combined with [7, Lemma 17.1(ii)
and Lemma 17.2]. Also, for a nonzero 2-form Ωx chosen as in the last paragraph,
Dx is the image of Ωx, that is, Ωx equals the exterior product of two vectors in Dx
(treated as 1-forms, with the aid of gx). Now (d) follows since, by (a), Ωx(ux, · ) = 0
if u is a section of D.
4 A. DERDZINSKI AND W. ROTER
Next, suppose that d = 1. Replacing M by a neighborhood of any given point,
we may assume that D is spanned by a vector field u. If u were not null, we would
have W (u, v, u, v ′) = 0 for any sections v, v ′ of D⊥, as one sees contracting the
twice-covariant tensor field W ( · , v, · , v ′) = 0, at any point x, in an orthogonal basis
containing the vector ux. (We have already established (e) for W.) Combined with
(e) for W and the symmetries of W, the relation W (u, v, u, v ′) = 0 for v, v ′ in D⊥
would then give W = 0, contrary to the assumption that d = 1. Thus, u is null,
which yields (a). Now
(4) we choose, locally, a null vector field u′ with g(u, u′) = 1.
For any section v of D⊥ one sees that W (u, · , u′, v) = 0 by contracting the tensor
field W ( · , · , · , v) = 0 in the first and third arguments, at any point x, in
(5) a basis of TxM formed by ux, u
x and n− 2 vectors orthogonal to them,
and using (e) for W, along with the inclusion D ⊂ D⊥, cf. (a). Since u′ and D⊥
span TM , assertion (e) for W thus implies (d).
To prove (b) and (c) when d = 1, we distinguish two cases: (M, g) is either es-
sentially conformally symmetric, or locally symmetric. For (c), it suffices to establish
vanishing of the scalar curvature s (cf. (2)). Now, in the former case, s = 0 accord-
ing to [5, Theorem 7], while (b) follows since, as shown in [6, Theorem 7 on p. 18],
for arbitrary vector fields v, v ′ and v ′′ on an essentially conformally symmetric pseu-
do-Riemannian manifold, ξ ∧ Ω = 0, where ξ = ρ(v, · ) and Ω = W (v ′, v ′′, · , · ). In
the case where g is locally symmetric, (b) and (c) are established in Appendix II.
Assertion (e) for R is now obvious from (e) for W and (c), since, by (b), ρ(v, · ) =
0 for any section v of D⊥. The claim about E in (f) is in turn immediate from (1)
and (e) for R, which states that R(w,w ′)v, for arbitrary vector fields w,w ′ and any
section v of D⊥, is orthogonal to all sections of D⊥ (and hence must be a section
of D). Finally, to prove (f) for D, with d = 1, let us fix a section u of D, a vector
field v, and define a differential 2-form ζ by ζ(w,w ′) = (n−2)R(w,w ′, u, v) for any
vector fields w,w ′. By (c) and (e), ζ = g(u, · )∧ ρ(v, · ), as D ⊂ D⊥ (cf. (a)), and so
ρ(u, · ) = 0 in view of (b) and symmetry of ρ. However, by (b), both g(u, · ) and
ρ(v, · ) are sections of the subbundle of T ∗M corresponding to D under the bundle
isomorphism TM → T ∗M induced by g, so that ζ = 0 since the distribution D is
one-dimensional. �
3. The case d = 2
For more details of the construction described below, we refer the reader to [7].
Let there be given a surface Σ, a projectively flat torsionfree connection D on Σ
with a D-parallel area form α, an integer n ≥ 4, a sign factor ε = ±1, a real vector
space V of dimension n− 4, and a pseudo-Euclidean inner product 〈 , 〉 on V .
CONFORMALLY SYMMETRIC MANIFOLDS 5
We also assume the existence of a twice-contravariant symmetric tensor field T on
Σ with divD(divDT ) + (ρD, T ) = ε (in coordinates: T jk,jk + T
jkRjk = ε). Here
divD denotes the D-divergence, ρD is the Ricci tensor of D, and ( , ) stands for
the obvious pairing. Such T always exists locally in Σ. In fact, according to [7,
Theorem 10.2(i)] combined with [7, Lemma 11.2], T exists whenever Σ is simply
connected and noncompact.
For T chosen as above, we define a twice-covariant symmetric tensor field τ on
Σ, that is, a section of [T ∗Σ]⊙2, by requiring τ to correspond to the section T of
[TΣ]⊙2 under the vector-bundle isomorphism TΣ → T ∗Σ which acts on vector fields
v by v 7→ α(v, · ). In coordinates, τjk = αjlαkmT
Next, we denote by hD the Patterson-Walker Riemann extension metric [14] on
the total space T ∗Σ, obtained by requiring that all vertical and all D-horizontal
vectors be hD-null, while hDx (ζ, w) = ζ(dπxw) for x ∈ T
∗Σ, any vector w ∈ TxT
any vertical vector ζ ∈ Ker dπx = T
Σ, and the bundle projection π : T ∗Σ → Σ.
Finally, let γ and θ be the constant pseudo-Riemannian metric on V correspond-
ing to the inner product 〈 , 〉, and the function V → R with θ(v) = 〈v, v〉.
Our Σ,D, α, n, ε, V , 〈 , 〉 now give rise to the pseudo-Riemannian manifold
(6) (T ∗Σ × V, hD− 2τ + γ − θρD) ,
of dimension n, with the metric hD− 2τ + γ − θρD, where the function θ and
covariant tensor fields τ, ρD, hD, γ on Σ, T ∗Σ or V are identified with their pull-
backs to T ∗Σ × V . (Thus, for instance, hD− 2τ + γ is a product metric.)
We have the following local classification result, in which d stands for the dimen-
sion of Olszak distribution D.
Theorem 3.1. The pseudo-Riemannian manifold (6) obtained as above from any
data Σ,D, α, n, ε, V , 〈 , 〉 with the stated properties is conformally symmetric and has
d = 2. Conversely, in any conformally symmetric pseudo-Riemannian manifold such
that d = 2, every point has a connected neighborhood isometric to an open subset of
a manifold (6) constructed above from some data Σ, D, α, n, ε, V , 〈 , 〉.
The manifold (6) is never conformally flat, and it is locally symmetric if and only
if the Ricci tensor ρD is D-parallel.
Proof. See [7, Section 22]. Note that, in view of Lemma 2.1(iii), the condition
rankW=1 used in [7] is equivalent to d = 2. �
The objects Σ,D, α, n, ε, V , 〈 , 〉 are treated as parameters of the above construc-
tion, while T is merely assumed to exist, even though the metric g in (6) clearly
depends on τ (and hence on T ). This is justified by the fact that, with fixed
Σ,D, α, n, ε, V , 〈 , 〉, the metrics corresponding to two choices of T are, locally, iso-
metric to each other, cf. [7, Remark 22.1].
6 A. DERDZINSKI AND W. ROTER
The metric signature of (6) is clearly given by −− . . .++, with the dots standing
for the sign pattern of 〈 , 〉.
4. The case d = 1
Let there be given an open interval I, a C∞ function f : I → R, an integer n ≥ 4,
a real vector space V of dimension n − 2 with a pseudo-Euclidean inner product
〈 , 〉, and a nonzero traceless linear operator A : V → V , self-adjoint relative to 〈 , 〉.
As in [16], we then define an n-dimensional pseudo-Riemannian manifold
(7) (I ×R× V, κ dt2 + dt ds + γ) ,
where products of differentials represent symmetric products, t, s denote the Carte-
sian coordinates on the I × R factor, γ stands for the pullback to I × R × V of
the flat pseudo-Riemannian metric on V that corresponds to the inner product 〈 , 〉,
and the function κ : I ×R× V → R is given by κ(t, s, ψ) = f(t)〈ψ, ψ〉+ 〈Aψ, ψ〉.
The manifolds (7) are characterized by the following local classification result,
analogous to Theorem 3.1. As before, d is the dimension of the Olszak distribution.
Theorem 4.1. For any I, f, n, V , 〈 , 〉, A as above, the pseudo-Riemannian man-
ifold (7) is conformally symmetric and has d = 1. Conversely, in any conformally
symmetric pseudo-Riemannian manifold such that d = 1, every point has a connected
neighborhood isometric to an open subset of a manifold (7) constructed from some
such I, f, n, V , 〈 , 〉, A.
The manifold (7) is never conformally flat, and it is locally symmetric if and only
if f is constant.
A proof of Theorem 4.1 is given at the end of the next section.
Obviously, the metric κ dt2+ dt ds + γ in (7) has the sign pattern − . . .+, where
the dots stand for the sign pattern of 〈 , 〉.
Remark 4.2. A classification result of the same format as Theorem 4.1 cannot
be true just for essentially conformally symmetric manifolds with d = 1. Namely,
such manifolds do not satisfy a principle of unique continuation: formula (7) with f
which is nonconstant on I, but constant on some nonempty open subinterval I ′ of I,
defines an essentially conformally symmetric manifold with a locally symmetric open
submanifold U = I ′ ×R × V . At points of U, the local structure of (7) does not,
therefore, arise from a construction that, locally, produces all essentially conformally
symmetric manifolds and nothing else.
As explained in [7, Section 24], an analogous situation arises when d = 2.
5. Proof of Theorem 4.1
The following assumptions will be used in Lemma 5.1.
(a) (M, g) is a conformally symmetric manifold of dimension n ≥ 4 and y ∈M .
CONFORMALLY SYMMETRIC MANIFOLDS 7
(b) The Olszak distribution D of (M, g) is one-dimensional.
(c) u is a global parallel vector field spanning D.
(d) t :M → R is a C∞ function with g(u, · ) = dt and t(y) = 0.
(e) dim V = n− 2 for the space V of all parallel sections of E = D⊥/D.
(f) ρ = (2−n)f(t) dt⊗dt for some C∞ function f : I ′ → R on an open interval
I ′, where ρ is the Ricci tensor and f(t) denotes the composite f ◦ t.
For local considerations, only (a) and (b) are essential. In fact, condition (e) (in
which ‘parallel’ refers to the connection in E induced by the Levi-Civita connection
of g), as well (c) and (d) for some u and t, follow from (a) – (b) if M is simply
connected. See Lemma 2.2(f). On the other hand, (c) – (d), Lemma 2.2(b) and
symmetry of ρ give ∇dt = 0 and ρ = χ dt⊗ dt for some function χ : M → R, so
that ∇ρ = dχ⊗ dt⊗ dt. However, ∇ρ is totally symmetric (that is, ρ satisfies the
Codazzi equation): our assumption ∇W = 0 implies the condition divW = 0, well
known [11, formula (5.29) on p. 460] to be equivalent to the Codazzi equation for the
Schouten tensor σ, while σ = ρ by Lemma 2.2(c). Thus, dχ equals a function times
dt, and so χ is, locally, a function of t, which (locally) yields (f).
For any section v of D⊥, we denote by v the image of v under the quotient-pro-
jection morphism D⊥ → E = D⊥/D.
The data required for the construction in Section 4 consist of I, f, n, V appearing
in (a) – (f), along with the pseudo-Euclidean inner product 〈 , 〉 in V , induced in an
obvious way by g (cf. Lemma 2.2(f)), and A : V → V characterized by 〈Aψ, ψ ′〉 =
W (u′, v, v ′, u′), for ψ, ψ ′ ∈ V , with a vector field u′ and sections v, v ′ of D⊥ chosen,
locally, so that g(u, u′) = 1, ψ = v and ψ ′ = v ′. (The resulting bilinear form
(ψ, ψ ′) 7→ 〈Aψ, ψ ′〉 on V is well-defined, that is, unaffected by the choices of u′, v or
v ′, as a consequence of Lemma 2.2(d),(e), while the function W (u′, v, v ′, u′) is in fact
constant, by Lemma 2.2(d), as ones sees differentiating it via the Leibniz rule and
noting that, since v and v ′ are parallel, the covariant derivatives of v and v ′ in the
direction of any vector field are sections of D.) That A is traceless and self-adjoint
is immediate from the symmetries of W. Finally, A 6= 0 since, otherwise, W would
vanish. (Namely, in view of Lemma 2.2(d),(e), W would yield 0 when evaluated on
any quadruple of vector fields, each of which is either u′ or a section of D⊥.)
Under the assumptions (a) – (f), with f = f(t), we then have
(8) R(u′, v)v ′ = [f g(v, v ′) + 〈Av, v ′〉]g(u′, u)u
for any sections v, v ′ of D⊥ and any vector field u′. In fact, ρ(v, · ) = ρ(v ′, · ) = 0
from symmetry of ρ and Lemma 2.2(b), so that, by Lemma 2.2(c), R(u′, v)v ′ =
W (u′, v)v ′ − (n − 2)−1g(v, v ′)ρu′, where ρu′ denotes the unique vector field with
g(ρu′, · ) = ρ(u′, · ). Now (8) follows: due to (d), (f) and the definition of A, both
sides have the same g-inner product with u′, and are orthogonal to u⊥ = D⊥ (with
R(u′, v)v ′ orthogonal to D⊥ in view of Lemma 2.2(e)).
8 A. DERDZINSKI AND W. ROTER
We fix an open subinterval I of I ′, containing 0, and a null geodesic I ∋ t 7→ x(t)
in M with x(0) = y, parametrized by the function t (in the sense that the function
t restricted to the geodesic coincides with the geodesic parameter). Namely, since
∇dt = 0, the restriction of t to any geodesic is an affine function of the parameter;
thus, by (d), it suffices to prescribe the initial data formed by x(0) = y and a null
vector ẋ(0) ∈ TyM with g(ẋ(0), uy) = 1.
As g(ẋ(0), uy) = 1, the plane P in TyM , spanned by the null vectors ẋ(0) and
uy (cf. Lemma 2.2(a)) is gy-nondegenerate, and so TyM = P ⊕ Ṽ , for Ṽ = P
⊥. Let
pr : TyM → Ṽ be the orthogonal projection. Since pr(Dy) = {0}, the restriction of
pr to Dy
⊥ descends to the quotient Ey = Dy
⊥/Dy, producing an isomorphism Ey → Ṽ ,
also denoted by pr. Finally, for ψ ∈ V , we let t 7→ ψ̃(t) ∈ Tx(t)M be the parallel
field with ψ̃(0) = pr ψy, and set κ(t, s, ψ) = f(t)〈ψ, ψ〉+ 〈Aψ, ψ〉, as in Section 4.
The formula F (t, s, ψ) = expx(t)(ψ̃(t) + sux(t)/2) now defines a C
∞ mapping F
from an open subset of R2× V into M .
Lemma 5.1. Under the above hypotheses, F ∗g = κ dt2+ dtds+ h.
Proof. The F -images w,w ′, F∗ψ of the constant vector fields (1, 0, 0), (0, 1, 0)
and (0, 0, ψ) in R2×V , for ψ ∈ V , are vector fields tangent to M along F (sections of
F ∗TM). Since D⊥ is parallel, its leaves are totally geodesic and, by Lemma 2.2(e), the
Levi-Civita connection of g induces on each leaf a flat torsionfree connection. Thus,
w ′ and each F∗ψ are parallel along each leaf of D
⊥, as well as tangent to the leaf,
and parallel along the geodesic t 7→ x(t). Therefore, w ′ = u/2, while the functions
g(w ′, F∗ψ) and g(F∗ψ, F∗ψ
′), for ψ, ψ ′ ∈ V , are constant, and hence equal to their
values at y, that is, 0 and 〈ψ, ψ ′〉. It now remains to be shown that g(w,w) = κ◦F ,
g(w, u/2) = 1/2 and g(w, F∗ψ) = 0. To this end, we consider the variation x(t, s) =
F (t, sa, sψ) of curves in M , with any fixed a ∈ R and ψ ∈ V . Clearly, w = xt along
the variation (notation of Section 1). Next, xts = xst is tangent to D
⊥, since so is
xs, while D
⊥ is parallel. Consequently, [g(xt, u)]s = 0, as u is parallel and tangent
to D. Thus, g(w, u) = g(xt, u) = 1. (Note that g(xt, u) = 1 at s = 0, due to (d),
as the geodesic t 7→ x(t) is parametrized by the function t.) However, xss = 0 and
xs is tangent to D
⊥, so that (3) and (8) now give xtss = [fg(xs, xs) + 〈Axs, xs〉]u,
which is parallel in the s direction, while xts = xst = 0 at s = 0. Hence xts =
s[fg(xs, xs) + 〈Axs, xs〉]u, and so g(xts, xts) = 0 (cf. (c) above and Lemma 2.2(a)).
This further yields [g(xt, xt)]ss/2 = g(xt, xtss) = fg(xs, xs) + 〈Axs, xs〉. The last
function is constant in the s direction, while g(xt, xt) = [g(xt, xt)]s = 0 at s = 0, and
so g(w,w) = g(xt, xt) = s
2[fg(xs, xs) + 〈Axs, xs〉] = κ. Finally, being proportional
to u at each point, xts is orthogonal to D
⊥, and hence to F∗ψ, which imples that
[g(xt, F∗ψ)]s = 0, and, as g(w, F∗ψ) = g(xt, F∗ψ) = 0 at s = 0, we get g(w, F∗ψ) = 0
everywhere. �
CONFORMALLY SYMMETRIC MANIFOLDS 9
We are now in a position to prove Theorem 4.1. First, (7) is conformally sym-
metric and has d = 1, as one can verify by a direct calculation, cf. [16, Theorem 3].
Conversely, if conditions (a) and (b) above are satisfied, we may also assume (c) – (f).
(See the comment following (f).) Our assertion is now immediate from Lemma 5.1.
Appendix I: Proof of Lemma 2.1
We prove Lemma 2.1 here, since Olszak’s paper [13] may be difficult to obtain.
The condition d = n is equivalent to conformal flatness of (M, g), since n > 2
and so Ω = 0 is the only 2-form ∧-divisible by all nonzero 1-forms ξ. At a fixed
point x, the metric gx allows us to treat the Ricci tensor ρx and any 2-form Ωx as
endomorphisms of TxM, so that we may consider their images (which are subspaces
of TxM). If W 6= 0, fixing a nonzero 2-form Ωx in the image of Wx acting on
2-forms at x we see that, for every u ∈ Dx, our Ωx is ∧-divisible by ξ = gx(u, · ),
and so the image of Ωx contains Dx. Thus, d ≤ 2, and (i) follows. (Being nonzero
and decomposable, Ωx has rank 2.) As shown in [6, Theorem 7 on p. 18], if (M, g)
is essentially conformally symmetric, the image of ρx is a subspace of Dx, so that
(i) yields (ii), since g in (ii) cannot be Ricci-flat. Next, if d = 2, the image of our
Ωx coincides with Dx (as rankΩx = 2). Every 2-form in the image of Wx thus is a
multiple of Ωx, being the exterior product of two vectors in Dx, identified, via gx,
with 1-forms. Hence rankW = 1. Conversely, if rankW = 1, all nonzero 2-forms
Ωx in the image of Wx are of rank 2, as Wx, being self-adjoint, is a multiple of
Ωx ⊗ Ωx, and so the Bianchi identity for W gives Ωx ∧ Ωx = 0. All such Ωx are
therefore ∧-divisible by ξ = gx(u, · ), for every nonzero vector u in the common
2-dimensional image of such Ωx, which shows that d = 2. Finally, (iv) follows if one
chooses Ωx 6= 0 equal to Wx(v, v
′, · , · ) for some v, v ′ ∈ TxM .
Appendix II: Lemma 2.2(b),(c) in the locally symmetric case
Parts (b) and (c) of Lemma 2.2 for locally symmetric manifolds with d = 1 could,
in principle, be derived from Cahen and Parker’s classification [1] of pseudo-Riemann-
ian symmetric manifolds. We prove them here directly, for the reader’s convenience.
Our argument uses assertions (a), (d) in Lemma 2.2, along with (e) for W, which
were established in the proof of Lemma 2.2 before Appendix II was mentioned.
Suppose that ∇R = 0 and d = 1. Replacing M by an open subset, we also
assume that the Olszak distribution D is spanned by a vector field u. By (1),
(9) i) R( · , · )u = Ω ⊗ u or, in coordinates, ii) ulRjkl
s = Ωjku
for some differential 2-form Ω, which obviously does not depend on the choice of u.
(It is also clear from (1) that Ω is the curvature form of the connection in the line
bundle D, induced by the Levi-Civita connection of g.) Being unique, Ω is parallel,
10 A. DERDZINSKI AND W. ROTER
and so are ρ and W, which implies the Ricci identities R · Ω = 0, R · ρ = 0, and
R ·W = 0. In coordinates: Rmlj
sτsk +Rmlk
sτjs = 0, where τ = Ω or τ = ρ, and
(10) Rqpj
sWsklm + Rqpk
sWjslm + Rqpl
sWjksm + Rqpk
sWjkls = 0.
Summing Rmlj
sΩsk + Rmlk
sΩjs = 0 against u
l, we obtain Ω ◦ Ω = 0, where the
metric g is used to treat Ω as a bundle morphism TM → TM that sends each
vector field v to the vector field Ωv with g(Ωv, v ′) = Ω(v, v ′) for all vector fields
v ′. Lemma 2.2(d) and (9.i) give W ( · , · , u, v) = R( · , · , u, v) = 0 for our fixed vector
field u, spanning D, and any section v of D⊥. Hence, by (2), g(u, · ) ∧ σ(v, · ) =
g(v, · ) ∧ σ(u, · ). Thus, σu = cu for the Schouten tensor σ and some constant c,
with σu defined analogously to Ωv. (Otherwise, choosing v such that u, σu and
v are linearly independent at a given point x, we would obtain a contradiction with
the equality between planes in TxM , corresponding to the above equality between
exterior products.) Consequently, g(u, · ) ∧ (σ + cg)(v, · ) = 0, and so σv + cv is a
section of D whenever v is a section of D⊥. Let us now fix u′ as in (4). Symmetry
of σ gives g(σu′, u) = c. In a suitably ordered basis with (5), at any point x, the
endomorphism of TxM corresponding to σx thus has an upper triangular matrix
with the diagonal entries c,−c, . . . ,−c, c, so that trgσ = (4 − n)c. Consequently,
(n − 2) s = 2(n − 1)(4 − n)c, for the scalar curvature s, and (n − 2)ρu = 2cu.
However, contracting (9.ii) in k = s, we get ρu = −Ωu, and so (n− 2)Ωu = −2cu.
The equality Ω ◦Ω = 0 that we derived from the Ricci identity R ·Ω = 0 now gives
c = 0. Hence s = 0 (which yields Lemma 2.2(c)), and ρu = 0.
As c = 0 and σ = ρ, the assertion about σv + cv obtained above means that
ρv is a section of D whenever v is a section of D⊥. Let λ, µ, ξ be the 1-forms
with λ = g(u, · ), µ = g(u′, · ), ξ(u′) = 0, and ρv = ξ(v)u for sections v of D⊥.
Transvecting (9.ii) with µs, we get Ω = R( · , · , u, u
′) = (n − 2)−1λ ∧ ρ(u′, · ) from
Lemma 2.2(c) with ρu = 0 and Lemma 2.2(d). However, evaluating ρ(u′, · ) on
u′, u and sections v of D⊥, we see that ρ(u′, · ) = hλ+ ξ, with h = ρ(u′, u′). (Note
that ξ(u) = 0 since ρu = 0, while D ⊂ D⊥ by Lemma 2.2(a).) Therefore,
(11) i) (n− 2)Ω = λ ∧ ξ , ii) ρ = hλ⊗ λ + λ⊗ ξ + ξ ⊗ λ.
In addition, if v ′ denotes the unique vector field with g(v ′, · ) = ξ, then u and v ′
are null and orthogonal, or, equivalently,
(12) the 1-forms λ and ξ are null and mutually orthogonal.
In fact, g(u, u) = 0 by Lemma 2.2(a), g(u, v ′) = 0 as ξ(u) = 0, and v ′ is null
since (11) yields (n−2)[ρ(Ωu′)−Ω(ρu′)] = 2g(v ′, v ′)u, while, transvecting the Ricci
identity Rmlj
sRsk + Rmlk
sRjs = 0 with u
l and using (9.ii), we see that ρ and Ω
commute as bundle morphisms TM → TM .
Furthermore, transvecting with µkµm the coordinate form Rmlj
sτsk+Rmlk
sτjs = 0
of the Ricci identity R · τ = 0 for the parallel tensor field τ = (n − 2)Ω + ρ =
CONFORMALLY SYMMETRIC MANIFOLDS 11
hλ⊗ λ + 2λ⊗ ξ (cf. (11)), we get 2λjblsξ
s = 0, where b = W (u′, · , u′, · ). Namely,
R = W + (n− 2)−1g ∧ ρ by Lemma 2.2(c), Wmlj
sτsk = 0 in view of Lemma 2.2(d),
µkµmWmlk
sτjs = 2λjblsξ
s since b(u, · ) = 0 (again from Lemma 2.2(d)), and the
remaining terms, related to g ∧ ρ, add up to 0 as a consequence of (12), (11.ii)
and the formula for τ . (Note that (12) gives Rj
sτsk = Rj
sτks = 0, and so four out
of the eight remaining terms vanish individually.) However, u 6= 0, and so λ 6= 0,
which gives b( · , v ′) = 0, where v ′ is the vector field with g(v ′, · ) = ξ. Thus,
W (u′, · , u′, v ′) = 0. As a result, the 3-tensor W ( · , · , · , v ′) must vanish: it yields
the value 0 whenever each of the three arguments is either u′ or a section of D⊥.
(Lemma 2.2(e) for W is already established.)
The relation W ( · , · , · , v ′) = 0 implies in turn that W ( · , · , · , ρv) = 0 (in coor-
dinates: Wjkl
sRsp = 0). In fact, by (11.ii), the image of ρ is spanned by u and v
while W ( · , · , · , u) = 0 according to Lemma 2.2(d).
As in [13, 1o on p. 214], we have W = (λ ⊗ λ) ∧ b (notation of (2)), where,
again, b = W (u′, · , u′, · ). Namely, by Lemma 2.2(e) for W, both sides agree on any
quadruple of vector fields, each of which is either u′ or a section of D⊥.
Finally, transvecting (10) with µkµm and replacing R by W + (n− 2)−1g ∧ ρ, we
obtain two contributions, one from W and one from g ∧ ρ, the sum of which is zero.
Since W = (λ⊗λ)∧ b, the W contribution vanishes: its first two terms add up to 0,
and so do its other two terms. (As we saw, b(u, · ) = 0, while, obviously, b(u′, · ) = 0.)
Out of the sixteen terms forming the g∧ρ contribution, eight are separately equal to
zero since Wjkl
sRsp = 0, and so, in view of (11.ii) and the relation W = (λ⊗ λ) ∧ b,
vanishing of the g∧ρ contribution gives λpSjlq = λqSjlp, for Sjlq = 2bjlξq−bqlξj−bqjξl.
Thus, Sjlq = ηjlλq for some twice-covariant symmetric tensor field η, which, summed
cyclically over j, l, q, yields 0 (due to the definition of Sjlq and symmetry of b). As
λ 6= 0 and the symmetric product has no zero divisors, we get η = 0 and Sjlq = 0.
The expression bjlξq− bqlξj is, therefore, skew-symmetric in j, l. As it is also, clearly,
skew-symmetric in j, q, it must be totally skew-symmetric and hence equal to one-
third of its cyclic sum over j, l, q. That cyclic sum, however, is 0 in view of symmetry
of b, so that bjlξq = bqlξj. Thus, ξ = 0, for otherwise the last equality would yield
b = ϕξ ⊗ ξ for some function ϕ, and hence W = (λ⊗ λ) ∧ b = ϕ(λ⊗ λ) ∧ (ξ ⊗ ξ),
which would clearly imply that the vector field v ′ with g(v ′, · ) = ξ is a section of
the Olszak distribution D, not equal to a function times u (as ξ(u′) = 0, while
g(u, u′) = 1), contradicting one-dimensionality of D. Therefore, ρ = hλ ⊗ λ by
(11.ii) with ξ = 0, which proves assertion (b) of Lemma 2.2 in our case.
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12 A. DERDZINSKI AND W. ROTER
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Department of Mathematics
The Ohio State University
Columbus, OH 43210
E-mail address : andrzej@math.ohio-state.edu
Institute of Mathematics and Computer Science
Wroc law University of Technology
Wybrzeże Wyspiańskiego 27, 50-370 Wroc law
Poland
E-mail address : roter@im.pwr.wroc.pl
http://arxiv.org/abs/math/0604568
	Introduction
	1. Preliminaries
	2. The Olszak distribution
	3. The case d=2
	4. The case d=1
	5. Proof of Theorem ??
	Appendix I: Proof of Lemma ??
	Appendix II: Lemma ??(b),(c) in the locally symmetric case
	References
