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arXiv:math/0510099v1 [math.DG] 5 Oct 2005
Second-Order Symmetric Lorentzian Manifolds
José M. M. Senovilla
Física Teórica, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spain.
Abstract. Spacetimes with vanishingsecond covariantderivativeof the Riemann tensor are studied.
Their existence, classification and explicit local expression are considered. Related issues and open
questions are briefly commented.
INTRODUCTION
Our aim is to characterize, as well as to give a full list of, the n-dimensional manifolds
V with a metric g of Lorentzian signature such that the Riemann tensor R
α
βγδ
of (V ,g)
locally satisfies the second-order condition
∇
µ
∇
ν
R
α
βγδ
= 0. (1)
It is quite surprising that, hitherto, despite their simple definition, this type of Lorentzian
manifolds have been hardly considered in the literature. Probably this is due to the clas-
sical results concerning these manifols in the proper Riemannian case, to the difficulties
arising in other signatures, and to the little reward: only very special cases survive.
Apart from their obvious mathematical interest, from a physical point of view they are
relevant in several respects: as a second local approximation to any spacetime (using for
instance expansions in normal coordinates); as examples with a finite number of terms
in Lagrangians; as interesting exact solutions for supergravity/superstring or M-theories;
for invariant classifications; for solutions with parallel vector fields or spinors.
A more complete treatment, with a full list of references, is given in [23].
SYMMETRIC SPACES AND ITS GENERALIZATIONS
Semi-Riemannian manifolds satisfying (1) are a direct generalization of the classical
locally symmetric spaces which satisfy
∇
µ
R
α
βγδ
= 0. (2)
These were introduced, studied and classified by E. Cartan [10] in the proper Rieman-
nian case
1
, see e.g. [11, 16, 15], and later in [6, 9, 7] for the Lorentzian and general semi-
Riemannian cases—see e.g. [8, 19] and references therein. They are themselves general-
izations of the constant curvature spaces and, actually, there is a hierarchy of conditions,
1
With a positive-definite metric.
TABLE 1. The hierarchy of conditions on the Riemann tensor
R
α
βγν
∝
δ
α
γ
g
βν
−
δ
α
ν
g
βγ
∇
µ
R
α
βγδ
= 0 ∇
µ
∇
ν
R
α
βγδ
= 0 ∇
[
µ
∇
ν
]
R
α
βγδ
= 0
constant curvature symmetric 2-symmetric semisymmetric
shown in Table 1, that can be placed on the curvature tensor. In the table, the restrictions
on the curvature tensor decrease towards the right and each class is strictly contained in
the following ones. The table has been stopped at the level of semi-symmetric spaces,
defined by the condition
2
∇
[
µ
∇
ν
]
R
α
βγδ
= 0, which were introduced also by Cartan [11]
and studied in [24, 25] as the natural generalization of symmetric spaces for the proper
Riemannian case—see also [4] and references therein.
Why was semisymmetry considered to be the natural generalization of local symme-
try? And, why not going further on to higher derivatives of the Riemann tensor? The
answer to both questions is actually the same: a classical theorem [17, 18, 27] states that
in any proper Riemannian manifold
∇
µ
1
...∇
µ
k
R
α
βγδ
= 0 ⇐⇒ ∇
µ
R
α
βγδ
= 0 (3)
for any k ≥ 1 so that, in particular, (1) is strictly equivalent to (2) in proper Riemannian
spaces. This may well be the reason why there seems to be no name for the condition
(1) in the literature. However, an analogous condition has certainly been used for the
so-called k-recurrent spaces [26, 12]; thus, I will call the spaces satisfying (1) second-
order symmetric, or in short 2-symmetric, and more generally k-symmetric when the left
condition in (3) holds—see [23] for further details.
Results at generic points
As a matter of fact, the equivalence (3) holds as well in “generic" cases of semi-
Riemannian manifolds of any signature. For some results on this one can consult [27,
12]. By “generic point" the following is meant: any point p ∈ V where the matrix
(R
αβ
γδ
)|
p
of the Riemann tensor, considered as an endomorphism on the space of 2-
forms Λ
2
(p), is non-singular. Then, for instance one can prove the following general
result, see [23] for a proof.
Proposition 1 For any tensor field T, and at generic
points, one has
k
z
}| {
∇· · ·· · ·∇ T = 0 ⇐⇒ ∇T = 0
for any k ≥ 1.
Of course, these results apply in particular to the Riemann tensor, and in fact sometimes
even stronger results can be proven. For instance, one can prove a conjecture in [12],
2
(Square) round brackets enclosing indices indicate (anti-)symmetrization, respectively.
namely, that all k-symmetric (and also all k-recurrent) spaces are necessarily of constant
curvature on a neighbourhood of any generic p ∈ V . As a matter of fact, a slightly more
general result is proven in [23]:
Theorem 1 All semi-symmetric spaces are of constant curvature at generic points.
Therefore, there is little room for spaces (necessarily of non-Euclidean signature) which
are k-symmetric but not symmetric nor of constant curvature. It is remarkable that there
have been many studies on 2-recurrent spaces, but surprisingly enough the assumption
that they are not 2-symmetric has always been, either implicitly or explicitly, made. The
paper [23] tries to fill in this gap for the case of 2-symmetry and Lorentzian signature.
LORENTZIAN 2-SYMMETRY
To deal with the problem of k-symmetric and k-recurrent spaces one needs to combine
several different techniques. Among them (i) pure classical standard tensor calculus by
using the Ricci and Bianchi identities; (ii) study of parallel (also called covariantly
constant) tensor and vector fields, and their implications on the manifold holonomy
structure; and (iii) consequences on the curvature invariants. I now present the main
points and results needed to reach the sought results. It turns out that the so-called
“superenergy" and causal tensors [22, 3] are very useful, providing positive quantities
associated to tensors that can be used to replace the ordinary positive-definite metric
available in proper Riemannian cases.
Identities in 2-symmetric semi-Riemannian manifolds
Of course, some tensor calculation is obviously needed, mainly to prove some helpful
quadratic identities. To start with, one needs a generalization of Proposition 1 to the case
of non-generic points.
Lemma 1 Let (V ,g) be an n-dimensional 2-symmetric semi-Riemannian manifold of
any signature. If ∇
λ
∇
µ
T
µ
1
...
µ
q
= 0 then
q
∑
i=1
∇
ν
R
ρ
α
i
λµ
T
α
1
...
α
i−1
ρα
i+1
...
α
q
− R
ρ
νλµ
∇
ρ
T
α
1
...
α
q
= 0, (4)
(∇
ν
R
ρ
τλµ
+ ∇
τ
R
ρ
νλµ
)∇
ρ
T
µ
1
...
µ
q
= 0, (5)
(∇
ν
R
ρ
µ
− ∇
µ
R
ρ
ν
)∇
ρ
T
µ
1
...
µ
q
= 0, (∇
ρ
R
µν
− 2∇
ν
R
ρ
µ
)∇
ρ
T
µ
1
...
µ
q
= 0. (6)
By using the decomposition of the Riemann tensor,
R
αβλµ
= C
αβλµ
+
2
n− 2
R
α
[
λ
g
µ
]
β
− R
β
[
λ
g
µ
]
α
−
R
(n− 1)(n− 2)
g
αλ
g
βµ
− g
αµ
g
βλ
(7)
a selection of the formulas satisfied in 2-symmetric manifolds are given next
Lemma 2 The Riemann, Ricci and Weyl tensors of any n-dimensional 2-symmetric
semi-Riemannian manifold of any signature satisfy
R
ρ
αλµ
R
ρβγδ
+ R
ρ
βλµ
R
αργδ
+ R
ρ
γλµ
R
αβρδ
+ R
ρ
δλµ
R
αβγρ
= 0 (8)
R
ρ
νλµ
∇
ρ
R
αβγδ
+ R
ρ
αλµ
∇
ν
R
ρβγδ
+ R
ρ
βλµ
∇
ν
R
αργδ
+
+R
ρ
γλµ
∇
ν
R
αβρδ
+ R
ρ
δλµ
∇
ν
R
αβγρ
= 0 (9)
∇
(
τ
R
ρ
ν
)
λµ
∇
ρ
R
αβγδ
= 0, ∇
(
τ
R
ρ
ν
)
λµ
∇
ρ
C
αβγδ
= 0, ∇
(
τ
R
ρ
ν
)
λµ
∇
ρ
R
αβ
= 0, (10)
R
ρ
(
µ
R
ρ
ν
)
αβ
= 0, R
ρ
µ
[
αβ
R
γ
]
ρ
= 0, C
ρ
µ
[
αβ
R
γ
]
ρ
= 0, R
ρσ
R
ρµσν
= R
µ
ρ
R
ρν
, (11)
R
ρ
αλµ
C
ρβγδ
+ R
ρ
βλµ
C
αργδ
+ R
ρ
γλµ
C
αβρδ
+ R
ρ
δλµ
C
αβγρ
= 0, (12)
(n− 2)
C
ρ
[
α
λµ
C
ρ
β
]
γδ
+C
ρ
[
γ
λµ
C
ρ
δ
]
αβ
− 2
R
[
α
[
λ
C
µ
]
β
]
γδ
+ R
[
γ
[
λ
C
µ
]
δ
]
αβ
−
−2
R
ρ
[
λ
δ
µ
]
[
α
C
ρ
β
]
γδ
+ R
ρ
[
λ
δ
µ
]
[
γ
C
ρ
δ
]
αβ
+ 2
R
n− 1
δ
[
λ
[
α
C
µ
]
β
]
γδ
+
δ
[
λ
[
γ
C
µ
]
δ
]
αβ
= 0 (13)
and their non-written traces, such as the appropriate specializations of (6). Actually, (8)
and (11-13) are valid in arbitrary semi-symmetric spaces.
Holonomy and reducibility in Lorentzian manifolds
Some basic lemmas on local holonomy structure are also essential. The classical
result here is the de Rham decomposition theorem [20, 16] for positive-definite metrics.
However, this theorem does not hold as such for other signatures, and one has to
introduce the so-called non-degenerate reducibility [28, 29, 30]. See also [1] for the
particular case of Lorentzian signature. To fix ideas, recall that the holonomy group [16]
of (V ,g) is called reducible (when acting on the tangent spaces) if it leaves a non-trivial
subspace of T
p
V invariant. And it is called non-degenerately reducible if it leaves a non-
degenerate subspace (that is, such that the restriction of the metric is non-degenerate)
invariant.
Only a simple result is needed. This relates the existence of parallel tensor fields to
the holonomy group of the manifold in the case of Lorentzian signature. It is a synthesis
(adapted to our purposes) of the results in [14] but generalized to arbitrary dimension n
(see [23] for a proof):
Lemma 3 Let D ⊂ V be a simply connected domain of an n-dimensional Lorentzian
manifold (V ,g) and assume that there exists a non-zero parallel symmetric tensor field
h
µν
not proportional to the metric. Then (D,g) is reducible, and further it is not non-
degenerately reducible only if there exists a null parallel vector field which is the unique
parallel vector field (up to a constant of proportionality).
Some important remarks are in order here:
1. If there is a parallel 1-form v
µ
, then so is obviously h
µν
= v
µ
v
ν
and the manifold
(arbitrary signature) is reducible, the Span of v
µ
being invariant by the holonomy
group. If v
µ
is not null, then (V ,g) is actually non-degenerately reducible. In this
case, the metric can be decomposed into two orthogonal parts as g
µν
= cv
µ
v
ν
+
(g
µν
− cv
µ
v
ν
), where c = 1/(v
µ
v
µ
) is constant. Thus, necessarily g
µν
is a flat
extension [21] of a (n− 1)-dimensional non-degenerate metric g
µν
− cv
µ
v
ν
.
2. If there is a parallel non-symmetric tensor H
µν
, then its symmetric part is also
parallel, so that one can put h
µν
= H
(
µν
)
in the lemma. In the case that H
µν
=
H
[
µν
]
6= 0 is antisymmetric, then in fact one can define H
µρ
H
ν
ρ
= h
µν
, which is
symmetric, parallel, non-zero and not proportional to the metric if n > 2. For these
last two statements, see e.g. [3].
3. Actually, the above can also be generalized to an arbitrary parallel p-form Σ
µ
1
...
µ
p
by defining h
µν
= Σ
µρ
2
...
ρ
p
Σ
ν
ρ
2
...
ρ
p
.
Curvature invariants in 2-symmetric Lorentzian manifolds
Recall that a curvature scalar invariant [13] is a scalar constructed polynomially
from the Riemann tensor, the metric, the covariant derivative and possibly the volume
element n-form of (V ,g). They are called linear, quadratic, cubic, etcetera if they are
linear, quadratic, cubic, and so on, on the Riemann tensor. This defines its degree. The
order can be defined for homogeneous invariants, that is, so that they have the same
number of covariant derivatives in all its terms. This number is the order of the scalar
invariant. Of course, all non-homogeneous invariants can be broken into their respective
homogeneous pieces, and therefore in what follows only the homogeneous ones will
be considered. Similarly, one can define curvature 1-form invariants, or more generally,
curvature rank− r invariants in the same way but leaving 1, ..., r free indices [13].
A simple but very useful lemma is the following [23]
Lemma 4 Let (D,g) be as before with arbitrary signature. Any 1-form curvature invari-
ant which is parallel must be necessarily null (possibly zero).
It follows that, in 2-symmetric spaces, either R is constant or ∇
µ
R is null and parallel.
This is a particular example of the following general important result [23].
Proposition 2 Let D ⊂ V be a simply connected domain of an n-dimensional 2-
symmetric Lorentzian manifold (V ,g). Then either
• all (homogeneous) scalar invariants of the Riemann tensor of order m and degree
up to m+ 2 are constant on D; or
• there is a parallel null vector field on D.
(Observe also that there will be no non-zero invariants involving derivatives of order
higher than one. Then, the degree is necessarily greater or equal than the order.)
The previous proposition has immediate consequences providing more information
about curvature invariants. For instance [23]
Corollary 1 Under the conditions of Proposition 2, either there is a parallel null vector
field on D or the following statements hold
1. All curvature scalar invariants of any order and degree formed as functions of the
homogeneous ones of order m and degree up to m+ 2 are constant on D;
2. All 1-form curvature invariants of order m and degree up to m+ 1 are zero.
3. All scalar invariants with order equal to degree vanish.
4. All rank-2 tensor invariants with order equal to degree are zero.
Remark: Of course, it can happen that the mentioned curvature invariants vanish and
there is a parallel null vector field too.
There is avery long list ofvanishingcurvature invariants asaresult of this Corollary—
if there is no null parallel vector field—. The list of the quadratic ones is (only an
independent set [13] is given, omitting those contaning ∇
µ
R = 0):
R
µν
∇
α
R
µν
= 0, R
µν
∇
µ
R
να
= 0, (14)
R
µνρα
∇
µ
R
νρ
= 0, R
µνρσ
∇
µ
R
νρσα
= 0 = R
µνρσ
∇
α
R
µνρσ
, (15)
∇
α
R
µν
∇
β
R
µν
= ∇
µ
R
νβ
∇
α
R
µν
= ∇
µ
R
να
∇
µ
R
ν
β
= ∇
µ
R
να
∇
ν
R
µ
β
= 0, (16)
∇
µ
R
νρ
∇
α
R
βρµν
= ∇
µ
R
νρ
∇
µ
R
ανβρ
= 0, (17)
∇
α
R
µνρσ
∇
β
R
µνρσ
= ∇
σ
R
µνρα
∇
σ
R
µνρβ
= 0 (18)
where of course the traces of (16-18) vanish, and one could also write the same expres-
sions using the Weyl tensor instead of the Riemann tensor.
MAIN RESULTS
All necessary results to prove the main theorems have now been gathered. Then, by
using the so-called future tensors and “superenergy" techniques [22, 3] one can prove
the following
3
[23]
Theorem 2 Let D ⊂ V be a simply connected domain of an n-dimensional 2-symmetric
Lorentzian manifold (V ,g). Then, if there is no null parallel vector field on D, (D, g) is
either Ricci-flat (i.e. R
µν
= 0) or locally symmetric.
Finally, one can at last prove that the narrow space left between locally symmetric and 2-
symmetric Lorentzian manifolds can only be filled by spaces with a parallel null vector
field.
Theorem 3 Let D ⊂ V be a simply connected domain of an n-dimensional 2-symmetric
Lorentzian manifold (V ,g). Then, if there is no null parallel vector field on D, (D, g) is
in fact locally symmetric.
Thus we have arrived at
Theorem 4 Let D ⊂ V be a simply connected domain of an n-dimensional 2-symmetric
Lorentzian manifold (V ,g). Then, the line element on D is (possibly a flat extension
3
It must be stressed that this proof is only valid for Lorentzian manifolds, as the definition of future
tensors requires this signature.
of) the direct product of a certain number of locally symmetric proper Riemannian
manifolds times either
1. a Lorentzian locally symmetric spacetime (in which case the whole (D,g) is locally
symmetric), or
2. a Lorentzian manifold with a parallel null vector field so that its metric tensor can
be expressed locally as an appropriately restricted case of formula (19) below.
Again, the following remarks are important:
1. Of course, the number of proper Riemannian symmetric manifolds can be zero, so
that the whole 2-symmetric spacetime, if not locally symmetric, is given just by a
line-element of the form (19) restricted to be 2-symmetric.
2. Although mentioned explicitly for the sake of clarity, it is obvious that the block
added in any flat extension can also be considered as a particular case of a locally
symmetric part building up the whole space.
3. This theorem provides a full characterization of the 2-symmetric spaces using
the classical results on the symmetric ones: their original classification (for
the semisimple case) was given in [2], and the general problem was solved
for Lorentzian signature in [9]. Combining these results with those for proper
Riemannian metrics [10, 11, 15], a complete classification is achieved.
Thus, the only 2-symmetric non-symmetric Lorentzian manifolds contain a parallel
null vector field. The most general local line-element for such a spacetime was discov-
ered by Brinkmann [5] by studying the Einstein spaces which can be mapped confor-
mally to each other. In appropriate local coordinates {x
0
,x
1
,x
i
} = {u, v, x
i
}, (i, j,k,... =
2,...,n− 1) the line-element reads
ds
2
= −2du(dv+ Hdu+W
i
dx
i
) + g
ij
dx
i
dx
j
(19)
where the functions H, W
i
and g
ij
= g
ji
are independent of v, otherwise arbitrary, and
the parallel null vector field is given by
k
µ
dx
µ
= −du, k
µ
∂
µ
=
∂
v
. (20)
It is now a simple matter of calculation to identify which manifolds among (19) are
actually 2-symmetric. Using Theorem 4 and its remarks, this will provide —by direct
product with proper Riemannian symmetric manifolds if adequate— all possible non-
symmetric 2-symmetric spacetimes. By doing so [23] one finds, among other results,
that (i) the g
ij
are a one-parameter family, depending on u, of locally symmetric proper
Riemannian metrics
4
; (ii) for a given choice of g
ij
in agreement with the previous point,
the integrability conditions provide the explicit form of the functions H and W
i
; (iii)
finally, the scalar curvature coincides with the corresponding scalar curvature
¯
R of g
ij
:
4
As these are classified in e.g. [11, 15], the part g
ij
of the metric is completely determined. For an explicit
formula, one only has to take any of them from the list and let any arbitrary constants appearing there to
be functions of u.
R =
¯
R. Due to (i), the function
¯
R depends only on u, and thus the 2-symmetry implies
R =
¯
R(u) = au+ b (21)
where a and b are constants. In particular, ∇
µ
R = −ak
µ
. Thus, we see that given any
locally symmetric proper Riemannian g
ij
and letting the constants appearing there to be
functions of u is too general, and these functions are restricted by the 2-symmetry so
that, for example, (21) holds.
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