A note on the coincidence of the projective and conformal Weyl tensors

0.08 MB·English

by  Christian Lübbe

A note on the coincidence of
the projective and conformal Weyl tensors
Christian Lu¨bbe ∗
Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK
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January 25, 2013
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n
a Abstract
J
Thisarticle examinesthecoincidenceoftheprojectiveandconformal Weyltensors asso-
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ciatedtoagivenconnection∇. TheconnectionmaybeageneralWeylconnectionassociated
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toaconformalclassofmetrics[g]. Themainresultforn≥4isthattheWeyltensorscoincide

iff ∇ is theLevi-Civita connection of an Einstein metric.
]
G
D
1 Introduction
.
h
t In 1918HermannWeyl introduced, whatis nowknownas Weyl geometries[5]. He observedthat
a
m the Riemann curvature has a conformally invariant component C k, which he referredto as the
ij l
conformalcurvature. In[6]Weyldiscussedbothconformalandprojectivegeometriesandshowed
[
that analogously the Riemann curvature has a projectively invariant component W k, referred
ij l
1
to as the projective curvature. The idea has been extend to parabolic geometries, (see e.g. [1],
v
[2]) and in the modern literature the invariant curvature component is simply referred to as the
9
5 Weyl tensor or the Weyl curvature, with the type of geometry typically implied by the context.
6 In this article we will be dealing with C k and W k simultaneously and we will refer to them
ij l ij l
5 as the conformal and projective Weyl tensors respectively.
1. In [4] Nurowski investigated when a given projective class of connections [∇] on M includes
0 a Levi-Civita connection of some metric g on M. An algorithm to check the metrisability of a
3 chosen projective structure was given. In proposition 2.5 of [4] it was shown that the projective
1
and conformalWeyl tensors coincide if and only if the Ricci tensor of the Levi-Civita connection
:
v satisfies
i M efR =0 (1)
X abcd ef
r where
a M ef =2g δeδf +2g g gef +2(n−1)g δfδe.
abcd a[c d] b a[d c]b b[d c] a
Corollary2.6of[4]deduces that the projectiveandconformalWeyltensorsofanEinsteinmetric
are equal. As a comment Nurowski raised the question whether there are non-Einstein metrics,
which satisfy condition (1).
This article proves that this is not the case. In particular, for a given connection ∇ on an
n ≥ 4 dimensional manifold the projective and conformal Weyl tensors associated to ∇ only
agreeif ∇ is the Levi-Civita connectionof anEinstein metric. The problemis addressedin more
generality by allowing for generalWeyl connections. This generalisationis of interest, due to the
fact that neither the Ricci curvature of a general Weyl connection nor the Ricci curvature of a
projective connection need be symmetric. Hence the possibility exists that the two Weyl tensors
agree when using a general Weyl connection that is not a Levi-Civita connection for a metric in
[g].
∗E-mailaddress:[email protected]
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2 Projective and conformal connection changes
We define the tensors
Σkl =δkδl +δlδk, Skl =δkδl +δlδk−g gkl
ij i j i j ij i j i j ij
Two connections ∇ and ∇ˇ are projectively related if there exists a 1-form ˇb such that the
i
connection coefficients are related by
Γˇ k =Γk +Σklˇb
i j i j ij l
We denote the class of all connections projectively related to ∇ by [∇] .
p
Supposefurtherthat∇isrelatedtotheconformalclass[g]. Bythiswemeanthatthereexists
a 1-form f such that
i
∇ g =−2f g (2)
i kl i kl
Thisholdsforg iffitholdsforanyrepresentativein[g]. Connectionsthatsatisfy(2)arereferred
ij
to as generalWeyl connections of [g]. Note that the Levi-Civitaconnection ofany representative
in [g] satisfies (2). However ∇ need not be the Levi-Civita connection for a metric in [g].
The connections ∇ and ∇ˆ are conformally related if there exists a 1-form ˆb such that the
i
connection coefficients are related by
Γˆ k =Γk +Sklˆb
i j i j ij l
We denote the class of all connections conformally related to ∇ by [∇] . Observe that all con-
c
nections in [∇] satisfy (2).
c
3 Decomposition of the Riemann curvature
Given a connection ∇ the Riemann and Ricci tensors are defined as
2∇ ∇ vk =R k vl, R =R k
[i j] ij l jl kj l
The projective and conformal Schouten tensor are related to the Ricci tensor of ∇ by [1], [3]
1 1
ρ = R + R
ij n−1 (jl) n+1 [jl]
1 1 R gkl
P = R + R − kl g
ij n−2 (jl) n [jl] 2(n−2)(n−1) ij
The Schouten tensors can be used to decompose the Riemann curvature as follows
R k =W k +2Σkmρ =C k +2SkmP , (3)
ij l ij l l[i j]m ij l l[i j]m
where W k and C k are the projective and conformal Weyl tensors respectively. Moreover the
ij l ij l
once contracted Bianchi identity ∇ R k =0 implies [1] that
k ij l
∇ W k = 2(n−2)∇ ρ =(n−2)y (4)
k ij l [i j]l ijl
∇ C k = 2(n−3)∇ P =(n−3)Y . (5)
k ij l [i j]l ijl
The tensor y and Y are known as the Cotton-York tensors.
ijl ijl
Underaconnectionchange∇ˇ =∇+ˇb respectively∇ˆ =∇+ˆb the Schoutentensorstransform
as
1
ρ −ρˇ = ∇ b + Σklˇb ˇb
ij ij i j 2 ij k l
1
P −Pˇ = ∇ b + Sklˆb ˆb
ij ij i j 2 ij k l
2
In both cases the Schouten tensors absorb all terms that arise in the Riemann tensor under
connection changes. It follows that the projective Weyl tensor W k and the conformal Weyl
ij l
tensor C k are invariants of the projective class [∇] and the conformal class [∇] , respectively.
ij l p c
The question we wish to address is for which manifolds these two invariants coincide.
We note that for n≤2 W k =0 and for n≤3 C k =0. Thereforeit follows trivially that:
ij l ij l
In n = 2 the Weyl tensors always agree. In n = 3 they agree if and only if the manifold is
projectively flat, i.e. the flat connection is contained in [∇]
p
Hence in the following we focus only on n>3.
4 Coincidence of the conformal and projective
Weyl tensors
The Ricci tensor can be decomposed into its symmetric trace-free, skew and trace components
with respect to the metric g :
ij
R
R = Φ +ϕ + g (6)
ij ij ij ij
n
Hence the Schouten tensors can be rewritten as
1 1 R
ρ = Φ + ϕ + g (7)
ij n−1 ij n+1 ij n(n−1) ij
1 1 R
P = Φ + ϕ + g (8)
ij n−2 ij n ij 2n(n−1) ij
The condition W k =C k is equivalent to
ij l ij l
2Σkmρ =2SkmP (9)
l[i j]m l[i j]m
Substitutions of (7) and (8) give
2 2R 2 2
2Σkmρ = δkΦ + δkg + δkϕ − δkϕ
l[i j]m n−1 [i j]l n(n−1) [i j]l n+1 [i j]l n+1 l ij
2 2 2R
2SkmP = δkΦ − g Φ gkm+ δkg
l[i j]m n−2 [i j]l n−2 l[i j]m n(n−1) [i j]l
2 2 2
+ δkϕ − g ϕ gkm− δkϕ
n [i j]l n l[i j]m n l ij
We observe that the scalar curvature terms are identical on both sides and hence only Φ and
ij
ϕ are involved in our condition. The scalar curvature can take arbitrary values.
ij
Taking the trace over il and equating both sides.
1 R 3
2Σkmρ gil = Φ k− δk+ ϕ k
l[i j]m n−1 j n j n+1 j
R 4−n 4
2ΣkmP gil = −Φ k− δk+ ϕ k =−R k+ ϕ k
l[i j]m j n j n j j n j
Comparing irreducible components we find that we require
n n2−4
Φ k =0 and ϕ k =0 (10)
n−1 j n(n+1) j
Thus under our assumption of n > 3, both Φ and ϕ must vanish. It follows that the Ricci
ij ij
tensor is pure trace and hence g is an Einstein metric. Note that the Bianchi identities (4), (5)
imply that R is constant.
The result can be formulated as follows
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Theorem 1. Let ∇ be a connection related to the conformal class [g].
• In n=2 the Weyl tensors always vanish and hence agree.
• In n=3 the Weyl tensors agree if and only if the manifold is projectively flat, i.e. the flat
connection is contained in [∇]
p
• Inn≥4theWeyltensorsagreeifandonlyiftheconnection∇istheLevi-Civita connection
of the metric g and the manifold is an Einstein manifold.
Corollary 1. If the projective and conformal Weyl tensor for n ≥ 4 coincide then the Cotton-
York tensors coincide as well. In fact they vanish identically.
TheresultfollowsimmediatelyfromthefactthattheconnectionistheLevi-Civitaconnection
ofanEinsteinmetric. HencetheSchoutentensorsareproportionaltothemetricandbothCotton-
York tensors vanish.
5 Conclusion
It has been shown that the coincidence of the projective and conformal Weyl tensors is closely
linked to the conceptof Einsteinmetrics. For metric connectionsin [∇] one could havededuced
c
the main result directly from(1) by using the above decompositionof the Ricci tensor and using
suitable traces of (1). However,the set-up given here allowed for a direct generalisationto Weyl
connections without requiring a more general form of (1). Moreover it was felt that the set-up
provided more clarity of role of the different types of curvatures involved.
References
[1] T. N. Bailey, M. G. Eastwood and A. R. Gover. Thomas’s structure bundle for conformal,
projective and related structures. Rocky Mountain J. Math., 24(4):1191–1217,1994.
[2] A.CˇapandJ.Slov´ak, Parabolic geometries I. Background and general theory, Mathematical
Surveys and Monographs 154 (2009)
[3] H. Friedrich, Conformal geodesics on vacuum spacetimes, Comm. Math. Phys. 235, 513
(2003).
[4] P. Nurowski, Projective versus metric structures, J. Geom. Phys. 62 (2012), 657674.
[5] H. Weyl, Reine Infinitesimalgeometrie, Mathematische Zeitschrift 2 (1918), 384-411.
reprinted in Gesammelte Abhandlungen ed. K.ChandrasekharanVol.2 1–28
[6] H. Weyl, Zur Infinitesimalgeometrie, Nachrichten der K¨oniglichen Gesellschaften der Wis-
senschaften zu G¨ottingen, Mathematisch-physikalische Klasse (1921), 99-112. reprinted in
Gesammelte Abhandlungen ed. K.ChandrasekharanVol.2 195-207
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