On the set of metrics without local limiting Carleman weights
aa r X i v : . [ m a t h . A P ] N ov ON THE SET OF METRICS WITHOUT LOCAL LIMITINGCARLEMAN WEIGHTS
PABLO ANGULO-ARDOY
Abstract.
In the paper [AFGR] it is shown that the set of Riemannianmetrics which do not admit global limiting Carleman weights is open anddense, by studying the conformally invariant Weyl and Cotton tensors.In the paper [LS] it is shown that the set of Riemannian metrics whichdo not admit local limiting Carleman weights at any point is residual,showing that it contains the set of metrics for which there are no localconformal diffeomorphisms between any distinct open subsets. Thispaper is a continuation of [AFGR], in order to prove that the set ofRiemannian metrics which do not admit local limiting Carleman weights at any point is open and dense. Introduction
The inverse problem posed by Calder´on asks for the determination of theconductivity of a medium by imposing different voltages in the boundary ofa domain and measuring the induced current at points in the boundary. It isunknown if the problem can been solved in this generality, but this voltageto current data is known to be enough to determine the conductivity in theinterior of the domain under some circumstances. In [DKSU] it was shownthat a few inverse problems are solvable in a domain if the Riemannianmanifold induced from the conductivity coefficients is admissible . The mainlocal restriction is existence of a so called limiting Carleman weight (LCW).Local existence of LCWs admits a nice geometric interpretation:
Theorem 1.1 ([DKSU, Theorem 1.2]) . Let U be a simply-connected opensubset of the Riemannian manifold ( M, g ) . Then ( U, g ) admits a limitingCarleman weight if and only if some conformal multiple of the metric g admits a parallel unit vector field. In [LS], it was shown that the set of Riemannian metrics which do notadmit local limiting Carleman weights at any point is residual (for the C ∞ topology), showing that it contains the set of metrics for which there are nolocal conformal diffeomorphisms between any distinct open subsets. Theorem 1.2 ([LS, Corollary 1.3]) . Let ( U, g ) be an open submanifold ofsome compact manifold ( M, g ) without boundary, having dimension n > .There is a residual set of Riemannian metrics on M (for the C ∞ topology)which do not admit limiting Carleman weights near any point of U . The author was supported by research grant ERC 301179.
However, it is very hard to check if a manifold admits local conformaldiffeomorphisms. Indeed, until [AFGR] appeared, it was difficult to tell ifa given metric admits LCWs or not. In that paper, it is shown that if ametric on a manifold of dimension n > eigenflag property at every point. Thus if, for a givenmetric of dimension n >
4, the Weyl tensor at one point does not satisfy theeigenflag property, then the metric cannot admit a global LCW.
Definition 1.3.
Let W be a Weyl operator on a vector space V . We saythat W has the eigenflag property if and only if there is a vector v ∈ V suchthat W ( v ∧ v ⊥ ) ⊂ v ∧ v ⊥ . In other words, for any w , w , w ∈ v ⊥ , we have W ( v ∧ w , w ∧ w ) = 0. Theorem 1.4 ([AFGR, Theorem 1.3]) . Let ( M, g ) be a Riemannian man-ifold of dimension n > . Assume that a metric ˜ g ∈ [ g ] admits a parallelvector field. Then for any p ∈ M , the Weyl tensor at p has the eigenflagproperty. The authors also provide a similar criterion in dimension n = 3, but thisone involves the Cotton-York tensor: if a metric on a manifold of dimension n = 3 admits a global LCW, then its Cotton-York is singular. Theorem 1.5 ([AFGR, Theorem 1.6]) . Let n = 3 . If a metric ˜ g ∈ [ g ] admits a parallel vector field, then for any p ∈ M det( CY p ) = 0 . This allows the authors to provide many explicit examples of Riemann-ian manifolds which do not admit local limiting Carleman weights, and toshow that the set of Riemannian metrics which do not admit global limitingCarleman weights contains an open and dense set in the C topology. Theorem 1.6 ([AFGR, Theorem 1.9]) . Let ( U, g ) be an open submanifold ofsome compact manifold ( M, g ) without boundary, having dimension n > .The set of Riemannian metrics on M which do not admit limiting Carlemanweights near one fixed point of U contains an open and dense subset of theset of all metrics, endowed with the C topology for n = 3 , and the C topology for n > . In this paper we show that the set of metrics on a compact manifold withboundary which do no admit local LCWs is an open and dense set in the C ∞ topology, building on the techniques in [AFGR].We make use of some basic results in transversality theory . Transversalitytheory is a very general framework that is often useful to prove that the setof certain objects (e.g. Riemannian metrics on a given manifold) for whicha certain derived object (e.g. its Weyl tensor) avoids a certain non typicalset (e.g. the set of Weyl tensors with the eigenflag property) is generic, andsometimes even open and dense, in a certain topology. N THE SET OF METRICS WITHOUT LOCAL LIMITING CARLEMAN WEIGHTS 3
In our specific situation, applying transversality theory is a little bit moreinvolved than usual, since we work with sections of vector bundles insteadof plain maps, and we want to study transversality to stratified sets insteadof plain smooth manifolds. We have dealt with this extra difficulties in ageneral way that may work in other similar situations, and we have tried tomake the presentation as self-contained as possible, so that we only dependon the standard reference [H] and the paper [KS]. We believe that thestrategy presented here may work if the metric is known to belong to somespecial family of metrics, for which the existence of local LCWs is not known.Each Riemannian metric of dimension at least 4 has a
Weyl section , thatassigns to a point p the Weyl tensor of the metric at p . This is a sectionof a suitable vector bundle, the Weyl bundle . The eigenflag bundle is thesubset of the Weyl bundle consisting of the Weyl tensors with the eigenflagproperty. This fiber bundle is no longer a vector bundle, but instead itsfibers are semialgebraic subsets of the space of Weyl tensors. A metric is eigenflag-transverse if its Weyl section is transverse to the eigenflag bundle.These definitions are made precise in definitions 4.2 and 4.3. Theorem 1.7.
Let ( U, g ) be an open submanifold of some compact manifold ( M, g ) without boundary, having dimension n > . Then: • The set of Riemannian metrics on M which do not admit limitingCarleman weights near any point of U contains the set of eigenflagtransverse metrics. • The set of eigenflag transverse metrics is an open and dense subsetof the set of all metrics, with the strong C ∞ topology. For manifolds of dimension 3, we use the Cotton-York tensor instead ofthe Weyl tensor. The
Cotton-York section assigns to a point p the Cotton-York tensor at p . This is a section of the vector bundle of symmetric tracelessbilinear operators on the tangent space S ( T p M ). This vector bundle has afiber sub-bundle consisting of the operators in S ( T p M ) whose determinantis zero. The fiber is a singular algebraic set. A metric is CY-transverse if itsCotton-York section is transverse to the singular Cotton-York bundle. See5.1 for the precise definitions.
Theorem 1.8.
Let ( U, g ) be an open submanifold of some compact manifold ( M, g ) without boundary, having dimension n = 3 . Then: • The set of Riemannian metrics on M which do not admit limit-ing Carleman weights near any point of U contains the set of CY-transverse metrics. • The set of
CY-transverse metrics is an open and dense subset of theset of all metrics, with the strong C ∞ topology. This paper is actually a companion to [AFGR], and we refer the readerto that paper for a more detailed introduction.
P.ANGULO-ARDOY
Acknowledgments:
We thank Daniel Faraco and Luis Guijarro, forconversations on this topic. We also thank an anonymous referee who madea careful reading of a previous version of this document.2.
Weyl and Cotton-York tensors
Let us define the (0 ,
4) curvature tensor R ( z, u, v, w ) = g (cid:0) z, ∇ u ∇ v w − ∇ v ∇ u w − ∇ [ u,v ] w (cid:1) , whose trace is the Ricci tensor Ric ( u, v ) = X i R ( u, e i , v, e i ) , where e i is any orthonormal frame { e i } . The trace of Ric is the scalarcurvature s = X i Ric ( e i , e i )and the Schouten tensor is given by(1) S = 1 n − (cid:18) Ric − n − sg (cid:19) . The symmetries of the curvature tensor at a point p allow to consider itas a symmetric bilinear operator ρ of the space of bivectors Λ ( T p M ): ρ ( x ∧ y, z ∧ t ) = R ( x, y, z, t ) . Here we have used a different letter to distinguish the (0 ,
4) tensor R fromthe curvature operator ρ ∈ S (Λ ( T p M )), but we will use the same letter R in the rest of the paper, when the context will make clear which one we use.The Kulkarni-Nomizu product ? : S ( V ) × S ( V ) → S (Λ ( V )) of twosymmetric 2-tensors is defined by( α ? β ) ijkl = α ik β jl + β ik α jl − α il β jk − β jk α il . With these definitions, we can define the
Weyl operator W :(2) W = R − S ? g. There is another way of looking at the curvature and the Weyl operators.Let V be an euclidean space with an inner product h·i . We define the Bianchimap for V b V : S (Λ ( V )) → Λ ( V ):(3) b V ( R )( x, y, z, t ) = 13 ( R ( x ∧ y, z ∧ t ) + R ( y ∧ z, x ∧ t ) + R ( z ∧ x, y ∧ t )) . The first Bianchi identity says the curvature operator R p is always in thekernel of b T p M , which justifies that R ( V ) = ker( b V ) is called the space of curvature operators for V . N THE SET OF METRICS WITHOUT LOCAL LIMITING CARLEMAN WEIGHTS 5
Note that the Bianchi map is defined without using the inner product of V . The Ricci contraction r V : S (Λ ( V )) → S ( V ) is the map:(4) r V ( R )( x, y ) = T r [ R ( x, · , y, · )] = X i R ( x ∧ e i , y ∧ e i )for an orthonormal frame { e i } . The space of Weyl operators for V is a linearsubspace of the space of curvature operators:(5) W ( V ) = ker ( b V ) ∩ ker ( r V ) . It can be checked easily that b V ( S ? g ) = 0 and r V ( S ? g ) = Ric = r V ( R ),hence the Weyl operator at p always lies in W ( T p M ). Lemma 4.9 andidentity (2) show that for any element in a neighborhood of 0 ∈ W ( T p M ),we can find a metric whose Weyl operator at p is that element.The Weyl tensor is widely used in conformal geometry, since its (1 , n > ,
3) tensor defined as(6) C ijk = ( ∇ i S ) jk − ( ∇ j S ) ik . where the notation ( ∇ a S ) bc stands for ( ∇ ∂ a S )( ∂ b , ∂ c ), so that( ∇ a S ) bc = ∂ a ( S ( ∂ b , ∂ c )) − S ( ∇ a ∂ b , ∂ c ) − S ( ∂ b , ∇ a ∂ c ) . The Cotton tensor has the following symmetries:(7) C ijk = − C jik C ijk + C jki + C kij = 0 g ij C ijk = 0 g ik C ijk = 0 . The Cotton tensor is conformally invariant only in dimension 3, and in-deed, in dimension 3, a metric with vanishing Cotton tensor is conformallyflat.The Cotton tensor is equivalent to the so called
Cotton-York tensor . Thisnew tensor is defined by considering the Cotton tensor as a map C p : T p M → Λ ( T ∗ p M ) (thanks to the anti-symmetry of C with respect to its first twoentries) and composing with the Hodge star operator ∗ : Λ ( T ∗ p M ) → T ∗ p M .This gives a (0 ,
2) tensor that turns out to be symmetric and trace-free, butnot conformally invariant, given by(8) CY ij = 12 C kli g jm ǫ klm √ det g = g jm ( ∇ k S ) li ǫ klm √ det g . It follows from (7) that this tensor is symmetric and its trace is zero: CY ij = CY ji P.ANGULO-ARDOY g ij CY ij = CY ii = 0 . Lemma 5.5 shows that for any sufficiently small symmetric traceless ma-trix, we can find a metric whose Cotton-York tensor at p has that matrixin the canonical basis. In particular, the Cotton-York tensor has no morepointwise symmetries, and we call the space S ( V ) of symmetric tracelessbilinear operators on an euclidean space V the space of Cotton-York tensors of V . 3. Transversality theory
In this work we want to study the transversality properties of the Weylsections and Cotton-York sections, in order to prove Theorems 1.7 and 1.8.Let us recall the main results of transversality theory. The reader can findmore details in the very accessible book [H].3.1.
Weak topology in the space of smooth functions.Definition 3.1. [H, section 2.1] Let M and N be C r manifolds, and C r ( M, N )be the set of C r maps from M to N .Given f ∈ C r ( M, N ), charts ( ϕ, U ) for U ⊂ M and ( ψ, V ) for V ⊂ M ,a compact set K ⊂ U such that f ( K ) ⊂ V , and ε >
0, we define the set N r ( f ; ( φ, U ) , ( ψ, V ) , K, ε ), consisting of all functions g ∈ C r ( M, N ) suchthat g ( K ) ⊂ V and k D k ( ψ ◦ g ◦ ϕ − )( x ) − D k ( ψ ◦ f ◦ ϕ − )( x ) k ε for all points x ∈ K , k = 0 , . . . r .The weak , or compact-open C r topology, on C r ( M, N ), is the topologygenerated by the basis sets N r ( f ; ( φ, U ) , ( ψ, V ) , K, ε ). Definition 3.2.
Let P → M be a C r smooth bundle, and Γ r ( P → M ) bethe set of C r sections of P → M .The weak , or compact-open C r topology, on Γ r ( P → M ), is the topologyinduced by the inclusion of Γ r ( P → M ) into C r ( M, P ), with the weak C r topology. Remark . Both topologies agree for the trivial bundle P = M × N , if weidentify Γ r ( M × N → M ) with C r ( M, N ) in the usual way: C r ( M, N ) → Γ r ( M × N → M ) f → u ( p ) = ( p, f ( p ))Γ r ( M × N → M ) → C r ( M, N ) u → f ( p ) = π ◦ u ( p )where p is a point in M and π : M × N → N is the projection onto thesecond factor. Remark . There is also a different, natural topology for C r ( M, N ) andΓ r ( P → M ), called the strong topology, but it agrees with the weak topologywhen M is compact, as we assume in this paper. N THE SET OF METRICS WITHOUT LOCAL LIMITING CARLEMAN WEIGHTS 7
Definition 3.5. [H, section 2.1] If M and N are smooth manifolds, the C ∞ ( M, N ) (or C ( M, N )) topology is defined as the union of all the C k ( M, N )topologies.The Γ ∞ ( P → M ) can be defined either as the subspace topology inheritedfrom C ∞ ( M, N ), or as the union of the topologies Γ r ( P → M ).We will use the C ∞ topologies for the rest of the paper.3.2. Transverse maps and sections.Definition 3.6. [H, pg 22] Let f : M → N be a smooth map, A ⊂ N asmooth submanifold and K ⊂ M an arbitrary subset. We say f is transverse to A at x ∈ M if and only if either f ( x ) / ∈ A or T y A + d x f ( M x ) = T y N. We say f is transverse to A along K (and write f ⋔ K A ) if and only if f is transverse to A at every x ∈ K . We write f ⋔ A for f ⋔ M A .We define ⋔ K ( M, N ; A ) as the set of all maps f : M → N transverse to A along K , and ⋔ ( M, N ; A ) as ⋔ M ( M, N ; A ).Let Q be a sub-bundle of P whose typical fiber is a smooth manifold. Asection u : M → P is transverse to Q at x ∈ M if and only if u is transverseto Q at x as a map from M into P to the total space of Q , which is a smoothmanifold. Remark . A section u : M → M × N of a trivial bundle is transverse tothe sub-bundle M × A , for a smooth submanifold A of N if and only if theassociated function π ◦ u : M → N is transverse to A .3.3. Stratified sets and stratified bundles.
So far, we have only definedtransversality to a smooth submanifold A ⊂ M . For most applications oftransversality, this is enough, but for the results in this paper we will have toconsider the less common notion of transversality to a smooth stratification.The reason is that the set of Weyl operators with the eigenflag property isnot a smooth manifold, but it has the structure of smooth stratification.The required definitions and theorems for stratifications are indeed quitesimilar to those for smooth manifolds, and in this paper we will only useresults about transversality to submanifolds, which can be found in thestandard reference [H]. Definition 3.8.
A smooth stratification of a closed set S ⊂ N , for a mani-fold N , is collection of disjoint smooth submanifolds S j of dimension j suchthat S ∪ · · · ∪ S k = S (some of the S j may be empty). S is called a strati-fied set , and the S j are called strata . The dimension of S is the maximumdimension of a non-empty strata.We further require that S ∪ · · · ∪ S j is a closed set for each j = 0 . . . k .We will not make use of the following property, but it is central in thetheory of stratifications, and is mentioned in all the references (though some-times with a different name): P.ANGULO-ARDOY
Definition 3.9.
A smooth stratification is regular (or satisfies
Whitney’s Acondition ) if and only if whenever x n → y , for x n ∈ S j and y ∈ S j − , andthe tangents to S j at x n converge to a space τ , then τ contains the tangentto S j − at y . Remark . A smooth manifold with boundary is a regular stratification,with two strata consisting of the interior and the boundary. Any semialge-braic or semianalytic subset S of R n can be stratified, and the stratificationis regular (see [W, page 336]). Definition 3.11.
Let π : P → M be a smooth vector bundle with typicalfiber L , and let S ⊂ L be a subset invariant under the action of the structuregroup of the vector bundle.Then the sub-bundle of P → M associated to S is the subset R of thetotal space consisting of the points that map to S by any trivialization. Forany two trivializations ψ : π − ( U ) → U × L and φ : π − ( V ) → V × L , theinduced map on the fiber over p ∈ U ∩ V is φ ◦ ψ − | { p }× L , which belongs tothe structure group of P → M . Hence R is well defined, and it is clear thatthe restriction of π to R defines a fiber bundle with typical fiber S .A smooth stratified sub-bundle is a sub-bundle associated to a subset S that admits a stratification where each stratum is invariant under the actionof the structure group. Remark . A stratified sub-bundle is the union of the fiber sub-bundles R j → M associated to the strata S j of S . Since the typical fiber of each R j is the smooth manifold S j , the total space R of the bundle is stratified bythe total spaces R j of the sub-bundles. Definition 3.13.
A smooth map f : M → N is transverse to a smoothstratification of a set S ⊂ N if and only if it is transverse to each strata S j .We define ⋔ K ( M, N ; S ) as the set of all maps transverse to S along K ⊂ M .A smooth section u : M → P of P → M is transverse to a stratifiedbundle R → M if and only if it is transverse to each of the sub-bundles R j → M that stratify R → M .We define ⋔ K ( P ; R ) as the set of all sections of P that are transverse to R along K ⊂ M , and ⋔ ( P ; R ) as ⋔ M ( P ; R ).The following lemma is straightforward: Lemma 3.14.
Let u : M → P be a smooth section of a smooth vector bundle π : P → M with typical fiber V , and let R → M be a stratified sub-bundleof P → M associated to S ⊂ V . The following are equivalent: • u is transverse to R → M . • For any trivialization of the bundle ψ : π − ( U ) → U × V , the map π ◦ ψ ◦ u | U : U → V is transverse to each strata S j of S . • u is transverse to the total space R of the bundle as a smooth map(recall that by definition, this means that u is transverse to the totalspace of each R j ). N THE SET OF METRICS WITHOUT LOCAL LIMITING CARLEMAN WEIGHTS 9
The importance of transversality is clear from the following lemma. Onlyits last item is not standard, and this is all that we will need for stratifica-tions. The reader can find a similar result in exercises 3 and 15 of section3.2 of [H].
Lemma 3.15.
Let M and N be smooth manifolds, with M compact, A ⊂ N a smooth submanifold, S ⊂ N a stratified set, and f : M → N a smoothmap. • If f is transverse to A ⊂ N , then f − ( A ) is either empty, or a smoothsubmanifold of M with the same codimension of A . In particular, if dim ( M ) < codim ( A ) , then f − ( A ) is empty. • If f is transverse to S , then f − ( S ) is either empty, or a smoothstratification of M with the same codimension of S . In particular, if dim ( M ) < codim ( S ) , then f − ( S ) is empty. • Assume that only the highest dimensional stratum S k has a codi-mension smaller or equal than dim ( M ) . If f is transverse to S ,then f − ( S ) is either empty, or a smooth compact submanifold of M with codimension codim ( S k ) = dim ( N ) − k .Proof. The first result can be found in [H, Section 1, Theorem 3.3].For the second point, we remark that each f − ( S j ) is a submanifold bythe previous point, so f − ( S ) = ∪ j f − ( S j ) is partitioned into submanifoldswhose codimension is dim ( N ) − j . For any j , f − ( S ) ∪ · · · ∪ f − ( S j ) = f − ( S ∪ · · · ∪ S j ), which is closed.Assume now that S = S ∪ . . . ∪ S j ∪ S k is a smooth stratification wherethe dimension of each stratum, except the top dimensional one, is less that dim ( N ) − dim ( M ), and let f : M → N be a smooth map transverse toeach stratum. By the previous item, f does not intersect the closed set S ∪ · · · ∪ S j , hence f − ( S ) = f − ( S k ). It follows that f − ( S k ) is a closedsubmanifold of M , so it is also compact. (cid:3) The following is a straightforward extension of the above lemma for strat-ified bundles:
Lemma 3.16.
Let R → M be a stratified sub-bundle of P → M and u : M → P a smooth section. Let R → M be a stratified sub-bundle associatedto a set S which admits a smooth stratification S = S ∪ · · · ∪ S j ∪ S k , where j < dim ( P ) − dim ( M ) .Then u − ( R ) is a smooth compact submanifold of M of codimension dim ( P ) − dim ( M ) − k .Proof. By Lemma 3.14, u : M → P is transverse as a map to the total spaceof R , which is a manifold stratified by the total spaces of the sub-bundles R j , whose dimension is dim ( M ) + j . The result follows by the last item ofthe previous lemma. (cid:3) The set of transverse sections.
We recall that a set S ⊂ R n is residual if it can be expressed as a countable intersection of open sets. Adiffeomorphism carries residual sets to residual sets, which makes it possibleto translate this notion to open sets of manifolds. Indeed, this notion makessense in any topological space such that every intersection of a countablecollection of open dense sets is dense. These spaces are called Baire spaces ,and the space C ∞ ( M, N ) is a Baire space (see [H, section 2.4]).This is the main result of transversality theory:
Theorem 3.17.
Let M and N be smooth manifolds, and A be a smoothsubmanifold of N . (1) ⋔ ( M, N ; A ) is residual in C ∞ ( M, N ) . (2) If M is compact and A is closed in N , then ⋔ ( M, N ; A ) is open anddense.Proof. See [H, 3.2.1]. (cid:3)
It follows immediately from the first item that the set of maps transverseto a stratified set is residual. However, the individual strata of a smoothstratification may not be closed, so we cannot use the second item in theprevious lemma. Exercise 15 in section 3.2 of [H] asserts that ⋔ ( M, N ; S ) isopen if the stratification is regular. However, we will not use that hypothesis,so we prove directly that ⋔ ( M, N ; S ) is open under some hypothesis thathold in our situation: Theorem 3.18.
Let M and N be smooth manifolds, with M compact. Let S ∪ . . . S i . . . ∪ S k be a smooth stratification of a closed set S ⊂ N where dim ( S i ) = i (1) Let A be a smooth submanifold of N such that dim ( M ) < codim ( A ) .Then ⋔ ( M, N ; A ) is open and dense. (2) Assume that only the highest dimensional stratum S k has a codimen-sion smaller or equal than dim ( M ) . Then ⋔ ( M, N ; S ) is open anddense.Proof. We know from 3.15 that for f in ⋔ ( M, N ; A ), f ( M ) is disjoint with A , so the distance between them is some ε >
0. We can cover M byfinitely many compact sets K i such that each K i is contained in an chart( U i , φ i ) and f ( K i ) is contained in a chart ( V i , ψ i ). For any r >
1, the set T i N r ( f ; ( φ, U i ) , ( ψ, V i ) , K i , ε/
2) is open in the C ∞ ( M, N ) topology. Forany g in this set, g ( M ) is disjoint with A .This proves that ⋔ ( M, N ; A ) is open, and it is dense by Theorem 3.17.Assume now that S = S ∪ · · · ∪ S k is a closed smooth stratification wherethe dimension of each stratum, except the top dimensional one, is less that dim ( N ) − dim ( M ), and let f : M → N be a smooth map transverse to eachstratum. Then f ( M ) is disjoint to the closed set S ∪ · · · ∪ S j , hence theirdistance is some ε > M by finitely many compact sets K i such that each K i is contained in a chart ( U i , φ i ) and f ( K i ) is contained in a chart ( V i , ψ i ). N THE SET OF METRICS WITHOUT LOCAL LIMITING CARLEMAN WEIGHTS 11
For any element g of U = T i N r ( f ; ( φ, U i ) , ( ψ, V i ) , K i , ε/ g ( M ) is dis-joint with S ∩ · · · ∩ S j . It follows that U ⊂ T jh =0 ⋔ ( M, N ; S h ). We cancompute U ∩ ⋔ ( M, N ; S ) = U ∩ ⋔ ( M, N ; S k )= U ∩ T i ⋔ K i ( M, N ; S k )= T i ( N r ( f ; ( φ, U i ) , ( ψ, V i ) , K i , ε/ ∩ ⋔ K i ( M, N ; S k )) . But for each i N r ( f ; ( φ, U i ) , ( ψ, V i ) , K i , ε/ ∩ ⋔ K i ( M, N ; S k ) = N r ( f ; ( φ, U i ) , ( ψ, V i ) , K i , ε/ ∩ ( ι ∗ U i ) − ( ⋔ K i ( U i , N ; S k )) = N r ( f ; ( φ, U i ) , ( ψ, V i ) , K i , ε/ ∩ ( ι ∗ U i ) − ( ⋔ K i ( U i , V i ; S k )) . where ι U i : U i → M is the inclusion and ι ∗ U i : C ( M, N ) → C ( U i , N ) sends f to f ◦ ι U i , so that ( ι ∗ U i ) − ( ⋔ ( U i , N ; S k )) is the set of maps whose restrictionto U i is transverse to S k along K i .The map ι U i is continuous and ⋔ K i ( U i , V i ; S k ) is open by lemma [H,3.2.3]. Thus U ∩ ⋔ ( M, N ; S ) is an open neighborhood of f . It follows that ⋔ ( M, N ; S ) is open, and it is dense since by Theorem 3.17, each ⋔ ( M, N ; S i )is residual and C ( M, N ) is a Baire space. (cid:3)
Finally, we also need the so called parametric transversality results:
Theorem 3.19.
Let B be a smooth manifold, S ⊂ N a stratified set. Let F : B × M → N be a smooth map transverse to S .Define the functions F b : M → N by F b ( x ) = F ( b, x ) .Then the set ⋔ ( F ; S ) = { b ∈ B : F b ⋔ S } is residual.Proof. By definition, ⋔ ( F ; S ) = T j ⋔ ( F ; S j ), and [H, 3.2.7] shows that eachset ⋔ ( F ; S j ) is residual in V . (cid:3) Proof of Theorem 1.7
It may be tempting to think that a generic metric has at every point aWeyl tensor without the eigenflag property. This is actually true in dimen-sion 5 and above, but not in dimension 4, as we shall see later. However, ifa metric admits a local LCW at one point p , then the Weyl tensor will havethe eigenflag property at all points near p . Thus, we only need to provethat for a generic metric, the set of points whose Weyl tensor does not havethe eigenflag property is dense. Both in dimension 4 and higher, the resultfollows by the transversality arguments in the previous section.Let V be an euclidean space, Λ V the associated space of bivectors, S (Λ V ) the symmetric operators on the space of bivectors, and W ( V ) bethe intersection of the kernels of the Bianchi map b V (3) and the Ricci map r V (4) on S (Λ V ).We recall the following purely algebraic statement from [AFGR]: Theorem 4.1 ([AFGR, Theorem 6.1]) . The subset EW ( V ) of the Weyltensors on V that has the eigenflag property is a semialgebraic subset of thespace of Weyl tensors.Its codimension is exactly: n − n − n + 2 . In particular, the codimension is for n = 4 and for n = 5 . It is greaterthan n for n > . Semialgebraic sets are defined by any combination of polynomial equa-tions and inequalities. They also appear as projections of algebraic sets. Aprojection of a real algebraic set need not be an algebraic set, but it alwaysis semialgebraic, by the Tarski-Seiderberg theorem. As an example, if weproject the circle { ( x, y ) ∈ R : x + y = 1 } onto the x axis, we get theclosed interval [ − , Definition 4.2.
Given a Riemannian manifold (
M, g ), the curvature bundle
R → M is the vector bundle whose fiber at p is the kernel R p of the Bianchimap b p : S (Λ ( T p M )) → Λ ( T p M ).The Weyl bundle
W → M is the vector sub-bundle of the curvaturebundle whose fiber at p is the kernel W p of the Ricci contraction r p : R p → S ( T p M ). Definition 4.3.
The eigenflag bundle
EW → M is the sub-bundle of W → M associated to the subset of W ( R n ) where the Weyl operator has theeigenflag property.The Weyl section of a metric g is the section of the Weyl bundle thatmaps the point p to its Weyl operator W p at p .The curvature bundle is defined for a smooth manifold, and does notdepend on the choice of a Riemannian metric on the manifold. The Weylbundle, however, does depend on g . This is an inconvenient property forthe purposes of this paper: if we define the map that sends a metric g to itsWeyl section p → W p , we would have to use as target space the set Γ( R ) ofsections of the full curvature bundle. Then the property of a metric beingeigenflag-transverse would not be identified with transversality of that mapto the set of sections of a fixed sub-bundle of R , and we could not applyeasily the standard results in transversality theory, like theorems 3.17 and3.18.In order to overcome this technical difficulty, we define the Weyl map from the space of Riemannian metrics into the space of sections of a fixed extended vector bundle, and study when the section that corresponds to ametric is transverse to a fixed stratified sub-bundle.
N THE SET OF METRICS WITHOUT LOCAL LIMITING CARLEMAN WEIGHTS 13
Definition 4.4.
A Riemannian metric is a section of the vector bundlewhose fiber at p is the set S ( T p M ) of symmetric operators on T p M . Fur-thermore, Riemannian metrics must be positive definitive, and this amountsto restricting to an open set (in the C topology) of sections of the bundle S ( T M ). This is the space of Riemannian metrics G ( M ), and we alwaysconsider it with the topology inherited from the compact open C ∞ topologyin Γ ∞ ( S ( T M )).The extended Weyl bundle is the vector bundle whose fiber at p is: f W p = { ( g p , W p ) ∈ S ( T p M ) × R p : r g p ( W p ) = 0 } , where r g p is the Ricci map for T p M and the metric g p .The extended Weyl section of a metric g is the section of the extendedWeyl bundle that sends p to the pair ( g p , W p ).The Weyl map sends a metric g to its extended Weyl section, and is acontinuous map from the space of Riemannian metrics into the space ofsmooth sections of the extended Weyl bundle W : G ( M ) → Γ ∞ ( f W ) . Definition 4.5.
The extended eigenflag bundle g EW → M is the sub-bundleof f W → M associated to the subset of S ( T p M ) × W p where the Weyloperator has the eigenflag property. We will see in Lemma 4.7 that g EW → M is a stratified bundle.A metric is eigenflag-transverse if its extended Weyl section is transverseto the extended eigenflag bundle. Remark . For a fixed metric g , the Weyl bundle for g can be identifiedwith a subset of the extended Weyl bundle in a natural way. Its intersectionwith the extended eigenflag bundle is the eigenflag bundle for g . The tangentto the Weyl bundle for g contains the tangent to the factor W p of the fiberto the extended Weyl bundle. The tangent to the extended eigenflag bundlecontains the tangent to the other factor S ( T p M ). Hence the Weyl bundlefor g is transverse to the extended eigenflag bundle.Thus, the eigenflag bundle for g inherits a stratification from a stratifica-tion of the extended eigenflag bundle, consisting of the intersections of thestrata of the extended eigenflag bundle with the eigenflag bundle for g .The Weyl section of a metric g is transverse to the eigenflag bundle for g if and only if the metric is eigenflag-transverse. Lemma 4.7.
The extended eigenflag bundle is a stratified sub-bundle of theextended Weyl bundle.Proof.
The general linear group GL ( T p M ) acts on S ( T p M ) × R ( T p M ) andpreserves the fiber f W p of the extended Weyl bundle: ρ L ( R )( x ∧ y, z ∧ t ) = R ( L ( x ) ∧ L ( y ) , L ( z ) ∧ L ( t )) ρ L ( g )( x, y ) = g ( L ( x ) , L ( y )) . The set of pairs ( g, W ) where W has the eigenflag property is clearly invari-ant under this action.By Theorem 4.1 and lemma 4.8, the set of Weyl operators with the eigen-flag property admits a stratification that is invariant under the action ofthe structure group of the tangent bundle. The lemma follows from thedefinition of stratified bundle. (cid:3) There are many proofs in the literature (see [W] or [BCR, 9.2.1] for in-stance) that a semialgebraic set such as EW ( V ) admits a smooth stratifica-tion (and in fact, the stratification is regular). However, this stratificationmay not be invariant under the action of GL ( V ). Lemma 4.8.
Let S ⊂ V be a closed semialgebraic subset of a real vectorspace. Let G be a group and let ρ : G × V → V be an action of G on V bysmooth maps such that for every g ∈ G , we have g ( S ) = S .Then S admits a smooth stratification S = S ∪ . . . ∪ S k such that g ( S j ) = S j for every j = 1 . . . k and g ∈ G .Proof. The proof is inspired in the simple proof of a similar result in pages336 and 337 of [W].We proceed by induction on d = dim( S ). We say a point p ∈ S is regularif S ∩ U is a C manifold for some neighborhood U of p . We split S into theset of regular points S reg and its complement S sing . Since the action ρ is byglobally invertible diffeomorphisms, S sing and S reg are invariant under theaction. It is clear that S sing is closed.We can use theorem 2.1 in [KS] to show that S sing is a semialgebraicsubset of S , if we consider any semialgebraic set N (e.g., a point), and let f : S → N be a constant map. Then f − ( f ( x )) = S , and Σ = S sing .Since it is a semialgebraic set, S sing is a finite disjoint union of Nashmanifolds N α . The dimension of each N α can be at most d −
1, sinceotherwise any point of a strata N α with dimension d that is not in theclosure of the other strata would be regular (see [KS, 2.4]).By induction on the dimension of the semialgebraic set, S sing admits asmooth stratification S sing = S ∪ . . . ∪ S d − where all the strata are invariantunder the action ρ .Then S can be stratified by the disjoint subsets S , . . . , S d − , S reg , whichare smooth and invariant under the action. For j = 0 . . . d , the union S ∪ . . . ∪ S j is closed by the induction hypothesis. (cid:3) We also need the following lemma from [AFGR]:
Lemma 4.9 ([AFGR, Lemma 6.5]) . Let M be a Riemannian manifold withmetric g and p any point in M , with R p the curvature of the metric g at p . Let ϕ be a cutoff function with support contained in a neighborhood of p which admits normal coordinates x h .Then there is a number ε > such that, for any algebraic curvatureoperator R ∗ whose norm as a (4 , -tensor is smaller than ε , the following N THE SET OF METRICS WITHOUT LOCAL LIMITING CARLEMAN WEIGHTS 15 defines a Riemannian metric g ′ ij = g ij − X k,h R ∗ ihjk x h x k ϕ ( x ) , whose curvature at p is R p + R ∗ .Proof of Theorem 1.7. We first remark that it is enough to give the prooffor U = M .Let G g EW ( M ) be the subset of G ( M ) consisting of eigenflag transversemetrics, and let ⋔ ( f W , g EW ) be the set of sections of the extended Weylbundle that are transverse to the extended eigenflag bundle.By the transversality Theorem 3.17 and Lemma 4.7, ⋔ ( f W , g EW ) is residualin Γ ∞ ( f W ). However, we remark that G g EW ( M ) = ⋔ ( f W , g EW ), because anarbitrary section of the extended Weyl bundle may not be the Weyl sectionof any metric. As an example, let g E be the euclidean metric in R n , andchoose any nonzero bilinear operator W ∈ W ( R n ). Then x → ( g E , W ) isnot the Weyl section of any metric, since W should be the Weyl tensor ofthe euclidean metric, which is zero. Proof that ⋔ ( f W , g EW ) is open in Γ ∞ ( f W ).If the dimension is 5 or greater, Theorem 4.1 implies that the codimensionof g EW in f W is greater than the dimension of n . The total space of g EW is afinite union of submanifolds { A j } j ∈ J of the total space of f W . For fixed j , theset of maps from M into the total space of f W which are transverse to { A j } j ∈ J is open by the first part of Theorem 3.18, since codim ( A j ) > dim ( M ). Since J is finite, the set ⋔ ( f W , g EW ) is the intersection of an open set with the setof maps that are sections of the bundle, hence ⋔ ( f W , g EW ) is open in Γ( W ).For dimension 4, we start with the results in section 6.1 of [AFGR] (see[AFG, 2.1] for a similar result). The set of Weyl operators with the eigenflagproperty has the following decomposition: • A nonsingular stratum of codimension 2 in W where the operatorsdiagonalize in a basis of simple bivectors, and the Weyl operator hasthree different eigenvalues, each of multiplicity 2. • A stratum of operators with two different eigenvalues: λ of multi-plicity 2 and − λ/ • The zero operator.The second stratum is parameterized by λ and a 2-plane. Since the dimen-sion of the Grassmannian of 2-planes is 2 · (4 −
2) = 4, we deduce thatthe set of Weyl operators with the eigenflag property has a top dimensionalstratum of codimension 2, a stratum of codimension 5 (since the space ofWeyl operators in dimension 4 has dimension 10) and a point.Once again, the total space of g EW is a finite union of submanifolds { A j } j ∈ J of the total space of f W , but this time exactly one of the strata A j has codimension smaller than dim ( M ) = 4. Then by the second part of Theorem 3.18 and the same argument we used for dimension greater than4, we see that the set of sections of f W that are transverse to g EW is open. Proof that G g EW ( M ) is open .Let g ∈ G g EW ( M ). Its extended Weyl section has a neighborhood U inΓ ∞ ( f W ) consisting only of eigenflag-transverse sections. Continuity of theWeyl map ensures that there is an open neighborhood of g in G ( M ) suchthat any metric g ′ in it has an extended Weyl section in U , that is thuseigenflag-transverse. Proof that G g EW ( M ) is dense .We will use the parametric transversality Theorem 3.19. Fix a Riemann-ian metric g ∈ G ( M ). We will build a finite dimensional space of Riemann-ian metrics on the manifold M containing g and parameterized by a map G defined on a smooth manifold B and continuous for the compact open C ∞ topology. Since we intend to use Theorem 3.19, the map G must be suchthat GW ( s, p ) = W ( G ( s )) p is transverse to g EW . Indeed, the map GW willbe a submersion onto f W .For any p ∈ M , let ϕ p be a smooth function such that ϕ p ( p ) = 1 and suchthat U p = supp ( ϕ p ) admits normal coordinates x k .We consider the map G p defined from S ( T p M ) × R ( T p M ) into the spaceof symmetric bilinear operators on M , where G p ( h, R ) agrees with g outside U p and is defined in U p by: G p ( h, R ) ij = ( g ) ij + h ij ϕ p ( x ) − X R ihjk x h x k ϕ p ( x ) . Clearly, G p ( h, R ) is a Riemannian metric if both h and R are small enough(in the sense that all the numbers | h ij | and | R ijkl | are smaller than some ε >
0, for example). By lemma 4.9, the map that assigns to every R ∈ R ( T p M )the curvature of G p (0 , R ) at p has a surjective differential. Its compositionwith the projection onto the Weyl part of the curvature is also surjective.It follows that the differential of ( h, R ) → ( G p ( h, R ) p , W ( G p ( h, R )) p ), issurjective at (0 , O p ⊂ U p such thatthe differential of ( h, R ) → W ( G p ( h, R )) q = ( G p ( h, R ) q , W ( G p ( h, R )) q ) issurjective for any q ∈ O p .We remark that if M admits global coordinates, we can choose ϕ ( x ) = 1for all x ∈ M , but the differential of ( h, R ) → W ( G p ( h, R )) q may still notbe surjective for all q ∈ M . However, since M is compact, there is a finitefamily of points { p λ } , for λ = 1 . . . L , such that M ⊂ ∪ O p λ . The differentialof ( h, R ) → W ( G p λ ( h, R )) q at (0 ,
0) is surjective for any q ∈ O p λ .We can now build the set B and the map G . The map G is defined fromΠ Ll =1 ( S ( T p l M ) × R ( T p l M )) into the space of symmetric bilinear operatorson M : G ( h , R , . . . , h L , R L ) = g + L X λ =1 ( G p λ ( h λ , R λ ) − g ) . N THE SET OF METRICS WITHOUT LOCAL LIMITING CARLEMAN WEIGHTS 17
The tensor G ( s ) is a Riemannian metric if s = ( h , R , . . . , h L , R L ) is suf-ficiently small. Let ˜ B ⊂ Π Ll =1 ( S ( T p l M ) × R ( T p l M )) be a neighborhood of { (0 , } L such that G ( s ) is a Riemannian metric for any s ∈ B . G inducesthe map: GW : ˜ B × M → f W ( s, p ) → W ( G ( s )) p = ( G ( s ) p , W ( G ( s )) p ) . If, for example, p ∈ O p , we restrict GW to ˜ B ∩ S ( T p M ) × R ( T p M ) ×{ (0 , } L − ×{ p } and we recover the map ( h, R, , . . . , , p ) → W ( G p ( h, R )) p .We know from our construction that this map has a surjective differential at h = 0 , R = 0 and it follows that the differential of GW at { (0 , } L × { p } issurjective for any p . In particular, GW is transverse to the extended eigen-flag bundle if we restrict it to a sufficiently small neighborhood B ⊂ ˜ B of { (0 , } L .Thus the parametric transversality theorem 3.19 shows that we can finda parameter s as small as we need, so that the metric G ( s ) is eigenflag-transverse, and as close to g as we want. The second part of Theorem 1.7follows. Proof that no metric in G g EW ( M ) admits a local LCW . It followsfrom the above and Lemma 3.15 that in dimension 5 and above, the Weyltensor of an eigenflag-transverse metric never has the eigenflag property. Indimension 4, the Weyl tensor of an eigenflag-transverse metric may havethe eigenflag property at the points of a 2 dimensional compact manifold.In both situations, the subset of M consisting of points whose Weyl tensordoes not have the eigenflag property is dense, and thus there cannot be anyLCW defined in an open set. The first part of Theorem 1.7 follows. (cid:3) Proof of Theorem 1.8.
In dimension 3, the Weyl tensor vanishes at every point, so we use theCotton-York tensor instead. We saw in section 2 that the Cotton-Yorkoperator is symmetric and traceless. The symmetric operators on
T M aredefined independently of the metric, but the definition of the trace requiresuse of the metric. We start with definitions analogous to the ones in 4.2:
Definition 5.1.
Given a Riemannian manifold (
M, g ), the
Cotton-York bun-dle
CY → M is the vector bundle whose fiber at p is the set of symmetrictraceless operators S ( T p M ).The singular Cotton-York bundle is the sub-fiber bundle SCY → M of CY → M whose fiber at p is the subset of the operators in S ( T p M ) withzero determinant.The Cotton-York section of a metric g maps a point p to its Cotton-Yorktensor CY p .For the same reasons as in the previous section, we define extended bun-dles whose definitions do not require a particular metric. Definition 5.2.
Given a manifold M , the extended Cotton-York bundle f CY → M is the vector bundle whose fiber at p is { ( g p , Y p ) ∈ S ( T p M ) × S ( T p M ) : X g ij Y ij = 0 } . The extended singular Cotton-York bundle is the sub-fiber bundle ] SCY → M of f CY → M whose fiber at p consists of those pairs ( g, Y ) where det ( Y )is zero.We will see in 5.4 that it is a stratified bundle.The extended Cotton-York section of a metric g is the section of the ex-tended Cotton-York bundle that assigns to each point p the pair ( g p , CY p ),where CY p is the Cotton-York tensor of g at p .A metric is SCY-transverse if its Cotton-York section is transverse to thesingular Cotton-York bundle.For the same reasons mentioned in Remark 4.6, we only need to definetransversality in the extended bundles.
Definition 5.3.
The extended Cotton-York map sends a metric g to itsextended Cotton-York section, and is defined from the space of smooth Rie-mannian metrics into the space of smooth sections of the extended Cotton-York bundle: CY : G ( M ) → Γ ∞ ( CY ) . Lemma 5.4.
The extended singular Cotton-York bundle is a stratified sub-bundle of the extended Cotton-York bundle.Proof.
The proof can be done using the exact same ideas used in 4.7, sinceboth the set of Cotton-York tensors and the subset of singular Cotton-Yorktensors are invariant under the action of the general linear group: ρ L ( g )( x, y ) = g ( L ( x ) , L ( y )) ρ L ( Y )( x, y ) = Y ( L ( x ) , L ( y )) . (cid:3) The following Lemma plays the role of Lemma 4.9 for the Cotton tensor:
Lemma 5.5 ([AFGR, Theorem 6.7]) . Let M be a Riemannian manifoldof dimension with metric g and p any point in M . Let ϕ be a cutofffunction with support contained in a neighborhood of p which admits normalcoordinates x k .Define the vector space of tuples of numbers a klmij invariant under permu-tations of the lower, and of the upper indices: A = { ( A klmij ) : i, j, k, l, m ∈ { , , } , A klmij = A σ ( klm ) π ( ij ) , π ∈ S , σ ∈ S } . Then for any algebraic Cotton-York tensor CY close enough to CY p ,there is ( A jklij ) ∈ A such that for the metric g ′ : g ′ ij = g ij + ϕ X A klmij x k x l x m N THE SET OF METRICS WITHOUT LOCAL LIMITING CARLEMAN WEIGHTS 19 the Cotton-York tensor at p is CY .Proof of Theorem 1.8. The argument is now similar to the one for dim( M ) >
4. Let G ] SCY ( M ) be the subset of G ( M ) consisting of SCY-transverse metrics,and let ⋔ ( f CY , ] SCY ) be the set of sections of the extended Cotton-Yorkbundle that are transverse to the singular Cotton-York bundle.
Proof that ⋔ ( f CY , g SCY ) is open in Γ ∞ ( f CY ).We have to stratify the space of singular traceless symmetric operators(see [AFG, 3.2] for a similar result). This time there are only two strata: • A nonsingular stratum where the operator Y has eigenvalues λ, − λ, • The zero operator.For an operator in the first stratum, since the eigenvalues are different, theoperator diagonalizes in a unique orthonormal basis (up to reordering theelements). We can cover the stratum with a mapping( λ, Q ) → Q · λ − λ
00 0 0 · Q t defined on R × SO (3). Thus, the set of singular Cotton-York operators hasa top dimensional strata of dimension 4, and hence codimension 1, and apoint (of codimension 5). Then the second part of Theorem 3.18 proves thatthe set of sections of f CY that are transverse to ] SCY is open.
Proof that G g SCY ( M ) is open .Let g ∈ G ] SCY ( M ). Its extended Cotton-York section has a neighborhood U in Γ ∞ ( f CY ) consisting only of SCY-transverse sections.Continuity of CY ensures that there is an open neighborhood of g in G ( M )such that any metric g ′ in it has an extended CY section in U , that is thusSCY-transverse. In other words, the set of SCY -transverse metrics is openin G ( M ). Proof that G g SCY ( M ) is dense .Let g be an arbitrary Riemannian metric on M .For any p ∈ M , let ϕ p be a smooth function such that ϕ p ( p ) = 1, suchthat U p = supp ( ϕ p ) admits normal coordinates x k .For every h ∈ S ( T p l M ) and A ∈ A , we define a metric G p ( h, A ) thatagrees with g outside U p and is defined in U p by:( h ij , A klmij ) → G p ( h, A ) ij = ( g ) ij + ϕ p ( x ) h ij + X ϕ p ( x ) A klmij x k x l x m By Lemma 5.5, there is an open neighborhood O p ⊂ U p of p such that( h, A ) → CY ( G p ( h, A )) q has a surjective differential for any q ∈ O p .We collect a finite family of points { p λ } , for λ = 1 . . . L , such that M ⊂∪ O p λ . We define ˜ B to be a neighborhood of { (0 , } L in Π Ll =1 ( S ( T p l M ) × A ) such that G ( h , A , . . . , h L , A L ) = g + X λ ( G U λ ( h λ , A λ ) − g )is a Riemannian metric for any ( h , A , . . . , h L , A L ) ∈ ˜ B . G induces a map:GCY : ˜ B × M → f CY ( s, p ) → CY ( G ( s )) p = ( G ( s ) p , CY ( G ( s )) p ) . If, for example, p ∈ O p , we restrict GCY to S ( T p M ) × A ×{ (0 , } L − ×{ p } , and recover the map ( h, A, , . . . , , p ) → CY ( G p ( h, A )) p . We knowfrom our construction that this map has a surjective differential at h =0 , A = 0 and it follows that the differential of GCY at { (0 , } L × { p } issurjective for any p . In particular, GCY is transverse to the extended singu-lar Cotton-York bundle if we restrict it to a sufficiently small neighborhood B ⊂ ˜ B of { (0 , } L .Thus the parametric transversality theorem 3.19 shows that we can find aparameter s as small as we need, so that the metric G ( s ) is SCY-transverse,and as close to g as we want. The second part of Theorem 1.8 follows. Proof that no metric in G g SCY ( M ) admits a local LCW .It follows from Lemma 3.15 that the Cotton-York tensor of an SCY-transverse metric is singular in a (possibly empty) compact manifold ofdimension 2, and it never vanishes. Thus, the subset of M consisting ofpoints whose Cotton-York tensor is not singular is dense, and thus therecannot be any LCW defined in an open set. The first part of Theorem 1.8follows. (cid:3) References [AFGR] Angulo-Ardoy P., Faraco D., Guijarro L., Ruiz A.,
Obstructions to the existenceof limiting Carleman weights , Analysis & PDE 9-3 (2016), 575–595. (arXiv:1411.4887)[AFG] Angulo-Ardoy P., Faraco D., Guijarro L.,
Sufficient conditions for the existence oflimiting Carleman weights , (Preprint arXiv:1603.04201)[BCR] Bochnak J., Coste M., Roy M-F. Real algebraic geometry (Springer, 1998)no. 8, 549552.[DKSU] Dos Santos Ferreira, D., Kenig, C. E., Salo, M., Uhlmann, G.,
Limiting Carlemanweights and anisotropic inverse problems , Invent. Math. 178 (2009), no. 1, 119–171.[H] Hirsch, M.,
Differential Topology , Graduate Texts in Mathematics, 33. Springer-Verlag, New York, 1976[KS] Koike, S. and Shiota M.,
Non-smooth points set of fibres of a semialgebraic mapping ,J. Math. Soc. Japan, Vol. 59, No. 4 (2007) pp. 953969[LS] Liimatainen, T., Salo, M.,
Nowhere conformally homogeneous manifolds and limitingCarleman weights , Inverse Probl. Imaging 6 (2012), no. 3, pp. 523–530.[W] Wall, C.T.C.,
Regular stratifications , Proceedings of “Applications of Topology andDynamical Systems” at Warwick, Lecture Notes in Mathematics 468, pp. 332-345
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