Limiting Carleman weights and conformally transversally anisotropic manifolds
Pablo Angulo-Ardoy, Daniel Faraco, Luis Guijarro, Mikko Salo
aa r X i v : . [ m a t h . A P ] N ov LIMITING CARLEMAN WEIGHTS AND CONFORMALLYTRANSVERSALLY ANISOTROPIC MANIFOLDS
PABLO ANGULO, DANIEL FARACO, LUIS GUIJARRO, AND MIKKO SALO
Abstract.
We analyze the structure of the set of limiting Carlemanweights in all conformally flat manifolds, 3-manifolds, and 4-manifolds.In particular we give a new proof of the classification of Euclidean lim-iting Carleman weights, and show that there are only three basic suchweights up to the action of the conformal group. In dimension three weshow that if the manifold is not conformally flat, there could be one ortwo limiting Carleman weights. We also characterize the metrics thathave more than one limiting Carleman weight. In dimension four weobtain a complete spectrum of examples according to the structure ofthe Weyl tensor. In particular, we construct unimodular Lie groupswhose Weyl or Cotton-York tensors have the symmetries of conformallytransversally anisotropic manifolds, but which do not admit limitingCarleman weights. Introduction
Background.
A version of the inverse problem of Calder´on [Ca80]asks for the determination of a potential q from boundary measurements(given by the Dirichlet-to-Neumann map Λ q ) for the Schr¨odinger operator − ∆ g + q in a compact Riemannian manifold ( M, g ) with boundary. Thereis an extensive literature for the case where (
M, g ) is a domain in Euclideanspace (see the survey [Uh14]). The corresponding problem for a compactmanifold (
M, g ) has been solved in two dimensions [GT11], or when theunderlying structures are real-analytic (see [LU89, LU02, GS09, LLS18]).The problem remains open in general when dim ( M ) > M, g ) belongs to theclass of conformally transversally anisotropic (CTA) manifolds . A manifoldis said to be CTA if there exists a conformal factor c ( x ) and an ( n − M , g ) such that ( M, g ) is isometric to a domain in R × M withthe metric c ( x )( dx ⊕ g ). In this article will always assume that n = dim ( M ) > . The first three authors were supported by research grants MTM2014-57769-1-P,MTM2014-57769-3-P and MTM2017-85934-C3 from the Ministerio de Ciencia e Inno-vaci´on (MCINN), by ICMAT Severo Ochoa projects SEV-2011-0087 and SEV-2015-0554(MINECO), and by the ERC 301179. The fourth author was supported by the Academy ofFinland (grants 284715 and 309963) and by ERC under Horizon 2020 (ERC CoG 770924).
If (
M, g ) is a CTA manifold, it turns out that one can construct complexgeometrical optics solutions for the equation ( − ∆ g + q ) u = 0 in M as in theclassical approach of [SU87] in the Euclidean case. This is based on the factthat the function ϕ ( x ) = x is a so called limiting Carleman weight (LCW),see [KSU07, DKSU09]. It was proved in [DKSU09] that the existence of anLCW is locally equivalent to the manifold begin CTA. Moreover, if ( M, g )is a CTA manifold, one can solve the inverse problem of determining thepotential q from boundary measurements if additionally • ( M , g ) is simple [DKSU09]; • ( M , g ) has injective geodesic X-ray transform [DKLS16]; or • ( M, g ) is conformal to a subdomain of R × M , instead of just R × M (this follows by combining [DKLS16] with [Sa17, Theorem 1.3]).Unfortunately, due to the conformal invariance it is not always easy todecide whether a manifold is CTA. In [AFGR16, AFG17] conformal sym-metries were exploited to investigate whether a manifold is CTA or not. Inparticular, [AFGR16] gave necessary conditions for a manifold to be CTAby proving that the Weyl or Cotton tensor of a CTA manifold needs to havecertain specific algebraic structure. This work also gave the first explicit ex-amples of manifolds (e.g. C P , Nil, ^ SL ( R )) that do not admit LCWs; it hadbeen shown earlier that generic manifolds do not admit LCWs [LS12, An17].The article [AFG17] complemented [AFGR16] by giving several sufficientconditions for a manifold to admit an LCW in dimensions 3 and 4.The works [AFGR16, AFG17] were concerned with necessary and suffi-cient conditions for a manifold to admit at least one LCW. The first goalof the present article is to analyze and classify the set of all possible LCWsin a given Riemannian manifold. In particular we prove that having sev-eral LCWs imposes very strong symmetries in the manifold. The secondaim is to further clarify the difference between the necessary conditions in[AFGR16] and the presence of an actual LCW.For the first goal we start by revisiting the case where LCWs are mostabundant, i.e. Euclidean space.1.2. LCWs in Euclidean space.
LCWs in a domain Ω ⊂ R n , n >
3, werecharacterized in [DKSU09, Theorem 1.3] where it was proved that any LCWin Ω belongs to one of six different families up to translation and scaling.The proof was based on two facts: • the level sets of an LCW are umbilical hypersurfaces; and • any umbilical hypersurface in R n is part of a hyperplane or sphere.The result then followed by an ODE analysis of the parameters that definethe level sets.We begin by giving a new proof of the classification of LCWs in R n .Instead of using the fact that the level sets of LCWs are umbilical hypersur-faces, we will start from the observation (also made in [DKSU09]) that anyLCW ϕ has an associated conformal Killing vector field |∇ ϕ | − ϕ . Thus one IMITING CARLEMAN WEIGHTS AND CTA MANIFOLDS 3 could try to classify LCWs by first classifying all conformal Killing vectorfields, and then checking which conformal Killing vector fields give rise toLCWs.It is well known (and recalled in Lemma 2.3) that in R n with n >
3, anyconformal Killing vector field is of the form X ( x ) = ( α · x ) x − α | x | + cx + Bx + γ for some α, γ ∈ R n , c ∈ R , and some skew-symmetric matrix B . We willwrite X = ( α, c, B, γ ) for short. Using this characterization, the fact that d ( | X | − X ) = 0 for X arising from an LCW, and arguments based on theconformal invariance of the problem, we obtain the following restatement of[DKSU09, Theorem 1.3]. Theorem 1.1.
Let Ω ⊂ R n , n > , be a connected open set, and let ϕ bean LCW in Ω . Then ϕ ( x ) = aψ ( x − x ) + b for some a ∈ R \ { } , b ∈ R and x ∈ R n , where ψ is one of the following six functions and X = |∇ ψ | − ∇ ψ is the conformal Killing vector field for ψ : ψ ( x ) = γ · x, X = (0 , , , γ ) ,ψ ( x ) = log | x | , X = (0 , , , ,ψ ( x ) = arc tan γ · xσ · x , X = (0 , , γ ∧ σ, ,ψ ( x ) = − γ · x | x | , X = ( γ, , , ,ψ ( x ) = √ s arctan − γ · x/ √ s | x | /s − , X = ( γ, , , s γ ) ,ψ ( x ) = √ s arctanh − γ · x/ √ s | x | /s +1 , X = ( γ, , , − s γ ) . Above γ, σ ∈ R n satisfy | γ | = | σ | = 1 and γ · σ = 0 , and s > . The six families above are the same as in [DKSU09]. However, we alsoprove that the last three families can be obtained from the first three byconformal mappings, thus reducing the number of basic LCWs to three:
Theorem 1.2.
The group of conformal transformations in R n acts on theset of LCWs by ψ F ∗ ψ = ψ ◦ F . This action, combined with the action ψ aψ + b where a ∈ R \ { } and b ∈ R , has exactly three orbits, givenby the following representatives ( e and e are the first two vectors of thecanonical basis): ψ ( x ) = e · xψ ( x ) = log | x | ψ ( x ) = arc tan e · xe · x . In other words, for any LCW ϕ defined in an open set Ω ⊂ R n , there isexactly one i ∈ { , , } , a conformal transformation F defined on the onepoint compactification of R n , and two real numbers a, b with a = 0 , suchthat ϕ is the restriction of a ( ψ i ◦ F ) + b to Ω . P.ANGULO, D. FARACO, L. GUIJARRO, AND M.SALO
If we only consider affine conformal mappings (without the inversion),there are exactly six orbits, corresponding to the six families in Theorem1.1.
LCWs on general manifolds.
Next, we turn our attention to generalmanifolds. We say that LCWs ϕ , . . . , ϕ m are orthogonal if their gradientsare orthogonal at each point. The next theorem describes the underlyinggeometries and shows that orthogonal LCWs can be chosen as coordinates. Theorem 1.3.
Let ( M, g ) be an n -dimensional Riemannian manifold. Let p ∈ M , and suppose that in an open neighbourhood of p , the conformal class [ g ] admits m different orthogonal LCWs ϕ , . . . , ϕ m . Then there are localcoordinates Φ = ( z , . . . , z n ) near p such that(1) z = ϕ , . . . , z m = ϕ m ;(2) some conformal multiple of g has the local expression . . . f ( z m +1 , . . . , z n ) . . . ... . . . f m ( z m +1 , . . . , z n ) 00 0 . . . D ( z m +1 , . . . , z n ) where D is an ( n − m ) × ( n − m ) symmetric matrix.Conversely, if in some local coordinates ( z , . . . , z n ) the metric has the aboveform up to a conformal multiple, then z , . . . , z m are orthogonal LCWs. The above Theorem has the following relevant corollary:
Corollary 1.4. An n -dimensional manifold with n orthogonal LCWs isconformally flat. As a matter of fact, Theorem 1.3 in combination with the analysis of[AFG17, AFGR16] yields a complete understanding of 3-manifolds. Firstwe obtain the following theorem:
Theorem 1.5.
Let ( M, g ) be a -manifold. Locallyi) ( M, g ) admits a limiting Carleman weight if and only if ( M, g ) isconformal to R × M , where M is a -manifold;ii) ( M, g ) admits two limiting Carleman weights with linearly indepen-dent gradients if and only if ( M, g ) is conformal to R × S , where S is a surface of revolution;iii) ( M, g ) admits three limiting Carleman weights with linearly indepen-dent gradients if and only if ( M, g ) is conformally flat. Note that if ϕ and ϕ are two LCWs with linearly independent gradients,then ϕ is not of the form aϕ + b for real numbers a = 0 , b .For 3-manifolds, the necessary condition from [AFGR16] for a manifoldto be CTA is det( CY ) = 0, where CY is the Cotton-York tensor. It was stillopen whether this condition is sufficient. Unimodular Lie groups admit left IMITING CARLEMAN WEIGHTS AND CTA MANIFOLDS 5 invariant metrics which are easy to work with. Equipped with these metrics,they become good candidates for counterexamples as their curvature tensors,and thus the eigenflag directions, are also left invariant. Lie brackets canbe prescribed so that the left invariant metric has a Cotton tensor of thedesired type, but the left invariant distributions spanned by the eigenflagdirections are not integrable. It is possible to make these heuristics preciseand to find a specific example of a Lie group whose Cotton-York tensor isnowhere zero, has the symmetries that correspond to a CTA manifold, andyet the manifold is not CTA. This is the content of our next theorem.
Theorem 1.6.
There exists a unimodular three dimensional Lie group G such that det ( CY ) = 0 but G is not CTA. As discussed in Section 6, the analysis on three manifolds is now com-pleted. Let us turn to four manifolds. In dimension four the conformalsymmetries are described by the Weyl tensor. In [AFGR16] we found thatthe gradients of LCWs need to be related to so called ”eigenflag” directionsfor the Weyl tensor.
Definition 1.7 ([AFGR16]) . Let W be a Weyl tensor in S (Λ V ). Wesay that W satisfies the eigenflag condition if and only if there is a nonzerovector v ∈ V such that W ( v ∧ v ⊥ ) ⊂ v ∧ v ⊥ , where by v ∧ v ⊥ we denote theset of bivectors { v ∧ w : w ∈ V, h v, w i = 0 } . A one-dimensional subspace of V is called an eigenflag direction if it isspanned by some v satisfying the above condition.In [AFG17] we classified the Weyl tensor in types A, B, C, D accordingto the number of eigenflag directions. Lemma 1.8.
The algebraic Weyl operators W in a vector space of dimen-sion fall into one of the following types: A: W has no eigenflag directions. B: W has at least one eigenflag direction and three different eigenspaces ofdimension . In this case, W has exactly four eigenflag directions. C: W has at least one eigenflag direction and two eigenspaces with dimen-sions four and two. In this case, the eigenflag directions for W consist of the union of two orthogonal 2-planes. D: W is null. All directions are eigenflag. We also showed that the product of two scalene ellipsoids had Weyl tensorof type C but it was not CTA, showing that for four manifolds our neces-sary condition was not sufficient. However, this left the question whetherthe product structure was important in the counterexample and genuinefour dimensional examples exist. With this aim, we generalize unimodularLie groups, and equip them with left invariant metrics, to find examples of
P.ANGULO, D. FARACO, L. GUIJARRO, AND M.SALO
Riemannian manifolds whose Weyl tensors have the symmetries that corre-spond to a CTA manifold, or even to a product of surfaces, but still theyare not conformal to a product of lower dimensional manifolds in any way.For the case of Weyl tensors of type B , there are four eigenflag direc-tions that could admit LCWs; we provide examples of manifolds where onlyone, two or three of the eigenflags correspond to actual limiting Carlemanweights. The case of four LCWs is settled by the following observation thatfollows immediately from Corollary 1.4 after observing that for Weyl tensorsof type B, the eigenflag directions form an orthogonal basis. Corollary 1.9.
Let ( M, g ) be a -manifold with a Weyl tensor of type Bwhich admits four limiting Carleman weights. Then ( M, g ) is conformallyflat. It remains to find an example of a metric with a Weyl tensor of typeB, but not conformally transversally anisotropic. Inspired by dimensionthree, we look for them among the significantly more complex structure of4-dimensional Lie groups.
Theorem 1.10.
There exists a generalized unimodular Lie group G witha left invariant metric with Weyl tensor of type B, but such that G is notconformally anisotropic. Thus Weyl tensors of type B are fully understood. Finally we discuss thecase of Weyl tensors of type C . We recall that this is the case appearingwhen M is the Riemannian product of surfaces. This was already consideredin [AFG17]. Theorem 1.11.
Let ( S , g ) and ( S , g ) be open subsets of R with Rie-mannian metrics. Assume that the Weyl operator of the product metric doesnot vanish at any point. The following are equivalent: • ( S , g ) is locally isometric to a surface of revolution. • ( S × S , g × g ) admits a LCW that is everywhere tangent to thefirst factor. This theorem was behind our example of two scalene ellipsoids havingeigenflag directions but not being CTA. In Section 8.3 we construct a Liegroup whose Weyl tensor is type C, has no LCWs, and it is not a productof surfaces.To help the reader, we finish the introduction by including a few tablesthat summarize some of the results from this paper and from [AFG17] inregard to the question of which eigenflag directions can be realized by LCWs.
Dimension 4, Weyl type B:
Structure of eigenflags: 4 directions, forming an orthogonal basis. • Number of LCWs: – Zero: example given in Section 8.2; – One: M = R × N , where the metric in N has no LCWs; IMITING CARLEMAN WEIGHTS AND CTA MANIFOLDS 7 – Two: M has a warped product structure, Section 7.1; – Three: M is an iterated warped product, Section 7.2; – Four: M is conformally flat, Corollary 1.4. Dimension 4, Weyl type C: • Structure of eigenflags: two orthogonal 2-planes. • They do not need to have LCWs: Theorem 1.10, proof in Section8.3; • They do not need to arise from product of surfaces.The rest of this article is organized as follows. Section 2 discusses confor-mal Killing fields, Section 3 deals with the action of the conformal group,and Section 4 characterizes Euclidean LCWs by proving Theorems 1.1 and1.2. Section 5 analyzes manifolds with several orthogonal LCWs, and Sec-tion 6 deals with three manifolds. The analysis of four manifolds is given inSection 7 (structural results) and Section 8 (counterexamples).2.
Conformal Killing vector fields
We begin with a result that follows from [DKSU09, Lemma 2.9]:
Lemma 2.1.
Let ( M, g ) be a simply connected open Riemannian manifold,and let ϕ ∈ C ∞ ( M ) satisfy dϕ = 0 everywhere. Then ϕ is an LCW in ( M, g ) if and only if |∇ ϕ | − ∇ ϕ is a conformal Killing vector field in ( M, g ) . This immediately implies the following result.
Lemma 2.2.
Let ( M, g ) be a simply connected open Riemannian manifold.The map ϕ
7→ |∇ ϕ | − ∇ ϕ gives a bijective correspondence between the set of LCWs in ( M, g ) moduloadditive constants and the set of nonvanishing conformal Killing vector fields X in ( M, g ) satisfying d ( | X | − X ♭ ) = 0 . For any such X , there is a uniqueLCW ϕ up to an additive constant that satisfies | X | − X = ∇ ϕ .Proof. If ϕ is an LCW, then X = |∇ ϕ | − ∇ ϕ is a nonvanishing conformalKilling vector field by Lemma 2.1 and d ( | X | − X ♭ ) = 0. Conversely, if X isas stated, then | X | − X = ∇ ϕ for some ϕ ∈ C ∞ ( M ). Then |∇ ϕ | − ∇ ϕ = X is a conformal Killing vector field, and ϕ is an LCW by the previous lemma.The correspondence is bijective, since if | X | − X = ∇ ϕ = ∇ ϕ , then ϕ and ϕ differ by a constant. (cid:3) It follows that the classification of LCWs reduces to the determination ofall nonvanishing conformal Killing vector fields satisfying d ( | X | − X ♭ ) = 0.The classification of conformal Killing vector fields in R n is well known, butwe give a proof for completeness. Lemma 2.3.
Let Ω ⊂ R n , n > , be open and connected. Then X ∈ C ∞ (Ω , R n ) is a conformal Killing vector field in (Ω , e ) if and only if X ( x ) = ( α · x ) x − α | x | + cx + Bx + γ P.ANGULO, D. FARACO, L. GUIJARRO, AND M.SALO for some α, γ ∈ R n , c ∈ R , and some skew-symmetric matrix B .Proof. The equations for a conformal Killing vector field are ∂ j X k + ∂ k X j = λδ jk in Ω , j, k n, where λ ∈ C ∞ (Ω) is a scalar function (actually λ = n div( X )). We begin byshowing that λ has zero Hessian. Taking derivatives in the previous equationgives ( ∂ a λ ) δ bc = ∂ ab X c + ∂ ac X b , ( ∂ b λ ) δ ac = ∂ ab X c + ∂ bc X a , ( ∂ c λ ) δ ab = ∂ ac X b + ∂ bc X a . These three equations imply that(1) 2 ∂ ab X c = ( ∂ a λ ) δ bc + ( ∂ b λ ) δ ac − ( ∂ c λ ) δ ab . Taking traces gives 2∆ X c = (2 − n ) ∂ c λ. Differentiating once gives (2 − n ) ∂ cd λ = 2∆ ∂ d X c , (2 − n ) ∂ cd λ = 2∆ ∂ c X d and thus (2 − n ) ∂ cd λ = ∆( ∂ c X d + ∂ d X c ) = (∆ λ ) δ cd . Taking traces gives that (2 − n )∆ λ = n ∆ λ which gives ∆ λ = 0. The previous equation gives ∂ cd λ = 0 for all c, d .Since Ω is connected, we have λ ( x ) = α · x + β for some α ∈ R n and β ∈ R . Now (1) implies2 ∂ ab X c = α a δ bc + α b δ ac − α c δ ab . Integrating gives that X j = 12 ( α k δ lj + α l δ kj − α j δ kl ) x k x l + m jk x k + γ j for some m jk , γ j ∈ R . We finally compute ∂ j X k + ∂ k X j = 2( α · x ) δ jk + m jk + m kj and this is of the form λ ( x ) δ jk if and only if m jk = cδ jk + b jk for some c ∈ R and some skew-symmetric matrix B = ( b jk ). (cid:3) By Lemmas 2.2 and 2.3, the classification of all LCWs in R n follows onceone checks which of the vector fields in Lemma 2.3 satisfy d ( | X | − X ♭ ) =0. Presumably one could use Lemma 2.2 to characterize LCWs in othermanifolds as well, provided that one can determine all conformal Killingvector fields. IMITING CARLEMAN WEIGHTS AND CTA MANIFOLDS 9 Action of the conformal group
The next lemma describes how the conformal group acts on LCWs andconformal Killing vector fields (in part (e) we also include multiplication byscalars).
Lemma 3.1.
Let ϕ be an LCW in a connected open set Ω ⊂ R n , n > ,and let X = dϕ | dϕ | e be the corresponding conformal Killing -form written as X ( x ) = ( α · x ) x − α | x | + cx + Bx + γ. Let F : ˜Ω → Ω be a conformal transformation from an open set ˜Ω ⊂ R n onto Ω , i.e. κF ∗ e = e for some smooth positive κ . Then ˜ ϕ = F ∗ ϕ is anLCW in ˜Ω corresponding to conformal Killing -form e X = κF ∗ X = d ˜ ϕ | d ˜ ϕ | e ,which be written as e X ( x ) = ( ˜ α · x ) x −
12 ˜ α | x | + ˜ cx + ˜ Bx + ˜ γ. (a) If F ( x ) = x − x , then ( ˜ α, ˜ c, ˜ B, ˜ γ ) = ( α, c − α · x , B + α ∧ x , γ + ( α · x ) x − | x | α − cx − Bx ) where α ∧ β is the skew-symmetric matrix ( α j β k − β j α k ) nj,k =1 .(b) If F ( x ) = x/r for some r ∈ R \ { } , then ( ˜ α, ˜ c, ˜ B, ˜ γ ) = ( 1 r α, c, B, rγ ) . (c) If F ( x ) = Rx for some matrix R with R t R = I , then ( ˜ α, ˜ c, ˜ B, ˜ γ ) = ( R t α, c, R t BR, R t γ ) . (d) If F ( x ) = x | x | , then ( ˜ α, ˜ c, ˜ B, ˜ γ ) = ( − γ, − c, B, − α ) . Moreover:(e) If k ∈ R \ { } , then ˜ ϕ = k − ϕ is an LCW corresponding to e X , where ( ˜ α, ˜ c, ˜ B, ˜ γ ) = ( kα, kc, kB, kγ ) . Proof.
The formulas in (a)–(d) hold for any conformal Killing 1-form (notnecessarily arising from an LCW). First note that if ω is a conformal Killing1-form in ( M, g ) and if F : ( ˜ M , ˜ g ) → ( M, g ) satisfies κF ∗ g = ˜ g for somesmooth positive κ , then κF ∗ ω is a conformal Killing 1-form in ( ˜ M , ˜ g ). Tosee this, write µ = F −∗ κ and compute the total covariant derivative ∇ ˜ g ( κF ∗ ω ) = ∇ F ∗ ( µg ) F ∗ ( µω ) = F ∗ ( ∇ µg ( µω )) . Now the general identities (valid for any 1-form ω ), ∇ e λ g ω = ∇ g ω − dλ ⊗ ω − ω ⊗ dλ + h dλ, ω i g, ∇ g ( µω ) = µ ∇ g ω + dµ ⊗ ω and the properties of ω and F imply that the symmetric part of ∇ ˜ g ( κF ∗ ω )is a multiple of ˜ g , so κF ∗ ω is indeed a conformal Killing 1-form.Now if X is conformal Killing in a subset of R n and e X is as above,then by Lemma 2.3 e X has the required representation in terms of ˜ α, ˜ c, ˜ B, ˜ γ .If we identify X and e X with the corresponding vector fields in Cartesiancoordinates, we have e X = κ ( DF ) t ( X ◦ F ) , κ ( DF ) t DF = I, κ = | det DF | − /n where DF = ( ∂ k F j ) nj,k =1 . The claims (a)–(d) are a consequence of the fol-lowing facts and short computations:(a) If F ( x ) = x − x , then e X ( x ) = X ( x − x ).(b) If F ( x ) = x/r , then e X ( x ) = rX ( x/r ).(c) If F ( x ) = Rx , then e X ( x ) = R t X ( Rx ).(d) If F ( x ) = x | x | , then DF ( x ) = | x | ( I − x ⊗ ˆ x ) where ˆ x = x | x | , so κ = | x | and e X ( x ) = | x | ( I − x ⊗ ˆ x ) X ( x | x | )= ( I − x ⊗ ˆ x ) (cid:20) | x | (( α · x ) x − | x | α ) + cx + Bx + | x | γ (cid:21) = 1 | x | (( α · x ) x − | x | α ) + cx + Bx + | x | γ − (cid:20) | x | (( α · x ) x −
12 ( α · x ) x ) + cx + ( Bx · ˆ x )ˆ x + ( γ · x ) x (cid:21) . Here Bx · ˆ x = 0 because of skew-symmetry. Thus e X ( x ) = − α − cx + Bx + | x | γ − γ · x ) x. Finally, if ϕ is an LCW corresponding to X = dϕ | dϕ | e and F if is conformal,it follows that ˜ ϕ = F ∗ ϕ is an LCW corresponding to e X = d ˜ ϕ | d ˜ ϕ | e = κF ∗ X .Part (e) is trivial. (cid:3) Classification of LCWs in the Euclidean space
We will now give the new proof of the classification of LCWs in R n , n > IMITING CARLEMAN WEIGHTS AND CTA MANIFOLDS 11
Before the proofs, we give two lemmas. The first lemma is a simpleconsequence of the equation d ( | X | − X ) = 0 that holds for conformal Killingvector fields X associated to LCWs. Lemma 4.1.
If a nonvanishing -form X satisfies d ( | X | − X ) = 0 , then dX ∧ X = 0 . Proof.
We compute d ( | X | − X ) = | X | − dX − | X | − d ( | X | ) ∧ X. Thus d ( | X | − X ) = 0 if and only if dX = | X | − d ( | X | ) ∧ X. This means that the two-form dX is proportional to a wedge product of1-forms. In particular, one obtains the three equations dX ∧ X = 0 ,dX ∧ d ( | X | ) = 0 , | d ( | X | ) ∧ X | = | X | | dX | . (Conversely, one can show that in any connected manifold these three equa-tions imply that dX = ±| X | − d ( | X | ) ∧ X .) (cid:3) Next we use the equation in Lemma 4.1 to derive some restrictions to theparameters of X = ( α, c, B, γ ) arising from an LCW. Lemma 4.2.
Let ϕ be an LCW in Ω ⊂ R n , n > . The correspondingconformal Killing vector field X = ( α, c, B, γ ) satisfies B ∧ γ = 0 ,cB = α ∧ γ where B = b kj dx j ∧ dx k .Proof. We write X = X k dx k , where X k = ( α j x j ) x k − α k | x | + cx k + b kj x j + γ k . It follows that dX = dX k ∧ dx k = ( α j x k − α k x j + b kj ) dx j ∧ dx k = α ∧ p + B where α = α j dx j , p = x j dx j and B = b kj dx j ∧ dx k . Similarly, in thisnotation, X = h α, p i p − | p | α + cp + i p ♯ B + γ. Thus Lemma 4.1 gives that0 = dX ∧ X = α ∧ p ∧ ( i p ♯ B + γ ) + B ∧ ( h α, p i p − | p | α + cp + i p ♯ B + γ ) . The components of this 3-form are second order polynomials in x and vanishfor all x ∈ Ω, hence also for all x ∈ R n . Evaluating at x = 0 yields theequation B ∧ γ = 0 . Moreover, looking at first order terms in x gives that α ∧ p ∧ γ + B ∧ ( cp + i p ♯ B ) = 0 . By the properties of the interior product we have B ∧ i p ♯ B = 0. In fact,0 = i p ♯ ( B ∧ B ) = ( i p ♯ B ) ∧ B + B ∧ i p ♯ B = 2 B ∧ i p ♯ B. Hence ( cB − α ∧ γ ) ∧ p = 0, and evaluating at x = e j for 1 j n we obtain cB = α ∧ γ. (cid:3) Proof of Theorem 1.1.
Let ϕ be an LCW in a connected open set Ω ⊂ R n , n >
3, and let X = ( α, c, B, γ ) be the corresponding conformal Killing vec-tor field. We divide the argument in several cases. In each case, we willuse Lemma 3.1 (in fact only the translation property in part (a)) to re-duce conformal Killing vector fields to simpler ones that can be realized byexplicit LCWs. We omit the routine computations required to show that d ( |∇ ϕ | − ∇ ϕ ) = 0 for each of the functions ϕ given below (so that indeedthese functions are LCWs). Case 1. α = 0.Since α = 0, from Lemma 4.2 we obtain that cB = 0. This gives twosubcases. Case 1a. α = 0 and B = 0.Then X = (0 , c, , γ ). If c = 0, then γ = 0 (since X is nonvanishing)and X = γ is realized by the LCW ϕ ( x ) = | γ | − γ · x . On the other hand,if c = 0, after a translation F ( x ) = x − γ/c we may bring X to the form(0 , c, , ϕ ( x ) = c − log | x | . Case 1b. α = 0 and c = 0.In this case X = (0 , , B, γ ). First suppose that γ = 0. Since then B = 0, we can choose a vector x with Bx = 0 and apply the translation F ( x ) = x + x to bring X to the form (0 , , B, Bx ). We may thus assumethat γ = 0. If B = 0 we are again in Case 1a, so we may assume that IMITING CARLEMAN WEIGHTS AND CTA MANIFOLDS 13 B = 0. From Lemma 4.2 we have B ∧ γ = 0, which implies that B = γ ∧ σ for some vector σ = 0, and we may further assume that σ is orthogonal to γ . After a translation F ( x ) = x − σ/ | σ | , X becomes (0 , , a ˆ γ ∧ ˆ σ,
0) where a = | γ || σ | >
0. This is realized by the LCW ϕ ( x ) = a − arctan ˆ γ · x ˆ σ · x . Case 2. α = 0.In this case X = ( α, c, B, γ ). If c = 0, we first use the translation F ( x ) = x − c | α | − α to reduce to the case X = ( α, , B, γ ) for some new γ . If γ = 0,then X = ( α, , B, x is a vector with α · x = 0 we apply thetranslation F ( x ) = x + x to bring X to the form ( α, , ˜ B, Bx − | x | α ) forsome skew-symmetric ˜ B . If | x | is large enough one has Bx − | x | α = 0.This means that we may assume that X is of the form ( α, , B, γ ) for some γ = 0.Now, let X = ( α, , B, γ ) with γ = 0. By Lemma 4.2 we have cB = α ∧ γ ,and since c = 0 this implies that γ = rα for some r ∈ R \ { } . Thus wemay assume that X = ( α, , B, rα ) for some r = 0. By Lemma 4.2 againwe have B ∧ rα = 0, which implies that B = α ∧ σ for some vector σ wherewe may assume that α · σ = 0. Thus we have X = ( α, , α ∧ σ, rα ). After atranslation F ( x ) = x + σ , the vector field reduces to X = ( α, , , sα ) where s = r + | σ | .It remains to find LCWs that realize X = ( α, , , sα ) for different valuesof s : • If s >
0, the LCW ϕ ( x ) = √ s | α | arctan − α · x/ √ s | x | /s − corresponds to X = ( α, , , s α ). • If s = 0, the LCW ϕ ( x ) = − | α | ˆ α · x | x | corresponds to X = ( α, , , • If s >
0, the LCW ϕ ( x ) = √ s | α | arctanh − α · x/ √ s | x | /s +1 yields the vectorfield X = ( α, , , − s α ).This concludes the proof. (cid:3) Proof of Theorem 1.2.
We have already shown that precomposition with atranslation F ( x ) = x − x and post-composition with a linear function re-duces any LCW to one of the six families of LCWs in Theorem 1.1. Aftercomposing with a rotation R , we can take γ = e and σ = e . The scalarparameter s > r = √ s and post-multiplication by a constant k = √ s .Therefore, any LCW on R n is reduced to one of the following six represen-tative LCWs: e · x, log | x | , arctan e · xe · x , − e · x | x | , arctanh − e · x | x | +1 , arctan − e · x | x | − . Observe that up to this moment, to arrive to these six representatives, wehave only used affine conformal transformations. Using also the inversion, we will show that the LCWs in the same column in the above listing belongto the same orbit.Composition of e · x with the inversion F ( x ) = x/ | x | gives e · x | x | . Therelation between log | x | and arctanh − e · x | x | +1 is seen more easily looking at thecorresponding conformal Killing vector fields: • The translation F ( x ) = x − e acts on (0 , , ,
0) (which correspondsto log | x | ) and gives (0 , , , − e ). • The inversion F ( x ) = x/ | x | acts on (0 , , , − e ) and gives (2 e , − , , • The translation F ( x ) = x + e acts on (2 e , − , ,
0) and gives(2 e , , , − e ). • The dilation F ( x ) = x/ e , , , − e ) and gives ( e , , , − e ),which corresponds to arctanh − e · x | x | +1 .A similar argument works for the arctan weights.It remains to prove that the three remaining orbits are different. Let usassume that ϕ = e · x is b + a ( ψ ◦ F ), for ψ = log | x | and F a conformaltransformation in Ω ⊂ R n . According to Liouville’s theorem (see e.g [IM02,Theorem 2.3.1]), F is the restriction to Ω of: F ( x ) = x + αA ( x − x ) | x − x | ǫ for x ∈ R n \ Ω , x ∈ R n , an orthogonal matrix A , α ∈ R \ { } , ǫ ∈ { , } .This is a composition of elementary transformations whose effect we knowthanks to Lemma 3.1, so we can apply them to the conformal Killing vectorfield X = (0 , , ,
0) that corresponds to ψ . We will show that F ∗ X is neverequal to the conformal Killing vector field of ϕ , which is X = (0 , , , e ).Let us consider the case of ǫ = 0 first: • The translation F ( x ) = x − x acting on (0 , , ,
0) gives (0 , , , − x ). • The rotation F ( x ) = Rx acting on (0 , , , − x ) gives (0 , , , − R t x ). • The dilation F ( x ) = x/r acting on (0 , , , − R t x ) gives (0 , , , − rR t x ). • Finally, the translation F ( x ) = x − x acting on (0 , , , − rR t x )must give a multiple of (0 , , , e ), but the constant e c = c = 1.If ǫ = 2: • The translation F ( x ) = x − x acting on (0 , , ,
0) gives (0 , , , − x ). • The inversion F ( x ) = x/ | x | acting on (0 , , , − x ) gives (2 x , − , , • The rotation F ( x ) = Rx acting on (2 x , − , ,
0) gives (2 R t x , − , , • The dilation F ( x ) = x/r acting on (2 R t x , − , ,
0) gives ((2 /r ) R t x , − , , • Finally, the translation F ( x ) = x − x acting on ((2 /r ) R t x , − , , , , , e ), which implies that (2 /r ) R t x = 0,and e c cannot be zero.We have proven that the orbits of e · x and log | x | by the conformalgroup are different. Analogous computations show that arctan e · xe · x also be-longs to a different orbit. Postcomposition with a linear function simply IMITING CARLEMAN WEIGHTS AND CTA MANIFOLDS 15 scales the conformal Killing vector field, and this action commutes with pre-composition with a conformal transformation, so this action cannot mix theorbits. (cid:3)
Remark 4.3.
The fact that there are only three orbits for the LCWs underthe action of the conformal group can also be seen directly as follows. Clearly e · x | x | = F ∗ ( e · x ) , F ( x ) = x | x | . Next define the conformal map F ( x ) = x − e | x − e | + 12 e . It is easy to see that F ∗ (arctan e · xe · x )( x ) = arctan 2 e · x | x | − . Moreover, since | F ( x ) | = | x + e | | x − e | , one has F ∗ (log 2 | x | ) = 12 log | x + e | | x − e | = 12 log 1 + x | x | +1 − x | x | +1 = arctanh 2 x | x | + 1 . On orthogonal LCWs
In this section, we show how the existence of several orthogonal LCWsresults in a special conformal product structure.
Definition 5.1.
We say that the LCWs ϕ , . . . , ϕ m for g are orthogonal iftheir gradients grad g ( ϕ ) , . . . , grad g ( ϕ m ) are orthogonal with respect to g . Theorem 5.2.
Let ( M, g ) be an n -dimensional Riemannian manifold. Let p ∈ M , and suppose that in an open neighbourhood of p , the conformalclass [ g ] admits m orthogonal LCWs ϕ , . . . , ϕ m . Then there is a set ofcoordinates about p , Φ = ( z , . . . , z n ) , such that(1) z = ϕ , . . . , z m = ϕ m ;(2) some conformal multiple of g has the local expression . . . f ( z m +1 , . . . , z n ) . . . ... . . . f m ( z m +1 , . . . , z n ) 00 0 . . . D ( z m +1 , . . . , z n ) where D is an ( n − m ) × ( n − m ) symmetric matrix.Conversely, if in some local coordinates ( z , . . . , z n ) the metric has the aboveform up to a conformal multiple, then z , . . . , z m are orthogonal LCWs. Proof.
Let ϕ , . . . , ϕ m be orthogonal LCWs. Using [DKSU09, Theorem 1.2and its proof], for each j = 1 , . . . , m there is a smooth conformal factor f j near p such that the gradient of ϕ j in the metric f j g is a unit parallel vectorfield with respect to f j g . In other words, writing X j = grad f j g ( ϕ j ) , we have ∇ f j g X j ≡ f j g ( X j , X j ) = 1. After passing to a conformalmultiple of g , we can assume without loss of generality that f ≡ ϕ j in the metric g is given by f j · X j , sincegrad g ( ϕ j ) = f j grad f j g ( ϕ j ) = f j X j . Using the formula for the Levi-Civita connection of a conformal metric,we have(2) ∇ f j gX Y = ∇ gX Y + ( Xh j ) Y + ( Y h j ) X − g ( X, Y )grad g ( h j )where h j = log f j . Thus in particular0 = ∇ f j gX k X j = ∇ gX k X j + X k ( h j ) · X j + X j ( h j ) · X k − g ( X j , X k ) grad g ( h j ) . Since g ( X j , X k ) = 0 for j = k , ∇ gX k X j ∈ span { X j , X k } . Interchanging j and k , we get also that ∇ gX j X k ∈ span { X j , X k } , and this implies that[ X j , X k ] = ∇ gX j X k − ∇ gX k X j ∈ span { X j , X k } . Thus the distribution generated by X j and X k is integrable, and it alsofollows that the whole distribution span { X , . . . , X m } is integrable.Using the Frobenius theorem, there is a chart ( y , . . . , y n ) near p such thatthe sets { ( y m +1 , . . . , y n ) = const } are integral manifolds of the distributionspan { X , . . . , X m } . We define another chart ( z , . . . , z n ) near p such that z j = ϕ j for 1 j m and z l = y l for l > m + 1. Since X j = grad f j g ( ϕ j )is tangent to the manifolds { ( y m +1 , . . . , y n ) = const } , and since the X j areorthogonal, it follows that the differentials of the functions z , . . . , z n arelinearly independent, and ( z , . . . , z n ) is a chart. We claim that for thischart,(3) X k = ∂ z k , k m. To prove (3), we note that for 1 j, k m one has X j ( ϕ k ) = dϕ k ( X j ) = g ( X j , grad g ( ϕ k )) = f k g ( X j , X k ) = δ jk . Here we used that grad g ( ϕ k ) = f k X k , the g -orthogonality of the X j , andthe fact that f k g ( X k , X k ) = 1. Moreover, X k ( z l ) = dz l ( X k ) = 0 whenever l > m + 1 and k m , since the manifolds { ( z m +1 , . . . , z n ) = const } areintegral manifolds of span { X , . . . , X m } . Thus X k = P nl =1 X k ( z l ) ∂ z l = ∂ z k for 1 k m , which proves (3).In particular, by (3) the Lie brackets [ X j , X k ] are zero. Next, we usethat the Levi-Civita connection is torsion free for any metric, that for any j , the vector field X j is parallel for the metric f j g , and finally the formula IMITING CARLEMAN WEIGHTS AND CTA MANIFOLDS 17 (2) for the conformal factor f j f k with h jk = log f j f k . These facts imply thatfor j = k one has0 = [ X j , X k ] = ∇ f j gX j X k − ∇ f j gX k X j = ∇ f j gX j X k = ∇ ( f j /f k ) f k gX j X k = ∇ f k gX j X k + X j ( h jk ) X k + X k ( h jk ) X j − f k g ( X j , X k ) grad f k g ( h jk )= X k ( h jk ) X j + X j ( h jk ) X k . Since X j = ∂ j and X k = ∂ k are linearly independent, we deduce that X k ( h jk ) = ∂ k ( h jk ) = 0 for j = k . Further, since h jk = h j − h k where h j = log f j , we obtain that ∂ k ( h j − h k ) = 0 , j, k m. One also has h ≡
0, since f ≡ h j , 1 j m , only depend on thevariables z m +1 , . . . , z n . First note that ∂ h j = ∂ ( h j − h ) = 0 , j m. Next we use that ∂ h = ∂ ( h − h ) = 0, which implies that ∂ h j = ∂ ( h j − h ) = 0 , j m. Repeating this argument shows that indeed h j , 1 j m , only depends onthe variables z m +1 , . . . , z n .Combining the above facts, the metric in the ( z , . . . , z n ) coordinates hasthe following form:(4) . . . f ( z m +1 , . . . , z n ) . . . . . . f m ( z m +1 , . . . , z n ) 00 0 . . . D ( z , . . . , z n ) where D is an ( n − m ) × ( n − m ) symmetric matrix.For any element D ab of D , where a, b > m + 1, and for any j m we cancompute ∂ j D ab = ∂ j ( g ( ∂ a , ∂ b )) = g ( ∇ g∂ j ∂ a , ∂ b ) + g ( ∇ g∂ j ∂ b , ∂ a )Using (2), the first term in the above sum can be computed as g ( ∇ g∂ j ∂ a , ∂ b ) = g ( ∇ g∂ a ∂ j , ∂ b )= g ( ∇ f j g∂ a ∂ j − ∂ a ( h j ) ∂ j − ∂ j ( h j ) ∂ a + g ( ∂ a , ∂ j ) grad g ( h j ) , ∂ b )= 0since the metric is of the form in (4). Similarly one has g ( ∇ g∂ j ∂ b , ∂ a ) = 0. We recall that we assumed f ≡
1, so the original metric in the ( z , . . . , z n )coordinates is a conformal multiple of . . . f ( z m +1 , . . . , z n ) . . . . . . f m ( z m +1 , . . . , z n ) 00 0 . . . D ( z m +1 , . . . , z n ) where D is an ( n − m ) × ( n − m ) symmetric matrix.Finally, to show the converse statement we assume that the metric isconformal to the above form in some local coordinates ( z , . . . , z n ). Using[DKSU09, Remark 2.8], z is an LCW. Similarly, after taking f as a factor,the metric is conformal to ˜ f ( z m +1 , . . . , z n ) 0 . . . . . . . . . ˜ f m ( z m +1 , . . . , z n ) 00 0 . . . D ( z m +1 , . . . , z n ) where ˜ f , ˜ f , . . . , ˜ f m , ˜ D only depend on z m +1 , . . . , z n . Thus z is an LCWagain by [DKSU09, Remark 2.8]. Similarly we get that z , z , . . . , z m are allLCWs, and they are clearly orthogonal. (cid:3) Corollary 5.3. An n -dimensional manifold with n orthogonal LCWs isconformally flat. Limiting Carleman weights in 3-manifolds
Conformal geometry in dimension three is usually studied through theuse of the Cotton tensor of the metric. This is a (3 , Theorem 6.1 ([AFGR16]) . Let n = 3 . If a metric ˜ g ∈ [ g ] admits a parallelvector field, then for any p ∈ M , there is a tangent vector v ∈ T p M suchthat CY p ( v, v ) = CY p ( w , w ) = 0 for any pair of vectors w , w ∈ v ⊥ . Moreover, if ϕ is an LCW in ( M, g ) ,then v = grad g ( ϕ ) | p has the above property. In analogy with the higher dimensional case we still call such a direction v ∈ T p M an eigenflag . Thus limiting Carleman weights line along eigenflag IMITING CARLEMAN WEIGHTS AND CTA MANIFOLDS 19 directions, but there is a priori no reason why eigenflag directions should berealized by limiting Carleman weights.Observe also that the above theorem has the simple algebraic consequencethat the determinant of CY p vanishes whenever CY p has the form of Theorem6.1 [AFGR16, Corollary 1.7]. Thus there is a sequence of implicationsExistence of LCW ⇒ existence of eigenflag ⇒ det CY p = 0.The above implications raise the following natural questions: • Are there metrics with eigenflag directions that do not correspondto limiting Carleman weights? • Does det( CY p ) = 0 in an open set imply the existence of a limitingCarleman weight? • How many different limiting Carleman weights can a metric admitwithout being conformally flat? • More generally, can we characterize metrics that have more than onelimiting Carleman weight?The rest of this section answers the above questions.6.1.
Here we will prove Theorem 1.6. We will providesuch examples in unimodular Lie groups. A good reference for such mani-folds is [Mi76].Let G be a 3-dimensional Lie group with Lie algebra g . Recall from [Mi76,Lemma 4.1] that if G is unimodular, there exists a self-adjoint map L : g → g such that for any u, v ∈ g , [ u, v ] = L ( u × v ) . The vector product in g is the one defined by an oriented orthonormal basis e , e , e consisting of eigenvectors of L , thus there are λ , λ , λ ∈ R suchthat [ e i , e j ] = λ k e k where i, j, k is an oriented set of indices.Following Milnor, to compute the curvature tensors of G , we define num-bers µ , µ , µ as µ i = λ + λ + λ − λ i . From [Mi76, Theorem 4.3], the Ricci tensor in the basis e , e , e diago-nalizes, withRic( e ) = 2 µ µ , Ric( e ) = 2 µ µ , Ric( e ) = 2 µ µ , and the scalar curvature is s = 2( µ µ + µ µ + µ µ ) . In order to compute the Cotton tensor, we calculate first the Levi-Civitaconnection for the vector fields e , e , e . [Mi76, equation (5.4)] gives that ∇ e i e j = X k
12 ( α ijk − α jki + α kij ) e k , with α ijk = h [ e i , e j ] , e k i = λ k . Writing this in terms of the λ i ’s yields ∇ e e = λ − λ + λ e , ∇ e e = λ − λ + λ e , ∇ e e = λ − λ + λ e , ∇ e e = − λ − λ + λ e , ∇ e e = − λ − λ + λ e , ∇ e e = − λ − λ + λ e . In what follows we will denote by n ijk = h∇ e i e j , e k i , and observe that n ijk = 0 whenever one of the indices repeats. Observe alsothat the n ijk are not antisymmetric in the first two indices, since this wouldotherwise force the Lie algebra to be abelian.The Schouten tensor in dimension 3 is S = Ric − s g, and the formula for the Cotton tensor (see [AFGR16, Section 2]) is given as C ijk = C ( e i , e j , e k ) = ∇ e i S ( e j , e k ) − ∇ e j S ( e i , e k ) . Since ∇ e i S ( e j , e k ) = e i ( S ( e j , e k )) − S ( ∇ e i e j , e k ) − S ( e j , ∇ e i e k ) and since S ( e j , e k ) is constant in x , the above formulas imply that for any i, j, k onehas C ijk = − n ijk S kk − n ikj S jj + n jik S kk + n jki S ii . Thus C ijk = 0 whenever two of the indices coincide. The only nonzero termsare C = − n S − n S + n S + n S C = − n S − n S + n S + n S C = − n S − n S + n S + n S . The Cotton-York tensor is obtained by dualizing the first two indices, thus CY = C , CY = C , CY = C , while the rest of the components are zero.As the reader can easily check by a tedious computation, by choosing λ = 6, λ = −
4, and λ = 5, one obtains CY = − , CY = 3152 , CY = 0 , Thus the directions e + e and e − e are eigenflag directions, that is theyparametrize the possible directions for the gradients of potential LCWs. IMITING CARLEMAN WEIGHTS AND CTA MANIFOLDS 21
However, they can not be realized by LCWs, since their orthogonal com-plements are not integrable (see [AFG17, Theorem 6]): in the first case,( e + e ) ⊥ is spanned by e − e and e , but[ e − e , e ] = 4 e − e which does not lie in the span of e − e and e . The other case is similar.6.2. In this section we determine metrics that admit several limitingCarleman weights. The example to keep in mind is the following:
Example . Assume M is a 3-dimensional Riemannian manifold admittinga limiting Carleman weight. Then M is, up to a conformal factor, containedin R × S for some surface S . We claim that:(1) if S is a surface of revolution different from the flat plane, then M has two LCWs;(2) if S is the flat plane, then M is conformally flat.Recall that a surface of revolution is a surface with a Riemannian metricadmitting a Killing vector field. Proof.
The second statement is obvious. For the first, observe that if S is asurface with a Killing field, its metric can be written as g = dt + dr + f ( r ) dθ , with a nonvanishing f . Here t is an LCW. Dividing by f , we observe that1 f g = dθ + 1 f ( dt + dr ) , and thus there is a second limiting Carleman weight in the θ -direction. (cid:3) In what follows, we want to see what restrictions appear in the metricwhen there are several limiting Carleman weights. We start with the casewhen there are three or more limiting Carleman weights, proving that thiscorresponds to the conformally flat situation.
Theorem 6.3.
Let M be a 3-dimensional Riemannian manifold that ad-mits three limiting Carleman weights whose gradients are pointwise linearlyindependent. Then M is conformally flat. Notice that by a standard density argument, the Theorem also works inthe case when the limiting Carleman weights are independent in a denseopen set . Proof.
We will show that CY p ≡ CY p . From Theorem 6.1, the associated matrix in an orthonor-mal base B = { e , e , e } with e an eigenflag, has the form a ba b with a, b ∈ R ; its eigenvalues are 0 and ±√ a + b .Suppose that CY p
0. Then there is a unit vector v ∈ e ⊥ ( v is a multipleof be − ae ) with CY p ( v ) = 0. Similarly, if e ′ and e ′′ are the other twoeigenflag directions such that { e , e ′ , e ′′ } are linearly independent, there areunit vectors v ′ ∈ ( e ′ ) ⊥ and v ′′ ∈ ( e ′′ ) ⊥ with CY p ( v ′ ) = CY p ( v ′′ ) = 0. Nowit is not possible that v, v ′ , v ′′ are collinear (since no unit vector can beorthogonal to e , e ′ , e ′′ ), so it follows that CY p ( w ) = 0 for all w in sometwo-dimensional subspace on T p M . Thus 0 is an eigenvalue of multiplicityat least two, but since CY p is trace free all eigenvalues must be zero. Thus CY p ≡
0, which is a contradiction. (cid:3)
Next we study the case with two LCWs.
Theorem 6.4.
Let M be a Riemannian 3-manifold admitting two limitingCarleman weights with linearly independent gradients. Suppose M is notconformally flat. Then M is conformal to R × S , where S is a nonflatsurface of revolution.Proof. Denote the two LCWs by ϕ and ϕ . Then the gradients of ϕ and ϕ give rise to eigenflag directions by Theorem 6.1, and consequentlydet( CY ) = 0 by [AFGR16, Corollary 1.7]. Since M is not conformally flat, CY is not null, and hence [AFG17, Lemma 14] states that M has exactly twoeigenflag directions. Moreover, the proof of [AFG17, Lemma 14] implies thatthese eigenflag directions are necessarily orthogonal. It thus follows that thegradients of ϕ and ϕ are orthogonal.Since ϕ and ϕ are orthogonal LCWs, by Theorem 1.3 there is a chart( x, y, z ) where x = ϕ , y = ϕ , and the metric is conformal to a ( z ) 00 0 b ( z ) . After a change of coordinates of the type ( x, y, z ) → ( x, y, v ( z )), the metricis conformal to a ( z ) 00 0 1 , and thus M is conformal to R × S for some surface of revolution S . (cid:3) Limiting Carleman weights in 4-manifolds
We start this section by determining the local structure of metrics in4-dimensional manifolds that admit several orthogonal LCWs. The caseof four orthogonal LCWs is covered by Corollary 1.4, and corresponds toconformally flat metrics. This leaves the cases of two and three LCWs, thatwe cover in the next two sections.
IMITING CARLEMAN WEIGHTS AND CTA MANIFOLDS 23
Structure of a 4D manifold with two orthogonal LCWs.
ByTheorem 1.3, the metric g is written in some coordinates ( t, x, y, z ), where t and x are LCWs, as a conformal multiple of a ( y, z ) 0 00 0 b ( y, z ) b ( y, z )0 0 b ( y, z ) b ( y, z ) . The above metric is the product R × ˜ M , where the metric ˜ g taken in ˜ M is a warped product over a surface ( S, ˜ h ) with fiber a line; i.e, at each point( x, y, z ), ˜ g = π ∗ ˜ h + a ( y, z ) dx where π : R → R is the projection onto the ( y, z )-coordinates and ˜ h isgiven by the 2 × b ij ).It is natural to ask to what extent the structure for the metric resemblesthis if we only assume existence of two LCWs, not necessarily orthogonal .This cannot happen for type B Weyl tensors, since eigenflags for that caseare always orthogonal. But the case of type C Weyl tensors allows thispossibility, although we have not been able to produce any example for thiscase.7.2. Structure of a 4D manifold with three orthogonal LCWs.
Inthis case, the metric is by Theorem 1.3 a conformal multiple of a ( z ) 0 00 0 b ( z ) 00 0 0 c ( z ) . If h ( z ) is a primitive of p c ( z ), we can define a new coordinate ¯ z = h ( z ) · z ;using ( t, x, y, ¯ z ) as the new system of coordinates, the above metric becomes a (¯ z ) 0 00 0 b (¯ z ) 00 0 0 1 . We can write this as the Riemannian product of R times a 3-dimensionalmanifold with a metric that is an iterated warped product; namely, if π ( x, y, ¯ z ) = ( y, ¯ z ), and π ( y, ¯ z ) = ¯ z , then the 3-dimensional factor hasthe metric ¯ g = a (¯ z ) dx + π ∗ ( b (¯ z ) dy + π ∗ d ¯ z ) . Product 4-manifolds and LCWs.
A 4-dimensional Riemannian man-ifold can split as a metric product only in two ways: either as R × N (andin that case, it would have a LCW), or as a product of surfaces. In thelatter case, the Weyl tensor is of type C, and as a consequence it has plentyof eigenflag directions. The first three authors proved in [AFG17] that a product of two surfaces has a LCW if and only if one of the factors is asurface of revolution. We now provide a simpler proof of Theorem 11 in[AFG17] (cf. Lemma 20 in [AFG17]). Theorem 7.1.
A product of two surfaces has a LCW if and only if one ofthe factors is a surface of revolution.Proof.
As pointed in [AFG17, proof of Theorem 11], there is a LCW when-ever one of the factors is a surface of revolution, so we need to prove onlyone implication.Let M = S × S be a product of two surfaces with the metric g = g ⊕ g ,and denote by ϕ a LCW for this metric. At any point, the gradient of ϕ isan eigenflag direction, and since the Weyl tensor of a product metric is oftype C, ∇ ϕ must be tangent to one of the two factors, say S (see [AFG17,Lemma 19]).Let x, y be coordinates in S and z, t be isothermal coordinates in S , sothat the metric in S is (cid:18) λ ( z, t ) 00 λ ( z, t ) (cid:19) . We first remark that ϕ does not depend on z or t , since ∂ z ϕ = dϕ ( ∂ z ) = h∇ ϕ, ∂ z i = 0 , as ∇ ϕ is everywhere tangent to S . This means that we can consider ϕ as afunction of the first factor S only. We now construct a companion function ψ : S → R , whose differential vanishes on ∇ ϕ . This is always possible bybasic ODE theory (there are coordinates ( x , x ) on S with ∇ ϕ = ∂ x , andone can take ψ = x ).We write the metric of M in the coordinates { x = ϕ, y = ψ, z, t } (whosecoordinate vector fields are denoted { ∂ , ∂ , ∂ , ∂ } ): a ( x, y ) 0 0 00 b ( x, y ) 0 00 0 λ ( z, t ) 00 0 0 λ ( z, t ) . In a multiple f · g of g , ∇ ϕ is a unit parallel vector field, and the metric iswritten, in the same coordinates { ϕ, ψ, z, t } (since ∂ , ∂ , ∂ are orthogonalto ∂ in g , they are also orthogonal in any multiple of g ): h ( y, z, t ) 0 00 0 h ( y, z, t ) 00 0 0 h ( y, z, t ) . IMITING CARLEMAN WEIGHTS AND CTA MANIFOLDS 25
Thus f ( x, y, z, t ) a ( x, y ) 0 0 00 b ( x, y ) 0 00 0 λ ( z, t ) 00 0 0 λ ( z, t ) = h ( y, z, t ) 0 00 0 h ( y, z, t ) 00 0 0 h ( y, z, t ) . Hence f = a depends only on x, y , but f = h λ does not depend on x . Hence a = f and b = h f do not depend on x . Hence g is written as a ( y ) 0 0 00 b ( y ) 0 00 0 λ ( z, t ) 00 0 0 λ ( z, t ) and S is a surface of revolution. (cid:3) Eigenflags vs. LCWs: the 4-dimensional case
In [AFG17], the first three authors classified Weyl tensors of 4-manifoldsdepending on the number of existing eigenflags (see Lemma 1.8). Some ex-amples in that paper showed that eigenflags may not necessarily be realizedby LCWs. It was not clear at that moment if there were examples witheigenflag directions but no LCWs at all.In this section we provide examples of Riemannian 4-manifolds, one witha Weyl tensor of type B, one with a Weyl tensor of type C, such that noeigenflag direction comes from a LCW. The latter case could be consideredas the most difficult, since the eigenflag directions fill the union of two 2-planes in each tangent space of M and we need to rule out the existence ofLCWs for all of them.8.1. General strategy.
Both examples are constructed using left invariantmetrics on 4-dimensional Lie groups. Denote by B = { e , e , e , e } a basisof left invariant vector fields. For each example, we will need to constructthe structure constants of the group, that determine its algebraic structure.These are constants c kij for 0 i, j, k e i , e j ] = X k c kij e k , c kij ∈ R . The metric g is then defined by asking that { e , e , e , e } forms an or-thonormal basis at every point of the group.The curvature tensor of M is computed with the formulas of [Mi76]; thecomputations could be done by hand, or be easily implemented using somemathematical software system as SAGE [SAGE]. Although [Mi76] only pro-vides formulas for the curvature and Ricci tensor components in the base B ,we can compute the Weyl tensor by calculating first the Schouten tensor(5) S = 1 n − (cid:18) Ric − n − sg (cid:19) and then using that(6) R = W + S ? g where ? is the well-known Kulkarni-Nomizu product of symmetric 2-tensorsdefined as ( α ? β ) ijkl = α ik β jl + α jl β ik − α il β jk − α jk β il where R and W are understood as (0 ,
4) tensors.Once we compute W , we can compute whether it has eigenflags, and thetype of the tensor.To check whether the metric has LCWs along these eigenflags, we will use[AFG17, Theorem 1.5] that we recall here: if a vector field e of eigenflagscan be realized with a LCW, then its orthogonal distribution D = e ⊥ isintegrable and umbillical: • Integrable:
Define the second fundamental form of the distribution D as II ( X, Y ) = P D ⊥ ( ∇ X Y )where P D ⊥ is the projection onto D ⊥ and X , Y , are vector fieldstangent to D ; the distribution is integrable if and only this form issymmetric. • Umbilical: there exists a vector field H ∈ D ⊥ , called the meancurvature vector field of D , such that for X, Y ∈ D and Z ∈ D ⊥ itholds that(7) g ( ∇ X Y, Z ) = g ( X, Y ) g ( Z, H )Umbilicity implies integrable.8.2.
A 4-dimensional metric with a type B Weyl tensor and noLCWs.
The structure constants all vanish, except for[ e , e ] = − e , [ e , e ] = − e , [ e , e ] = e , [ e , e ] = − e , [ e , e ] = − e . The Levi-Civita connection is determined by the above and the values ∇ e e = 0 , ∇ e e = − e , ∇ e e = 14 e , ∇ e e = 0 , ∇ e e = 0 , ∇ e e = − e + 34 e , ∇ e e = − e , ∇ e e = 0 , ∇ e e = − e , ∇ e e = 0 , that arise from applying the formulas in [Mi76].In the basis B , the components of the curvature tensor are given as R ijkℓ = g ( R ( e i , e j ) e k , e ℓ ) , IMITING CARLEMAN WEIGHTS AND CTA MANIFOLDS 27
Due to the symmetries of the curvature, we have that R ijkℓ = − R jikℓ = − R ijℓk = R kℓij . After carrying the computations, we get that (except for the above indexpermutations), the nonzero components f the curvature tensor are R = − , R = − , R = 116 , R = − R = 58 , R = − , R = − , R = − . The nonzero Ricci tensor components areRic = − , Ric = 98 , Ric = − , Ric = − , Ric = 98 , Ric = − . Observe that the metric is not Einstein due to the presence of a nonzeroterm off the diagonal.The scalar curvature is the trace of the Ricci tensor, resulting on s = − . The Schouten tensor can be now computed using formula (5) to get S = − , S = 916 , S = 116 S = 116 , S = 916 , S = − W = W = W = W = − , W = W = W = W = − W = W = W = W = 18 , W = W = W = W = 12 W = W = W = W = 58 , W = W = W = W = − , where we are seeing W as a (0 , B ′ = { e ∧ e , e ∧ e , e ∧ e , e ∧ e , e ∧ e , e ∧ e } , and obtain W = −
58 18 12 12 18 − Thus W diagonalizes in simple bivectors obtained from the canonicalbasis. Observe also that, since there are three different eigenvalues, theWeyl tensor is of type B and not C.The eigenflags for this metric are precisely the elements of the canonicalbasis B . To rule out that they could correspond to LCWs, we compute thesecond fundamental forms of the distributions orthogonal to each one of thevector fields e i : • D = e ⊥ : in the basis e , e , e the second fundamental form is ; • D = e ⊥ : in the basis e , e , e the second fundamental form is − − − ; • D = e ⊥ : in the basis e , e , e the second fundamental form is − ; • D = e ⊥ : in the basis e , e , e the second fundamental form is − − . Integrability means that the second fundamental form should be symmet-rical; this already rules out e , e , e as tangent to possible LCWs.Umbilicity implies that the second fundamental form should be a multipleof the identity; this rules out the left possibility e .8.3. A 4-dimensional metric that is not a product, has Weyl tensorof type C, and no LCWs.
Denote by B = { e , e , e , e } a basis of left invariant vector fields. Thegroup structure arises from[ e , e ] = − e − e , [ e , e ] = − e + e , [ e , e ] = − e − e , [ e , e ] = e + e , [ e , e ] = e − e , [ e , e ] = − e − e . The metric is defined by asking that { e , e , e , e } forms an orthonormalbasis.Once the computations are carried out, the nonzero components of W onthe basis B are (modulo trivial rearrangements of the indices) W = W = − W = W = W = W = 4 . IMITING CARLEMAN WEIGHTS AND CTA MANIFOLDS 29
This means that the planes π = e ⊕ e and π = e ⊕ e are entirely formedby eigenflags, thus giving a Weyl tensor of type C.To rule out the existence of LCWs, we will use [AFG17, Lemma 15].Suppose that some linear combination (where α : M → R is some function) X = cos( α ) e + sin( α ) e was a LCW. Then its orthogonal distribution (that is spanned by Y = − sin( α ) e + cos( α ) e , e and e ) would have to be umbilical, i.e, integrableand such that the second fundamental form B of its integral leaves is amultiple of the identity. But B ( e , Y ) = g ( ∇ e Y, X ) = 12 , instead of vanishing.An entirely similar argument shows that there are also no LCWs tangentto the planes spanned by e and e .To rule out the possibility that the metric were a Riemannian product, westart by observing that there can not be one dimensional factors, since theseare equivalent to LCWs, thus we only need to consider product of surfaces S × S .But in that case, the structure of the Weyl tensor implies that the 2-dimensional factors should be tangent at each point to the planes π = e ⊕ e and π = e ⊕ e . Thus, it is enough to check that such distributions arenot umbillical. A simple calculation shows that g ( ∇ e e , e ) = 14 , g ( ∇ e e , e ) = − , thus proving that the second fundamental form of the distribution e ⊕ e is not symmetrical. References [An17] P. Angulo-Ardoy,
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Inverse problems: seeing the unseen , Bull. Math. Sci. (2014),209–279. Department of Mathematics, ETS de Ingenieros Navales, Universidad Polit´ecnicade Madrid
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