Weyl-Ambient Geometries

Weyl geometry is a natural extension of conformal geometry with Weyl covariance mediated by a Weyl connection. We generalize the Fefferman-Graham (FG) ambient construction for conformal manifolds to a corresponding construction for Weyl manifolds. We first introduce the Weyl-ambient metric motivated by the Weyl-Fefferman-Graham (WFG) gauge. From a top-down perspective, we show that the Weyl-ambient space as a pseudo-Riemannian geometry induces a codimension-2 Weyl geometry. Then, from a bottom-up perspective, we start from promoting a conformal manifold into a Weyl manifold by assigning a Weyl connection to the principal $\mathbb{R}_+$-bundle realizing a Weyl structure. We show that the Weyl structure admits a well-defined initial value problem, which determines the Weyl-ambient metric. Through the Weyl-ambient construction, we also investigate Weyl-covariant tensors on the Weyl manifold and define extended Weyl-obstruction tensors explicitly.


Introduction
Conformal geometry is a very rich area of mathematics with its history deeply intertwined with that of physics. Historically, the subject was initiated at the beginning of the twentieth century with the work of Weyl [1], Cartan [2] and Thomas [3]. In physics, there have been numerous applications of conformal geometry, from conformal compactification [4] and conformal gravity [5] to the AdS/CFT correspondence [6,7].
The fundamental structure appearing in conformal geometry is a manifold M endowed with a conformal class of metrics [g]. Two metrics belong in the same conformal class [g] if one metric is a smooth positive multiple of the other. Local rescalings of the metric tensor by an arbitrary smooth positive function are called Weyl transformations. Compared to pseudo-Riemannian manifolds (M, g), conformal manifolds are endowed with an enlarged symmetry group with both diffeomorphisms and Weyl transformations, denoted by Diff(M ) Weyl. A tensor T on a conformal manifold (M, [g]) is said to be conformally covariant if it transforms covariantly under a Weyl transformation: where w T is the Weyl weight of the tensor T . On the physics side, conformal-covariant tensors appear as expectation values of operators in conformal field theories coupled to a background metric. As an important example, the expectation value of the trace of the energy-momentum tensor acquires an anomalous term after quantization, namely the celebrated Weyl anomaly [8].
By investigating the effective action in dimensional regularization, Deser and Schwimmer [9] made a conjecture regarding the possible candidates for the Weyl anomaly, which are global conformal invariants. This conjecture was later proven by Alexakis [10]. Just as diffeomorphism-covariant quantities, i.e., tensors, on pseudo-Riemannian manifolds can easily be constructed out of the metric, Riemann tensor and covariant derivatives, one might expect to find conformal-covariant tensors on conformal manifolds. However, unlike the abundance of diffeomorphism-covariant quantities on (M, g), it is significantly harder to construct conformalcovariant tensors on (M, [g]). Before the work of Fefferman and Graham, known examples of conformal tensors were the Weyl tensor in any dimension, the Cotton tensor [11] in 3d and the Bach tensor [12] in 4d. In their seminal work [13,14] Fefferman and Graham introduced the ambient metric construction based on previous work by Fefferman [15], which provided a systematic method of finding conformal-covariant tensors. The basic idea of the construction was to associate a (d+2)-dimensional "ambient" pseudo-Riemannian manifold to a d-dimensional conformal manifold. One can then find a specific class of ambient diffeomorphisms that induces Weyl transformations on the conformal manifold. In the context of AdS/CFT, diffeomorphisms that induce a Weyl transformation of the boundary metric are the Penrose-Brown-Henneaux (PBH) transformations [16]. Thus, conformal-covariant tensors can descend from ambient Riemannian tensors, and their Weyl transformations can be derived from certain ambient diffeomorphisms.
An important outcome of the ambient construction was to define extended obstruction tensors from covariant derivatives of the ambient Riemann tensor [17]. Obstruction tensors are the generalization to higher (even) dimension of the Bach tensor. For each even dimension, the corresponding obstruction tensor is the only irreducible conformal-covariant tensor in that dimension [18]. Defined through the ambient space, the k th extended obstruction tensor Ω (k) ij has a simple pole at d = 2k +2, whose residue is the obstruction tensor in that dimension. For example, the first obstruction tensor reads Ω (1) where B ij is the Bach tensor, namely the obstruction tensor in 4d. The extended obstruction tensors also play an integral role in the context of holography as the basic building blocks of the holographic Weyl anomaly [17,19], or equivalently the Q-curvature [18,20,21]. A different perspective on conformal geometry was introduced by Weyl [1], whose idea was to make the physical scale a local quantity. The Weyl connection was introduced so that one can transport the physical scale between two points of the manifold. Although Weyl's initial attempt to identify the Weyl connection with the electromagnetic gauge field failed, the consistent mathematical structure he introduced was developed further in [22,23]. In this approach, a Weyl connection a is introduced on the conformal manifold which transforms together with the metric g under a Weyl transformation. One can modify the conformal class [g] to a Weyl class [g, a], which is the equivalence class formed by the pairs (g, a) ∼ (B(x) −2 g, a − d ln B(x)). This defines a Weyl manifold (M, [g, a]), and the conformal geometry is promoted to Weyl geometry [22][23][24]. Equivalently, a Weyl connection can be thought of as a connection on the Weyl structure, which is a principal bundle with the Weyl symmetry group as the structure group [22].
Similarly to a conformal-covariant tensor, one can define a Weyl-covariant tensor T on a Weyl manifold (M, [g, a]) to be a tensor that transforms covariantly under a Weyl transformation: Although conformal-covariant tensors on a conformal manifold (M, [g]) are hard to find, Weylcovariant tensors on a Weyl manifold (M, [g, a]) can be constructed quite easily. Recall that on a pseudo-Riemannian manifold (M, g), one can define a Levi-Civita (LC) connection ∇, and it is well-known that diffeomorphism-covariant quantities can be constructed from the metric, Riemann curvature, and covariant derivatives of the Riemann curvature. On a Weyl manifold (M, [g, a]), one can define a Weyl-Levi-Civita connection∇, and a plethora of Weyl-covariant quantities can similarly be constructed from the metric, Weyl-Riemann curvature, and Weyl-covariant derivativeŝ ∇ of the Weyl-Riemann curvature. This indicates that the Diff(M ) Weyl symmetry is manifested more naturally on a Weyl manifold, and the representation has a similar structure as that of Diff(M ) on pseudo-Riemannian manifolds. There are corresponding notions of Weyl metricity, Weyl torsion and a uniqueness theorem giving a Weyl-LC connection [22]. A significant step towards incorporating the Weyl connection in the formalism of AdS/CFT was taken in [25]. By modifying the Fefferman-Graham (FG) ansatz of an asymptotically locally AdS (AlAdS) spacetime to the Weyl-Fefferman-Graham (WFG) form, it was shown [25] that the bulk LC connection induces a Weyl connection on the conformal boundary. Thus, the AlAdS bulk geometry in the WFG gauge induces a Weyl geometry instead of only a conformal geometry on the conformal boundary. Applying the holographic dictionary [7], they also calculated the holographic Weyl anomaly in the WFG gauge, and found it can be organized in a Weyl-covariant fashion. In the FG ambient construction, the conformal boundary (M, [g]) of a (d + 1)-dimensional AlAdS bulk is associated with a (d + 2)-dimensional ambient space, and the AlAdS bulk in the FG gauge can be considered as a hypersurface in the ambient space. A natural question to ask is whether such a construction exists for the conformal boundary as a Weyl manifold. In this paper we will provide such a construction. We introduce the Weyl-ambient space (M ,g) as a modification of the FG ambient space, in which the AlAdS bulk in the WFG gauge is a hypersurface and its boundary is associated with a codimension-2 Weyl manifold (M, [g, a]).
Following [25], the AlAdS bulk expansion in the WFG gauge was further investigated in [26]. Using the technique of dimensional regularization, the Weyl-obstruction tensors and extended Weylobstruction tensors were introduced as the poles in the on-shell metric expansion. It was also found in [26] that the holographic Weyl anomaly can be expressed in terms of extended Weyl-obstruction tensors. Although it is convenient to read off the extended Weyl-obstruction tensors from the pole of the AlAdS metric expansion, this should not be regarded as a precise definition since the pole may have an ambiguity when shifted by a finite term. One of the results of the present paper is to provide a definition of Weyl-obstruction tensors on a Weyl manifold (M, [g, a]) through the Weyl-ambient space (M ,g), in a way analogous to how extended obstruction tensors were defined in [14,17]. Many properties of the extended Weyl-obstruction tensors can also be derived from the Weyl-ambient space.
The main goal of this paper is to provide an ambient construction for Weyl manifolds. We start by introducing the Weyl-ambient metric as a modification of the FG ambient metric. We will then present two perspectives. The first one is a top-down approach. We will see that one naturally obtains a codimension-2 Weyl manifold (M, [g, a]). A more formal approach is the bottomup perspective, where we start from a d-dimensional conformal manifold (M, [g]), which is then enhanced into a Weyl manifold (M, [g, a]) by introducing a connection on the Weyl structure over M . A (d+2)-dimensional Weyl-ambient space can then be constructed by taking the Weyl structure as an initial surface, which follows the rigorous ambient space construction in [14]. This paper will be organized as follows. In Section 2 we first briefly review the ambient metric of Fefferman-Graham before we introduce the Weyl-ambient metricg at the end. To get some intuition on the construction, we start from the example of the flat ambient metric and then generalize to Ricci-flat ambient metrics. Some different coordinate systems presented in Section 2 are described in Appendix A. Then, from a top-down perspective, in Section 3 we demonstrate how (M ,g) induces a codimension-2 Weyl manifold (M, [g, a]). We also discuss how the Weyl-covariant tensors on (M, [g, a]) can be derived from the Riemann tensor of (M ,g), and define the extended Weyl-obstruction tensors as a special example. We work in first order formalism in Section 3 by introducing a null frame, with some details of the calculation given in Appendix B. Section 4 is the bottom-up construction of the Weyl-ambient metric. We show rigorously that the Weyl-ambient metric has a well-defined perturbative initial value problem, where the Ricci-flatness condition plays the role of the equation of motion. To show this we follow [14] closely using the second order formalism and generalize their definitions properly for the Weyl-ambient space. We also extend some major theorems of [14] with suitable modifications for the Weyl-ambient space. The details of some proofs are presented in Appendix C. After that we discuss Weyl-covariant tensors and extended Weyl-obstruction tensors from the point of view of the second order formalism, and prove the equivalence of the extended Weyl-obstruction tensors defined from both approaches. Finally, our results are summarized in Section 5.

Notation
We will label the indices in a d-dimensional manifold M by lowercase Latin letters i, j, · · · , in a (d + 1)-dimensional AlAdS bulk by lowercase Greek letters µ, ν, · · · , and in a (d + 2)-dimensional ambient spaceM by uppercase Latin letters I, J, · · · . The vectors on M are denoted by U , V , on the Weyl structure P W over M are denoted by u, v, and on the ambient manifoldM are denoted by U, V.
In Section 3, we mainly use the dual frame {e I }, and the ambient frame indices are I = +, 1, · · · , d, −. Unless otherwise indicated, in Section 4 we mainly use the ambient coordinate system {t, x i , ρ}, and the indices are I = 0, 1, · · · , d, ∞, where 0 labels the t-component and ∞ labels the ρ-component. The notation (0, x i , ∞) is also used for the components in a trivialization P W ×R R + ×M ×R, even without specifying a choice of coordinates on M . The above-mentioned notation is summarized in Table 1.

Ambient Metrics
In this section we will review the FG ambient metric and introduce the Weyl-ambient metric. To build up some intuition, we begin with the flat ambient metric and then generalize to Ricci-flat ambient metrics.

Flat Ambient Metrics
The simplest example of an ambient space is the flat ambient space. Consider the (d+2)-dimensional Minkowski spacetime R 1,d+1 with the metric One can describe (d + 1)-dimensional Euclidean AdS spaces as the following codimension-1 hyperboloids: 1 where L represents the AdS radius. The hyperboloids with different L form a one-parameter family of hypersurfaces foliating the interior of the future light cone, denoted by N + , emanating from the origin of the Lorentzian coordinate system {X 0 , X i }. Then, one can also write the Minkowski metric in the following "cone" form: where the coordinate = (X 0 ) 2 − R 2 , and g + is the (d + 1)-dimensional Euclidean AdS metric. Now the Euclidean AdS space is represented by the hyperbola = L. The metric g + can be expressed in the Fefferman-Graham (FG) form in the following different ways (see Appendix A for details): The metric (6) with g + = g + S or g + F is defined in the whole interior of the light cone N + , 2 while their AdS boundaries have different topologies. It is easy to see that the AdS boundary at z → 0 + of g + S in (7) is conformally a d-sphere while that of g + F in (8) is conformally flat. While the metric (6) is singular in the limit z → 0 + with fixed, it is well-defined when taking both z and to zero with z/ fixed. To make this evident we introduce a new coordinate system {t, x i , ρ}, called the ambient coordinate system, with t = /z and ρ = −z 2 /2. First we look at the metric (6) with g + S in (7), which in the ambient coordinate system becomes The coordinate patch of { , x i , z} which covers the interior of the light cone surface N + , corresponds to t ∈ (0, ∞), ρ ∈ (−2L 2 , 0) (see Figure 1). However, it is apparent now that the limit ρ → 0 − of the above metric is well-defined, and thus we can extend the coordinate patch of {t, x i , ρ} to include an open neighborhood of the surface N + at ρ = 0. Hence, N + is parametrized by {t, x i }, where t ∈ R + and x i are the coordinates of the d-sphere S d . In other words, N + can be regarded as a line bundle over S d whose fibres are parametrized by t.
Suppose φ is a function on R 1,d+1 , which defines a hypersurface Σ by the locus of points p ∈ R 1,d+1 such that φ(t, x i , ρ)| p = 0. In order to find the intersection Σ ∩ N + , one can set ρ = 0 and solve for t as a function t(x i ) of the d-sphere coordinates from φ(t, x, ρ = 0) = 0. The pullback metric on the intersection submanifold is η| Σ∩N + = t(x) 2 L 2 dΩ 2 d . The function t(x) depends on the choice of function φ (which is arbitrary) that defines Σ, and thus we see that the pullback metric is conformally equivalent to the metric of S d . An example is to take φ = ln t, and to consider the pull back of the metric at ρ = 0, t = 1, namely η| ρ=0,t=1 = L 2 dΩ 2 d . If we perform a diffeomorphism t = B(x) −1 t and pull back the metric at ρ = 0, t = 1, then we find η| ρ=0,t =1 = B(x) −2 L 2 dΩ 2 d . Therefore, at ρ = 0 we have a conformal class [g] of d-dimensional metrics, and the (d + 2)dimensional Minkowski metric expressed in (9) is said to be the ambient metric of [g]. This implies that the null surface N + at ρ = 0 is associated with a metric bundle, which will be important for the formal construction later in Section 4.
Similarly, the metric (6) with g + F in (8) can also be expressed in the ambient coordinates as In this case, the original coordinate patch of { , x i , z} corresponds to t ∈ (0, ∞), ρ ∈ (−∞, 0), and the null surface N + is again covered by the {t, x i , ρ} system at ρ = 0. Intersecting the null surface with a hypersurface and taking the pullback metric on the intersection, we now obtain a d-dimensional metric ds 2 = t(x) 2 δ ij dx i dx j that is conformally flat. This metric is also in the conformal class [g] but the topology is different from the d-dimensional metric obtained from (9). Note that the flat ambient metric in either (9) or (10) is homogeneous of degree 2 with respect to the t-coordinate; that is, under a constant scaling t → st the metric transforms as η → s 2 η, or in the infinitesimal form, 2 Note that for Lorentzian signature AdS spacetime, the metric (6) with g + F only covers half of the interior of the future light cone. c o n s t a n t t c o n s t a n t ρ P Figure 1: Sketch of a constant-ρ surface (red) and a constant-t surface (green) of the flat ambient metric (9) in the Lorentzian coordinate system {X 0 , X i }. Constant-t surfaces are past directed light cones. Changing t moves the apex P of the cone along the X 0 -axes. Constant-ρ surfaces are future directed timelike cones. When ρ → 0 − the constant ρ surface becomes the light cone N + (blue).
We will retain this property also for Ricci-flat ambient metrics and the Weyl-ambient metric. For relaxation of this homogeneity condition, see [27].

Ricci-Flat Ambient Metrics
The flat ambient metric combines hyperbolic metrics and their conformal boundaries in a unified framework. Before we describe its utility, we will review the generalization of flat ambient metrics to Ricci-flat ambient metrics. This will allow us to consider (d + 1)-dimensional asymptotically locally Anti-de Sitter (AlAdS) spaces which are especially relevant in holographic theories.
The main observation that allows an extension to Ricci-flat ambient metrics is that (6) can be generalized in the following form: where now g + (x) is an arbitrary (d + 1)-dimensional metric independent of . We will refer to this (d + 1)-dimensional geometry as the "bulk". The ambient Ricci tensorRic(g) can be decomposed in terms of the Ricci tensor of g + as [14,28] Ric(g) = Ric(g + ) + d The right-hand side of the above equation can also be written as G µν (g + ) + Λg + µν with Λ = − d(d−1) 2L 2 . Therefore, when the ambient metricg is Ricci-flat, g + is an Einstein metric and thus satisfies the vacuum Einstein equations.
According to the Fefferman-Graham theorem [13,28], any AlAdS Einstein metric can be expressed in the Fefferman-Graham form which is a generalization of (8). Then, by a coordinate transformation t = /z and ρ = −z 2 /2, the metric (12) takes the form We can see that the flat ambient metrics (9) and (10) are nothing but special cases of (15) wheñ g = η. The codimension-2 metric is now generalized to an arbitrary γ ij (x, z) whose corresponding g + in (14) is an Einstein metric. Note that the advantages of the ambient coordinate system {t, x i , ρ} mentioned before for the flat ambient space are now carried over to the Ricci-flat case. One can see that the surface at ρ = 0 is still a null hypersurface, denoted by N , which is a coordinate singularity in the original { , x i , z} coordinate system. Hence, the ambient coordinate system permits one to extend the spacetime region to include an open neighborhood of the null surface N . Denoting the extended spacetime manifold asM , then N is a hypersurface inM parametrized by {t, x i }, which furnishes a conformal class [γ] of codimension-2 metrics. Suppose M is a d-dimensional manifold equipped with the conformal class [γ], then (M ,g) is called the (d + 2)-dimensional ambient space of (M, [γ]).
Being part of the Ricci-flat ambient space, N can be regarded as an initial value surface. Then given the initial data γ ij (x, ρ)| ρ=0 , the Ricci-flatness condition can be used to "propagate" the metric beyond the initial surface to a neighborhood around ρ = 0. That is, the Ricci-flatness conditioñ Ric(g) = 0 is a set of differential equations forg ij (x, ρ), which can be solved iteratively in a series around ρ = 0 given the initial valueg(x, ρ)| ρ=0 . The initial value problem for the Ricci-flat ambient space has been defined and evaluated rigorously in [14], the results of which will be carried over to the Weyl-ambient space in later sections.

Weyl-Ambient Metric
Now we are ready to introduce the Weyl-ambient metric. We start from the (d + 2)-dimensional ambient metric in the form of (12). The expression of g + in (14) is the FG ansatz for an AlAdS spacetime, which is not preserved under a Weyl diffeomorphism z → z/B(x), x i → x i . This motivated the authors of [25] to introduce the Weyl-Fefferman-Graham (WFG) gauge by adding an additional mode a µ to (14) as follows: Now we substitute the g + in (12) with the WFG ansatz (16), then transforming back to the ambient coordinates {t, x i , ρ}, we obtain the Weyl-ambient metric 3 where g ij (x, ρ) := γ ij (x, ρ)−2ρa i (x, ρ)a j (x, ρ). We call the pseudo-Riemannian space (M ,g) a Weylambient space. Having the form of the Weyl-ambient metric, the ambient Weyl diffeomorphism 4 induces a change in the constituents a i and γ ij of the form If we regard the AlAdS bulk as a hypersurface of the Weyl-ambient space, the above transformation gives rise to the Weyl diffeomorphism which preserves the WFG ansatz. In addition, we want to point out that just as the ambient metric (15) is homogeneous with respect to t, the homogeneity property (11) also pertains for the Weyl-ambient metric (17) since both a i (x, ρ) and γ ij (x, ρ) are independent of t. This homogeneity property will be repeatedly used throughout this paper. In the following we use this property in order to show how an induced Weyl class arises from the Weylambient metric; it is also crucial for the bottom-up construction and for proving Propositions 3.1 and 4.6. The Ricci-flatness conditionRic(g) = 0 for the Weyl-ambient metric (17), similar to that for the ambient metric (15), is a set of differential equations forg ij (x, ρ) which can be solved order by order in a neighborhood of ρ = 0 given the initial valueg ij (x, ρ)| ρ=0 . To be precise, in a neighborhood of ρ = 0 we can expand γ ij and a i as 5 From the equationRic(g) = 0, one can solve for γ   i (x, ρ) as input data. This initial value problem will be examined in detail in Section 4 after the Weyl-ambient space is defined in terms of the Weyl structure and the ansatz in (17) will be shown to be the uniquely determined Weyl-ambient metric for any given γ ij (x) and a i (x, ρ). From the transformation (19) and the expansions (20) and (21), we can see that γ On the other hand, a Therefore, we should anticipate that a (0) i can be interpreted as a Weyl connection on the codimension-2 geometry. An important result of [25] was to show that the bulk metric of an AlAdS spacetime in the WFG gauge provides a Weyl geometry on the conformal boundary. In the next section we will show that by introducing a i (x, ρ) 4 In terms of the coordinates , z, the ambient Weyl diffeomorphism acts as ( , x i , z ) = ( , x i , B(x) −1 z). 5 There will be a second series starting from the ρ d/2 order in the expansion (20): However, to solve for the second series in γij order by order one needs the interior data π (0) ij of the ambient space. This is related to the obstruction tensors as explained in [26].
in the ambient metric, we indeed obtain a Weyl geometry at codimension-2, where γ (0) ij and a (0) i play the role of a metric and a Weyl connection, respectively.
Closing this section, we remark that the codimension-1 surface N at ρ = 0 is again a null surface parametrized by (t, x) with t ∈ R + , just like the case of the ambient metric (15). This surface in fact has the structure of a line bundle with each fibre parametrized by t, which turns out to be a principal bundle with the structure group R + . The new ingredient a i in the Weyl-ambient metric (17) induces naturally a connection on this principal bundle, represented by a (0) i = a i | ρ=0 . We will explore this in Section 4.

Induced Weyl Geometry
The goal of this section is to analyze the Weyl-ambient metric from a top-down perspective before we introduce the more formal bottom-up construction of the Weyl-ambient space in Section 4. We will show explicitly that the Weyl-ambient metric (17) leads to a Weyl geometry at codimension-2.
Define a dual frame {e P } on the (d + 2)-dimensional manifoldM as follows: where now P = {+, i, −}. In this frame the Weyl-ambient metric (17) can be written as It is easy to check that the 1-forms defined in (23) are covariant under (18) and (19), and thus the form ofg in (24) is preserved under an ambient Weyl diffeomorphism. The corresponding frame {D P } of (23) reads From (24) it is clear that D + and D − are null vectors.
It follows from (25) that The Frobenius theorem implies that the distribution C d is integrable when f ij = 0, though we will not generally assume this to be the case. One should note that the codimension-1 distribution spanned by {D i , D + } is integrable at ρ = 0, and thus defines a codimension-1 subspace (see Appendix B for relevant details). Suppose M is a d-dimensional manifold with a local coordinate system {y i } on U ⊂ M , and a pointp ∈M has coordinates (t, x i , ρ). One can consider the coordinate patchŨ of the ambient coordinate system {t, x i , ρ} as a fibre bundle with the projection π :Ũ → U such that π(p) = p ∈ M has coordinates y i = x i , i.e. each fibre inŨ is parametrized by (t, ρ). For simplicity, in what follows we will refer toŨ asM and U as M , and we will not distinguish {x i } and {y i }. Now that we have a bundle structure π :M → M , we can see that a i (x, ρ) plays the role of an Ehresmann connection that specifies the horizontal subspace Hp = C d |p ⊂ TpM , which defines the horizontal lift T p M → Hp with ∂ i → D i . In general then, we are describing an isolated surface.
Since we have a bundle structure π :M → M , each section defines an embedding φ : M →M such that a point p ∈ M with coordinates x i is mapped to φ(p) = (t(x), x i , ρ(x)). With the horizontal subspace defined, we have π * : from the subspace ofM at t = 1 and ρ = 0 similar to what we did for the flat ambient space: γ Under the coordinate transformation (18) inM induced by an ambient diffeomorphism, we can consider the pullback That is, under the ambient Weyl diffeomorphism inM , we obtain two induced metrics which are related by a Weyl transformation in M . Hence, the ambient Weyl diffeomorphisms acting on the surface ρ = 0, namely the null surface N , gives rise to a conformal class of metrics on M . 7 Having a conformal class of induced metrics on M , now let us look at how a connection is induced fromM onto M . Suppose∇ is the Levi-Civita connection of the ambient space (M ,g), i.e. it is torsion-free and has zero metricity∇ D Pg M N = 0. The ambient connection coefficientsΓ P M N of∇ are defined with respect to the frame D M of TM as: In the following discussion we will denote the covariant derivative∇ D P along D P as∇ P for brevity (P = +, i, −); we emphasize that these are not however the coordinate frame components. The ambient connection 1-formω M N =Γ M P N e P in this frame is then found to be (the matrix elements are arranged in the order of +, i, −) where the upper i, j indices are raised by γ ij ≡ (γ ij ) −1 , and We note that the Levi-Civita condition∇ igjk = 0 evaluates to Hence, if we interpret γ ij , i.e.g IJ restricted to the i, j indices, as giving rise to a metric on the distribution C d spanned by {D i } inM , then the connection ∇ on C d has a nonvanishing metricity 2a i γ jk . Equivalently, we say that this connection has vanishing Weyl metricity, and it is therefore convenient and natural to introduce a connection∇ on C d , such that The vanishing of the Weyl metricity is a Weyl-covariant condition, whereas the vanishing of the usual metricity ∇ i γ jk is not. More generally, for any tensor T defined on C d (i.e., T has no +, − components) that transforms covariantly under an ambient Weyl diffeomorphism as will also transform covariantly with the same weight. For example, it follows from the definitions in (33) , and thus we can write their Weyl-covariant derivatives aŝ From the above behavior of the induced connection on C d , we can naturally expect that the induced connection on M will give us a codimension-2 Weyl geometry. However, since {D i } is not an integrable distribution when a i is turned on, the connection coefficients (34) cannot be pulled back directly to M . As we will see below, this problem does not exist if we focus on the surface at ρ = 0. Notice thatΓ i jk does not depend on t, and thus at any value of t at ρ = 0, the induced connection coefficients can be expressed as To define an induced connection on M , let us take t = 1 as a representative, i.e. take φ(M ) to be a d-dimensional surface inM at ρ = 0 and t = 1. At first sight, the connection defined by (37) Then, ∇ (0) can also be defined on T M , which defines the parallel transport of a vector along a curve on M : In this way we get a connection ∇ (0) on M whose connection coefficients are given by (37). This is a connection that satisfies ∇ jk , i.e. it has vanishing Weyl metricity, and a (0) i plays the role of a Weyl connection on M . One can also define a metricity-free connection∇ (0) on M satisfying∇ jk = 0, which can be referred to as a Weyl-LC connection.
An ambient Weyl diffeomorphism inM induces on M a Weyl transformation γ 8 This means that we get a Weyl class [γ (0) , a (0) ], which is the equivalence class formed by all the pairs of γ (0) and a (0) that are connected by Weyl transformations, i.e., With the Weyl class defined on M , we obtain a d-dimensional Weyl manifold (M, [γ (0) , a 0 ]) (see [22,23,25]) induced by the Weyl-ambient space (M,g), where the geometric quantities defined in terms of the Weyl connection are Weyl covariant. For example, one can define on M the Weyl- [26] for a review on Weyl geometry and the details of these Weyl-covariant quantities.)

Weyl-Obstruction Tensors: First Order Formalism
A very useful property of the ambient metric introduced in [30] in the context of conformal geometry is the ability to construct conformal-covariant tensors from the ambient Riemann tensor, including the (extended) obstruction tensors, which were generalized to (extended) Weyl-obstruction tensors in [26]. The Weyl-obstruction tensors on a d-dimensional Weyl manifold (M, [γ (0) , a (0) ]) were introduced in [26] as the poles of the metric expansion of γ ij in the AlAdS bulk. However, they can also be defined in a more explicit way from the Weyl-ambient space (M ,g). Before we construct the Weyl-ambient space rigorously, we would like to demonstrate how the Weyl-obstruction tensors on M can be derived from (M ,g) in the first order formalism using the frame introduced in (23).
Starting from the metric (24), one can solveRic(g) = 0 order by order (which is equivalent to solving the Einstein equations in the AlAdS bulk [26]) to find 9 where f ij at d = 2k (P ij also represents the "pole" of γ (1) ij at d = 2, which identically vanishes in 2d). Later in this section we will recognize them as extended Weyl-obstruction tensors 8 If one considers a more general version of the diffeomorphism (18) through a precise definition. In terms of γ (0) ij , these quantities can be written aŝ jP ik is the Weyl-Cotton tensor. Note that indices are lowered with γ (0) ij as necessary. We first look at how the Weyl-Schouten tensorP ij is derived from the Weyl-ambient geometry. Consider the expansion of γ ij . At ρ = 0 and t = 1, the ambient connection 1-form (32) becomes Notice that the first term, which is the pullback ofω M (0) N from T * M to T * M , can be recognized as the Cartan normal conformal connection [31,32]. From here we can see that the Weyl-Schouten tensor of the boundary appears in the leading order (ρ = 0) of the ambient connection.
From the connection 1-form (32), we can also find the ambient curvature 2-form in the frame {e + , e i , e − }: Here we defined where∇ was introduced in (35), and Plugging in (41) and (42) from the ρ-expansion of γ ij , one obtains at the leading order Therefore, when pulled back fromM to M the Riemann curvature of the Weyl-ambient space gives us on M the Weyl tensorŴ i (0) jkl , Weyl-Cotton tensorĈ ijk and the tensorΩ (1) ij we obtained in (45) as follows: The corresponding curvature 2-form at ρ = 0, t = 1 can be expressed as As expected, the first matrix in (55), which represents the components ofR M (0) N in the e i ∧ e j directions, is the curvature 2form of the Cartan normal connection. The e i ∧ e − components, on the other hand, give rise to the tensorΩ (1) ij on M , which is expected to be the first extended Weyl-obstruction tensor. This implies that we can define the extended Weyl-obstruction tensors on the d-dimensional manifold M by means of the (d + 2)-dimensional Weyl-ambient space. Before getting to that, we first provide the following proposition, which shows that diffeomorphism-covariant tensors in the Weyl-ambient space are Weyl-covariant tensors when pulled back to M .
Proof. Under the ambient Weyl diffeomorphism (18), the vector basis {D P } transforms as where Hence,∇ Noticing the fact thatg is homogeneous in t with degree 2, and considering the t-dependence of D + and D − in (25), we havẽ Combining (59) and (60) we obtain (56).
Since diffeomorphism-covariant tensors can be constructed out of the Riemann tensor and its covariant derivatives [33], this proposition implies that the pullback of an ambient tensorT M 1 ···M k to M : can be read off to be 2, 0, and −2, respectively, which are indeed the correct Weyl weights (see Table 1 in Appendix A of [26]). As a special kind of Weyl-covariant tensor, we introduce the extended Weyl-obstruction tensors as follows. Definition 1. Suppose k is a positive integer. The k th extended Weyl-obstruction tensorΩ Some properties of Weyl-obstruction tensors can be readily seen from the above definition. From the symmetry of the Riemann tensor we can see thatΩ (k) ij is a symmetric tensor. It follows from Proposition 3.1 thatΩ (k) ij is Weyl covariant with Weyl weight 2k. Also, from the Ricci-flatness condition we obtain thatg IJ∇ M 1 · · ·∇ MrRIKJL = 0, which gives rise to γ ij ij is traceless.
We have seen in (53) that when k = 0, this definition gives theΩ (1) ij in (45). By computing ∇ −R−ij− , one also finds thatΩ (2) ij defined in this way gives exactly the expression in (46) (see Appendix B). Notice again that before Definition 1 (also in [26]), when we referred toΩ (k) ij as the k th extended Weyl-obstruction tensor, we simply meant that they denote the pole of γ (k+1) ij at d = 2k + 2; however, there is an ambiguity since the pole can be shifted by a finite term, and so that was not a precise definition for extended Weyl-obstruction tensors. Now theΩ (k) ij defined through the Weyl-ambient space is uniquely determined. The proposition below will show that eachΩ (k) ij defined through the Weyl-ambient space indeed has a pole at d = 2k + 2, whose residue is the same as the pole in γ  Res d=2kΩ More specifically,Ω (k−1) ij has the following form: where ∆ (0) ≡∇ (0) k∇ k (0) and the ellipsis represents the terms with fewer number of∇ (0) . The terms inside the brackets represent the Weyl-obstruction tensor.
Proof. First, let us show that γ (k 2) ij has a pole at d = 2k, which has the form We have seen this previously for k = 2 and 3. Using mathematical induction, now we will prove the following equation for k 2: where ∆ ≡∇ k∇ k . This relation leads to (65) when ρ = 0 since ψ ij = 1 2 (∂ ρ γ ij + f ij ) (the f ij in the left-hand side are combined in the ellipsis). Differentiating the Ricci-flatness condition of the form (B.24) with respect to ρ and use the expression (B.25) we can see that which is (66) in the case k = 2. Now we assume (66) holds for k = n. Differentiating both sides of (67) for n − 1 times with respect to ρ yields Note that ∂ ρ produces two∇ when acting on ψ, while it only produces one∇ when acting onΓ i jk , and thus when we commute ∂ ρ with∇, the new terms only contribute to the ellipsis. Hence, where we used (B.23) and the assumption that (66) holds for k = n. This is exactly (66) for k = n + 1, and thus (66) is proved for any k 2. Therefore, at ρ = 0 we have From (48) we can read off that Hence, the Weyl-obstruction tensorΩ ij has the form where finite terms at d = 2k are shifted into the pole. On the other hand, from (72) we also have Res d=2kΩ where in the second equality we considered that f ij does not contribute to the pole.
This proposition indicates that both the extended Weyl-obstruction tensorΩ ij are meromorphic functions, which are holomorphic in the whole complex plane except at even integers d = 4, 6, · · · , 2k. We have seen that the pole at d = 2k is a simple pole, while the pole at a lower even dimension could be of higher order. These two tensors only differ by terms that are finite at d = 2k. Therefore, we can express γ (k) ij in terms ofΩ (k−1) ij plus finite terms as we have seen for k = 1, 2 in (42) and (43).
Later in the next section, we will introduce the extended Weyl-obstruction tensors in the second order formalismà la [14] and show that the two definitions are equivalent. 4 Weyl-Ambient Space: Bottom-Up Perspective

Formal Theory
In this section we will present a geometric interpretation of the Weyl-ambient metric (17) as well as the Weyl connection therein in terms of a bottom-up construction. By "bottom-up" we mean to construct a (d+2)-dimensional Weyl-ambient space from a d-dimensional manifold M . The majority of this section is based on Section 2 and Section 3 of [14] where a more detailed exposition of the ambient construction can be found. We will generalize the main definitions and theorems there with the inclusion of a Weyl connection on the principal R + -bundle. The resulting Weyl structure together with the metric bundle, viewed as an associated bundle, will be then used to define the Weyl-ambient metric. For this section to be self-contained we repeat some of the definitions and proofs of [14] when necessary while generalizing them appropriately.
We start with a d-dimensional manifold M and introduce a principal R + -bundle P W over M that we call a Weyl structure. 10 Definition 2. Given a d-dimensional manifold M , a Weyl structure is a (d+1)-dimensional manifold P W together with the structure group R + , which is equipped with (2.1) a free right action δ : P W × R + → P W , such that δ s (p) = p · s, ∀p ∈ P W , s ∈ R + ; (2.2) a projection map π : P W → M , such that π(p) = π(p · s), ∀p ∈ P W , s ∈ R + ; For brevity, suppose U i ⊂ M has local coordinates {x i }, we can express a point p ∈ P W as (x, t) with t ∈ R + .
A connection on the Weyl structure can be described as follows. First we note that the push forward π * : T P W → T M defines the vertical sub-bundle V ⊂ T P W given at any point p ∈ P W by In the present case V p is a one-dimensional vector space spanned by the fundamental vector field which generates the group action along the fibres; in the local trivialization, it is expressed as T = t∂ t . From the perspective of P W , we can then think of the action of R + as corresponding to a dilatation of the fibres. To assign a connection on P W is to specify a horizontal sub-space H p ⊂ T p P W such that T p P W = H p ⊕ V p at any p. In the local trivialization given above, the horizontal bundle can be described as the span of vectors of the form D i = ∂ i − a i (x)t∂ t . 11 Equivalently, it can be described as the kernel of a form n := t −1 dt + a i (x)dx i ∈ T * P W , i.e.
We note that under the Abelian group action (x, t(x)) → (x, t (x)) = (x, t(x)s(x)), we have and so we see that the coefficients a i (x) transform as connection coefficients. Note also that it is natural to introduce the projector a : T P W → V as which is an alternative way to express the connection on P W . We will refer to both a and a i (x) as the Weyl connection. This line bundle has an important representation given by a conformal class of metrics. Indeed, all the non-trivial representations are one-dimensional, and thus a representation of R + is given by specifying a Weyl weight w. We call the corresponding associated bundle E w and its sections respond to the group action as Equivalently, this determines the transition functions on the associated bundle. Suppose a conformal class [g] of smooth metrics of signature (p, q) is given on M , in which any two representatives g and g are related by a smooth function B(x) as g x = B(x) −2 g x , where g x is the value of g at a point x ∈ M . Then, (M, [g]) is a conformal manifold. One can define a metric bundle G as follows [14]: This definition simply identifies a conformal class of metrics with a bundle associated to the Weyl structure given by the weight w = −2 representation of R + . We note that it is isomorphic to the Weyl structure P W , as is any non-trivial associated bundle of P W . 12 Under a trivialization, assigning an isomorphism between P W and the metric bundle G can be thought of as a choice of representative g of the conformal class [g] if we identify Given g ∈ [g], for any p ∈ P W , by means of the corresponding (x, h) ∈ G one can define a symmetric tensor g 0 of type (0, 2) called the tautological tensor that acts on vector fields w 1 , w 2 ∈ T p P W as follows: which can be expressed as g 0 = t 2 π * g under the identification in (79). If we pick another representative g x = B(x) −2 g x of the conformal class [g], following the identification in (79), we obtain another isomorphism between P W and G by identifying It is easy to see that the two isomorphisms are related by setting t = B(x)t. To preserve the horizontal subspace on P W , from (76) we can see that a i (x) satisfies In the present circumstances, it is natural to replace the notion of conformal class [g] by the Weyl class [g, a], with the property where d is the exterior derivative on M . Before we proceed to define the Weyl-ambient space based on the Weyl structure P W , we would like to make a few remarks. Recall that for the Weyl-ambient metric (17), the coordinates t and x i parametrize a codimension-1 null hypersurface N located at ρ = 0. One can see that this surface is exactly a Weyl structure. In Section 3, the degenerate "induced metric" ofg on N is the tautological tensor, the induced metric γ (0) on M is a representative g in the conformal class, and the Weyl connection a  [g, a]) defines a Weyl manifold. We will discuss more details of the role of the Weyl connection and the horizontal subspace it defines in Theorem 4.1 below. It is noteworthy that the projector in (77), which defines the Weyl connection on P W , is a special case of the construction presented in [34] with restricted diffeomorphisms. Now we will define a Weyl-ambient space for a Weyl manifold generalizing the definition of a Fefferman-Graham ambient space for a conformal manifold introduced in [14]. Consider a (d + 2)dimensional spaceM which looks at least locally like P W × R where each point can be labeled by (p, ρ) with ρ ∈ R. The inclusion map ι : P W →M is defined such that p → (p, 0). By letting the map δ s act only on p ∈ P W , we can extend δ s to a map onM , which commutes with ι. The vector field T which generates the Weyl group action is extended to a vector field T = ι * T = t∂ t onM .
Without condition (4.3), (M ,g) is called a Weyl pre-ambient space for (M, [g, a]). Note that the condition (3) in [14] is presented differently when d is even and odd, and Ric(g) has an obstruction in the order O(ρ d/2−1 ) for even d. Here we take the dimension to be a continuous complex variable, and so the Ricci-flatness condition always holds to infinite order. The obstruction at even dimension will be manifested by the pole of the expansion ofg at even d, which is identified as the extended Weyl-obstruction tensor. This formulation was described in [25] and the equivalence of the two formulations in the level of AlAdS bulk was addressed in [26]. Now we introduce the final ingredient in our Weyl-ambient construction-the Weyl-normal form, which is a generalization of the normal form defined in [14].
) is said to be in Weyl-normal form with acceleration A if (5.1) For each fixed p ∈ P W , the set of ρ ∈ R such that (p, ρ) ∈M is an open interval I p ∈ R containing 0.
(5.2) For each p ∈ P W , the parametrized curve C p : I p →M , ρ → (p, ρ) has a tangent vector U, whose acceleration A ≡∇ U U satisfiesg(T , A) = 0, where∇ is the Levi-Civita connection of (M ,g).
where g 0 is the tautological symmetric tensor defined in (80).
Definition 5 is engineered for the purpose of generating the Weyl-ambient metric from the "initial surface" at ρ = 0. At ρ = 0, the Weyl-ambient metric we have seen in (17) has the form (84), which motivates condition (5.3). Since T = t∂ t everywhere inM , condition (5.2) implies that the covector A of the acceleration does not have a t-component. Furthermore, one can also parametrize the accelerated curve C p such thatg(A, U) = 0, and let A have no ρ-component either. 13 We will assume that ρ is such a parametrization. Note that in the special case where A = 0, the ρcoordinate lines are geodesics, and condition (5.2) goes back to that of normal form in [14], while condition (5.3) will still be different as long as a i (x) are nonvanishing. The acceleration A encodes all the higher modes a (k 1) i (x) in the expansion (21) of a i (x, ρ), as we will see in Lemma 4.3. In fact, if both a i (x) and A are zero, the mode a i (x, ρ) in (17) vanishes.
The following Theorem is a generalization of Proposition 2.8 in [14]. This theorem indicates that given a representative pair (g, a), any Weyl pre-ambient space can be put into Weyl-normal form by a diffeomorphism φ. (M ,g) and (M , φ * g ) are also said to be ambient-equivalent (see Definition 2.2 in [14] for the precise definition of ambient equivalence). For a proof of the theorem, see Appendix C.1.
Before we move on to the main result of this section, namely Theorem 4.2, let us introduce some useful notation. Given a local coordinate system {x i } (i = 1, · · · , d) on M , the fibre coordinate t of P W and the parameter ρ naturally defines an ambient coordinate system {t, x i , ρ} onM . Later on, we will follow [14] and use I, J, · · · = (0, i, ∞) to label the ambient coordinate indices, where 0 labels the t-component and ∞ labels the ρ-component. It is also convenient to interpret the notations (0, i, ∞) as representing the components in a trivialization P W × R R + × M × R, even without specifying a choice of coordinates on M .
We will now present Theorem 4.2, which is a natural generalization of Theorem 2.9 of [14], based on our definition of Weyl-normal form. As a corollary of this theorem, we will show that for a Weyl-ambient space in Weyl-normal form, the Weyl-ambient metric (17) emerges from the initial surface uniquely under the Ricci-flatness condition. We emphasize again that we consider the dimension d of the manifold M formally as a complex parameter, and do not need to distinguish between even and odd dimensions. This is in alignment with previous considerations in [25,26].   [g, a]). Suppose for each p ∈ P W , the set of all ρ ∈ R such that (p, ρ) ∈M is an open interval I p containing 0. Let g be a metric in the representative (g, a) of the Weyl class, which provides a local trivialization P W × R R + × M × R. Then (M ,g) is in Weyl-normal form with acceleration A if and only if one has onM : Proof. Supposeg satisfies (85), then it follows from the condition ι * g = g 0 for the pre-ambient space thatg| ρ=0 must have the form (84). Thus, all we have to prove is that forg satisfying (84) at ρ = 0, the condition that the ρ-coordinate lines have acceleration A withg(T , A) = 0 is equivalent to (85). The fact that the ρ-coordinate lines have an acceleration A implies whereΓ IJK ≡g KLΓ L IJ . The conditiong(T , A) = 0 leads to A 0 = 0. As we have mentioned, one can also parametrize the curve C p : I p →M such thatg(U, A) = 0, then we also have A ∞ = 0, and thus A I = (A 0 , A i , A ∞ ) = 0, t 2 ϕ i (x, ρ), 0 . The functions ϕ i (x, ρ) are considered as external input and cannot be determined from the initial conditions. The factor t 2 is derived from the homogeneity property ofg and (86). If we set I = ∞ in (86) we get where in the last step we used the initial condition g ∞∞ | ρ=0 = 0. Similarly, setting I = 0 in (86) we find where we used the initial condition g 0∞ | ρ=0 = t. Finally, setting I = i yields where we used the initial conditiong ∞i | ρ=0 = t 2 a i (x).
The first property above is the homogeneity property which is still taken to be true for the Weylambient metric. Property (3) is equivalent to condition (5.2) of Definition 5 due to Lemma 4.3, which indicates thatg I∞ components are known, while the rest are now regarded as unknown functions. Property (2) can be considered as the initial data of these components at the initial surface at ρ = 0, while the Ricci-flatness property (4) is a system of partial differential equations that one can solve to find the metric components beyond the initial surface. We will show that this is a well defined initial value problem so that the unknown components of the Weyl-ambient metric can be uniquely determined in a series expansion in ρ, which will prove part (B) of Theorem 4.2.
The complete proof can be found in Appendix C.1.
As an important corollary, we now show in Theorem 4.5 that the metricg determined from Theorem 4.2 has exactly the form of the Weyl-ambient metric (17). First we need the following lemma.
Proof. Forg of the form (90), we can write the inverse metric as and the Christoffel symbolsΓ IJK =g KLΓ L IJ are given bỹ where Γ ijk = g kl Γ l ij are the Christoffel symbols of g ij (x, ρ), and F jk = ∂ j a k − ∂ k a j . Substituting (91) and (92) into (C.33) we can computeR 0I explicitly and find thatR 0I = 0.  (g, a), the uniquely determined metricg has the following form . This metric is exactly the Weyl-ambient metric introduced in (17).

Weyl-Obstruction Tensors: Second Order Formalism
In Section 3 we have seen that Weyl-obstruction tensors can be defined as the derivatives of the ambient Riemann tensor in the first order formalism. In this section we will follow the setup of the present section in the second order formalism and show that appropriate ambient tensors constructed from the Weyl-ambient Riemann tensor onM behave as Weyl-covariant tensors on M , through which Weyl-obstruction tensors can again be defined as a special case. Then we will show that the Weyl-obstruction tensors defined in this way agree with the Weyl-obstruction tensors we defined previously in Definition 1.
We have proven in the last subsection that for any pair of (g, a) on M , there exists a unique Weyl-ambient space (M ,g) for the Weyl manifold (M, [g, a]) whereg has the form of (17). In Section 3 we saw that the ambient Weyl diffeomorphism and Υ(x) ≡ − ln B(x), Υ j ≡ ∂ j Υ(x).R IJKL;M 1 ...Mr denotes covariant derivatives of the Riemann tensor ofg in the coordinates X I = (t , x i , ρ ) given by the trivialization provided by g .
Proof. The logic for the proof of this Proposition follows the proof of Proposition 6.5 in [14] closely. We start by observing that the ambient Weyl diffeomorphism ψ : (t , x i , ρ ) → (t, x i , ρ) has the following properties: where the Weyl-ambient metricg has the form of (17), and g ij = B(x) −2 g ij , a i = a i + Υ i . The where Υ(x) ≡ −lnB(x) and Υ i ≡ ∂ i Υ(x). At ρ = 0 the Jacobian matrix (98) reads The above matrix can be written as the following matrix product: with Since the Weyl-ambient metric is homogeneous of degree 2 under dilatations δ * sg = s 2g , it follows that the left-hand side of (95) satisfies Under the ambient Weyl diffeomorphism (94) the covariant derivatives of the ambient Riemann curvature components transform tensorially as Evaluating both sides of (103) at ρ = 0, t = e −Υ(x) and using (101) we havẽ Plugging (102) into (104), we obtain (95).
Theorem 4.6 helps us to find Weyl-covariant tensors on (M, [g, a]). First let us look at the case without derivatives. In the coordinate basis, the nonvanishing components of the Weyl-ambient Riemann tensorR IJKL areR ∞jk∞ ,R ∞jkl andR ijkl . Evaluating at ρ = 0 and t = 1, they arẽ HereĈ jkl andŴ ijkl are the Weyl-Cotton tensor and the Weyl curvature tensor on M , respectively, andΩ jk for now simply denotes the tensor defined in (45). Then, applying (95) we getĈ jkl =Ĉ jkl andŴ ijkl = B 2 (x)Ŵ ijkl under Weyl transformation as expected, we can also read off from (95) that the Weyl weight ofΩ (1) jk is +2. Now we will define Weyl-obstruction tensors as the derivatives ofR ∞jk∞ . Definition 6. Suppose k is a positive integer. The k th extended Weyl-obstruction tensorΩ For k = 1 we can see from (105) thatR ∞jk∞ | ρ=0,t=1 =Ω (1) jk is indeed the first extended Weylobstruction tensor.
From the symmetry of the Weyl-ambient Riemann tensor we can immediately see thatΩ (k) ij given by Definition 6 is symmetric. From the Ricci-flatness conditionRic(g) = 0 and the fact that R 0IJK = 0, we can see thatΩ (k) ij is traceless. Now we will show another important property of the extended Weyl-obstruction tensors defined in this way, namely that they are Weyl covariant.  ), then s ∞ = k + 1 and under a Weyl transformation we havẽ It follows from Lemma 4.7 that Therefore, we obtain from (108) thatΩ ij is a Weyl-covariant tensor with Weyl weight 2k.
Finally, we would like to show that Definition 6 and Definition 1 are equivalent; that is, the Weyl-obstruction tensors defined by the derivatives of the ambient Riemann tensor in the frame {e + , e i , e − } and the coordinate basis {dt, dx i , dρ} are equivalent. To start, let us look at the transformation between {e + , e i , e − } and the coordinate basis {dt, dx i , dρ}: Denote the transformation matrix as Λ, i.e. e J = Λ J I dx I (J = {+, i, −}, I = {0, i, ∞}), then the inverse matrix reads Comparing (54) and (105), we can see that the components R ijkl , R −ijk and R −ij− in the null frame match the corresponding components R ijkl , R ∞ijk and R ∞ij∞ in the coordinate basis when ρ = 0 and t = 1. Now let us show that any Weyl-obstruction tensor defined in (62) is equivalent to that in (106). First, notice that although the componentsR −+M N ofR IJKL in the frame {+, i, −} vanish, the components∇ PR−+M N are not necessarily zero. (Using the notation in Section 3, here we denote∇ D P as∇ P for P = +, i, −.) The following lemma will be used in the proof of Proposition 4.10.
where Λ M ∞ = tδ M − [see (110)] is used in the second equality. Using the symmetries of the Riemann tensor, we have where Λ i j = δ i j is used in the first equality and Lemma 4.9 is used in the second equality. Plugging (113) into (112) completes the proof.
From Proposition 4.10 we can directly see that theΩ (k) ij defined in (62) is equivalent to (106). Therefore, the descriptions of the Weyl-obstruction tensors in the first order and second order formalisms are equivalent. Each of these two formalisms have their own advantages. The first order formalism is suited for the top-down approach as the metricg has a simple form in the dual frame {e I }. It is also more convenient to construct Weyl-covariant tensors in the first order formalism since (56) gives a covariant transformation while (95) has the matrix p with an off-diagonal element. On the other hand, the second order formalism is designed for the bottom-up approach, as one can evaluate the initial value problem more naturally in the coordinate basis.

Conclusions
In this paper we have generalized the ambient construction for conformal manifolds to that for Weyl manifolds. Inspired by the WFG gauge for AlAdS [25], we introduced the Weyl-ambient metricg in (17). From a top-down perspective we showed how the Weyl-ambient space (M ,g) induces a Weyl geometry on a codimension-2 manifold M . The metricg and the LC connection onM give rise to a Weyl class [γ (0) , a (0) ] on M , in which a representative includes an induced metric γ From a bottom-up perspective, we formulated the (d + 2)-dimensional Weyl-ambient space from a d-dimensional Weyl manifold (M, [g, a]). We first introduced a Weyl structure P W on M together with a Weyl connection. We then generalized the definition of ambient spaces to Weyl-ambient spaces, and proved that any Weyl-ambient space can be put in Weyl-normal form by a diffeomorphism. Besides assigning the Weyl connection a i on P W , the ρ-coordinate lines of a Weyl-ambient space in Weyl-normal form are not required to be geodesics but can acquire an acceleration A. By taking the Weyl structure as an initial surface, we have shown that there exists a unique Weylambient space in Weyl-normal form for any given Weyl manifold provided the data (g ij , a i , A) is given. The metric generated order by order from the initial value problem is exactly theg we introduced in (17) from the top-down approach, where g ij corresponds to γ (0) ij , and (a i , A) corresponds to a i (x, ρ).
Based on the Weyl-ambient construction, we investigated Weyl-covariant quantities induced by the ambient tensors in both first and second order formalisms. As an important example, the extended Weyl-obstruction tensorΩ (k) ij is defined through covariant derivatives of the ambient Riemann tensor, and its definition in the first and second order formalisms are shown to be equivalent. We also proved thatΩ ij at d = 2k in the ambient metric expansion, which justifies the description of Weyl-obstruction tensors in [26]. Compared with the extended obstruction tensor Ω (k−1) ij , whose residue is only conformally covariant in d = 2k, the extended Weyl-obstruction tensorΩ (k−1) ij is Weyl covariant in any dimension. There are many possible extensions of this work. It would be interesting to reframe the Weylambient geometry in the framework of Atiyah Lie algebroids, as set up in [35]. Using the machinery of Lie algebroids, one can naturally formulate the BRST cohomology and quantum anomalies in a geometrical way [36]. Since obstruction tensors, which are defined canonically through the ambient construction, also appear in the type B Weyl anomaly, we hope that the Weyl-ambient space can provide a geometric interpretation for the Weyl anomaly in the Lie algebroid language. We will explore this in future work.
The Weyl-ambient space induces the Diff(M ) Weyl symmetry on the codimension-2 manifold M , which can be regarded as an asymptotic corner symmetry [37,38]. The algebra of corner symmetries and their Noether charges have been studied in [37,39] (see also [40]), it is possible to apply the results therein to the Weyl-ambient space and study the asymptotic corner symmetries of the Weyl-ambient space. Moreover, since the surface N at ρ = 0 of the Weyl-ambient space is null, there is an induced Carroll structure [34,41]. This is evident from the fact that the ambient Weyl diffeomorphism acts on the null surface as (a special case of) a Carrollian diffeomorphism. It would be interesting to see if the results of [42] for a Ricci-flat space with a Killing vector can be emulated for the Weyl-ambient space, which is Ricci-flat but possesses a conformal Killing vector T = t∂ t .
One also expects intriguing holographic applications of the Weyl-ambient construction, for ex-ample in the context of celestial holography [43][44][45] and codimension-2 holography [46,47]. In particular, the Diff(M ) Weyl symmetry on M corresponds to the Weyl-BMS symmetry onM [48] (with supertranslations turned off). Therefore, we expect that the Weyl-ambient construction will provide a new arena for realizing the holographic principle. The Weyl-ambient metric construction is part of a bigger program of introducing the Weyl connection back into physics. Viewed as an ordinary gauge symmetry, the Weyl symmetry can provide an organizing principle for constructing effective field theories (e.g., for conformal hydrodynamics). Weyl manifolds would be the proper geometric setup for such future investigations. More recently, the ambient construction was used to study correlators of CFTs on general curved backgrounds [49]. We hope the Weyl-ambient geometries can be utilized in similar contexts.

A Coordinate Systems of the Flat Ambient Space
In this appendix we demonstrate the transformation between the flat ambient metric in different coordinate systems introduced in Section 2.
Start with Minkowski spacetime R 1,d+1 in Lorentzian coordinates {X 0 , X i } with i = 1, . . . , d+1: First, we can define a stereographic coordinate system { , r, x i } as follows: 2 and L is a positive constant. In this system, the Minkowski metric (A.1) becomes where in the second equality we expressed {x i } in the spherical coordinates. The coordinate patch is > 0, 0 r < L, which covers the interior of the future light cone. Notice that in these coordinates the metric has a "cone" form (6), with g + given in (7), which is the (d + 1)-dimensional Euclidean AdS metric g + G in global coordinates. This AdS metric can be converted into the FG from by transforming the coordinate r to a coordinate z Then, the metric (A.3) takes the form .5) and the interior of the future light cone is now covered by > 0, 0 < z < 2L. We can further convert (A.5) into the ambient form (15) by setting = zt , z 2 = −2ρ , (A. 6) and the metric turns into the form shown in (9): Plugging (A.6) and (A.4) into (A.2) we find that From the above equation one can see that the constant-t and constant-ρ surfaces are indeed the cones depicted in Figure 1, with α the angle of the constant-ρ cone with respect to the X 0 -axis. The Minkowski metric (A.1) can also be written in the cone form with g + = g + P the Euclidean AdS metric in Poincaré coordinates given in (8). Introduce another coordinate system { , x i , z} as follows: The metric (A.1) becomes Define the ambient coordinate system {t, x i , ρ} as = zt , then the metric (A.10) will have the form shown in (10)

B Details of Null Frame Calculations
In Section 3 we introduced the following frame: (B.14) The metric (17) can be written in this frame as g = e + ⊗ e − + e − ⊗ e + + t 2 γ ij e i ⊗ e j , and the metric components read The commutation relations of the frame are as follows: where ϕ = ∂ ρ a i , and f ij = D i a j − D j a i . From the above commutators we can read off the commutation coefficients: Then, we can compute the connection coefficientsΓ P M N of the ambient LC connection: The nonvanishing components arẽ which constitute the connection 1-formω M N presented in (32). Then, using Cartan's second structure equationR we can find the ambient curvature 2-form, the nonvanishing components arẽ where∇ is introduced in (35), ψ ij ≡ 1 2 (∂ ρ γ ij + f ij ), and The components in (B.20) constitute the curvature 2-formR M N presented in (48). Now one can derive the extended Weyl-obstruction tensors according to Definition 1. For example,Ω (1) ij andΩ (2) ij can be computed as follows: Plugging the on-shell solution (41)- (43) in to the above expressions, one obtains the extended Weyl-obstruction tensorsΩ (1) ij andΩ (2) ij given in (45) and (46). From the components of the ambient Riemann curvature, we can also find the Ricci components in this frame: where B ij is defined in (49). The Ricci-flatness condition gives the following three equations: In the leading order when ρ = 0, the condition (B.22) leads to the fact thatΩ (1) ij is traceless, and (B.24) gives the Bianchi identity∇ (0) iP i j =∇ (0)P , whereP is the trace ofP ij . DifferentiatingR ij with respect to ρ yields which leads to (67) when differentiating (B.24).

C Proofs
C.1 Proof of Theorem 4.1 To prove Theorem 4.1, we first need to introduce a (g, a)-transversal vector (generalized from the concept of a g-transversal vector in [14]), where the horizontal subspace H p defined by the Weyl connection plays an important role. Once we pick a representative (g, a) in the Weyl class, g induces an isomorphism between P W and G through (79), which determines the fibre coordinate t of P W ; a defines for any p ∈ P W a horizontal subspace H p ⊂ T p P W given in (75), which can also be viewed as a subspace of T (p,0) (P W × R) via the inclusion map ι : P W → P W × R. We define a vector V ∈ T (p,0) (P W × R) to be a (g, a)-transversal vector forg if it satisfies the following three conditions at (p, 0): When a i (x) = 0 in (77), i.e., a = ∂ t ⊗ dt, the (g, a)-transversal vector forg goes back to the gtransversal vector forg defined in [14]. From (84) one can see that for (M ,g) in Weyl-normal form, ∂ ρ is (g, a)-transversal forg at (p, 0). Following the proof of Lemma 2.10 in [14], it is straightforward to show that the (g, a)-transversal vector is unique and dilatation-invariant (i.e. δ s * V p = V δs(p) ) for g at (p, 0). The proof of Theorem 4.1 proceeds similar to the proof of Proposition 2.8 in [14]; one only has to let the g-transversal vector V to be a (g, a)-transversal vector. Here we will not repeat all the details but only outline the proof and elaborate on the steps when the Weyl connection a is relevant.
Proof of Theorem 4.1. Suppose p ∈ P W and let V p be the (g, a)-transversal vector forg at (p, 0). One can parametrize the (non-geodesic) curve C p : λ → φ(p, λ) ∈M with initial conditions with the "equation of motion" ∇ U U = A, where U = d dλ is the tangent vector to the accelerated curve C p , and the acceleration vector A satisfiesg(T , A) = 0. Suppose the domain of φ isŨ 0 ⊂ P W × R, which is dilatation-invariant. Then φ :Ũ 0 →M is a smooth map commuting with dilatation, and it can be proved that there existsŨ 1 ⊂Ũ 0 as a dilatation-invariant neighborhood of P W × {0} such that φ :Ũ 1 →M is a diffeomorphism (see [14]).
Finally let us verify condition (5.3) of Definition 5. Since V satisfies the conditions in (C.26) and φ satisfies (C.27), under the identification R + × M × R P W × R induced by g we have at (λ = 0, p): For a given connection a = t∂ t ⊗ t −1 dt + a i (x)dx i on P W , the horizontal subspace H p at (p, 0) is spanned by D i = ∂ i − ta i ∂ t . Since (M , φ * g ) is a Weyl pre-ambient space for (M, [g, a]), ι * (φ * g) is the tautological tensor g 0 on P W . Then, the above equations give that φ * g | λ=0 = t 2 g 0 + 2t(dt + ta i (x)dx i )dλ. Therefore, all the conditions in Definition 5 are satisfied by (M , φ * g), which completes the existence part of the Proposition. The uniqueness part follows from the fact that the above construction of φ is forced. Suppose φ : M → M is a diffeomorphism such that (M , φ * g) is a pre-ambient space in Weyl-normal form, then V p must be (g, a)-transversal forg at (p, 0), and the whereP ij is the Weyl-Schouten tensor. One can observe that this agrees with (17), where g ij (x) corresponds to γ (0) ij in the expansion (20), and the order O(ρ) matches γ (1) ij given in (41). Note that the above components of a Weyl-ambient metric reduce to the components of an ambient metric in [14] when the Weyl connection a i is turned off.
The next stage of the proof is to carry out an inductive perturbation calculation for higher orders in ρ. The purpose of this calculation is to prove (inductively) that the Ricci-flatness condition can be used to determine the unknown components ofg in Weyl-normal form to infinite order in ρ.
Letg [k] represent a metric that includes the terms of the ρ-expansion ofg up to (including) order O(ρ k ), i.e.,g =g [k] + O(ρ k+1 ). Then, the Ricci-flatness condition ofg implies that the components R To carry out the induction, we assume thatg [m −1] has been uniquely determined from the condition (C.35) with k = m − 1. We have seen this is true for m = 2 above by explicit calculation. Now we want to show thatg [m] then can be uniquely determined from the condition (C.35) with k = m. Setg where a (m) i (x) is the m th order term of a i (x, ρ) [see (21)], and we have considered the fact thatg KL , respectively, and Γ Φ IJK ≡ 1 2 (∂ J Φ IK + ∂ I Φ JK − ∂ K Φ IJ ). The components of Γ Φ IJK can be expressed as follows:  i (x) appears throughout the calculation, it cancels itself out rather unexpectedly, except for the terms inR (m) ∞∞ . The inductive argument then proceeds in the same way as [14]. First we consider the Ricci components with I, J = ∞. From the first two equations in (C.39) one can uniquely determine φ 00 and φ 0i such thatR ij vanishes. Therefore,g [m] will be uniquely determined byR Note that when d = 2m, the situation becomes subtle because the term φ ij vanishes inR [m] . In [14], this is attributed to the obstruction of the Ricci-flatness condition at O(ρ d/2−1 ) when d is an even integer, and one has to carefully consider even and odd d separately. Nevertheless, since we consider the dimension d as a continuous parameter, we can always solve for φ ij from the Ricciflatness condition for any d, and the information regarding these obstructions is not lost but takes the form of poles in φ ij at d = 2m. As is shown in Proposition 3.2, since φ ij represents the order O(ρ m ) of g ij (x, ρ) in the Weyl-ambient metric (17), this pole represents exactly the Weyl-obstruction tensor.
So far we have proved that the unknown components ofg are determined to infinite order by the Ricci-flatness condition for I, J = ∞. To finish the analysis we also need to show that the remaining Ricci componentsR I∞ also vanish to infinite order when we plug in the solution forg We can see that the Weyl connection appears only in the last equation of (C. 41). Note that all the Ricci termsR IJ with I, J = ∞ has been dropped from (C.41) since they vanish to infinite order. SupposeR I∞ = γ I ρ m−2 . The first equation Proof of Lemma 4.9. We will prove this identity by induction. First, noticing thatR −+M N = 0, when n = 0 we havẽ where we used the fact thatΓ i M + = −ρΓ i M − andR +jM N = −ρR −jM N , which can be seen from (32) and (48), respectively. Thus, for n = 0 we have ∇ PR−+M N = − 1 t δ i PR−iM N . Assuming that this lemma holds for all n k − 1, now we show that it will hold for n = k > 0: Therefore,∇ P∇− · · ·∇ − nR −+M N = − 1 t δ i P∇− · · ·∇ − nR −iM N holds for n = k if it is valid for all n k − 1, which completes the proof.