Riemannian Anosov extension and applications

Let $\Sigma$ be a Riemannian manifold with strictly convex spherical boundary. Assuming absence of conjugate points and that the trapped set is hyperbolic, we show that $\Sigma$ can be isometrically embedded into a closed Riemannian manifold with Anosov geodesic flow. We use this embedding to provide a direct link between the classical Livshits theorem for Anosov flows and the Livshits theorem for the X-ray transform which appears in the boundary rigidity program. Also, we give an application for lens rigidity in a conformal class.


Introduction
A closed Riemannian manifold (M, g) is called Anosov if the corresponding geodesic flow on the unit tangent bundle T 1 M is an Anosov flow.For example, all closed manifolds with strictly negative curvature are Anosov.Special examples of manifolds which are not negatively curved, but carry Anosov geodesic flows are known.The first one is probably due to Eberlein [Ebe73] who performed a careful local deformation of a hyperbolic manifold to create a small disk of zero curvature.Due to the C 1 stability of the Anosov property, Eberlein's example can be perturbed further to create some positive curvature while keeping the Anosov property.Further examples were constructed by Gulliver [Gul75], using radially symmetric caps of positive curvature, and by Donnay-Pugh [DP03] who constructed Anosov surfaces embedded in R 3 .It is shown in a recent paper [DSW21] that for a geodesic billiard system whose trapped set is hyperbolic and non-grazing, it is possible to produce a smooth model of Axiom A flow for the discontinuous flow defined by the non-grazing billiard trajectories.
Our main result shows that one can embed certain Riemannian manifolds (Σ, g) with boundary and hyperbolic trapped sets isometrically into an Anosov manifold (Recall that the trapped set is the set of geodesics that are defined for all time, and a boundary is called strictly convex if its second fundamental form is positive definite everywhere).
Theorem A (Theorem 8.1).-Let (Σ, g) be a compact smooth Riemannian manifold with boundary.Assume that each component of the boundary is a strictly convex set diffeomorphic to a sphere.Also, assume that (Σ, g) has no conjugate points and the trapped set for the geodesic flow is hyperbolic.Then, there exists a codimension 0 isometric embedding (Σ, g) ⊂ (Σ ext , g ext ) such that (Σ ext , g ext ) is a closed Anosov manifold.
Remark.-We do not require Σ to be connected.If we do not insist on the embedding being codimensional 0 then it is not hard to apply Nash's embedding theorem to isometrically embed (Σ, g) into a high dimensional Euclidean space and then into a horosphere in a manifold of constant negative curvature (we owe this remark to Keith Burns).
To the best of our knowledge, the above theorem is the first general result on existence of Anosov extensions.We note that all assumptions except for convexity and diffeomorphism type of the boundary are necessary assumptions to admit an Anosov extension.One fact which immediately follows from Theorem A is that for any point in any Riemannian manifold, one can isometrically embed any sufficiently small neighborhood of the given point into a closed Anosov manifold.
Theorem A allows one to transfer some results from the setting of closed Riemannian manifolds to the setting of compact Riemannian manifolds with boundary.We proceed with a description of such applications.
Denote by ∂ − (respectively, ∂ + ) the unit inward (respectively, outward) vectors based on ∂Σ (precise definition are given in Section 2.3).The lens data consists of two parts: the length map ℓ g : ∂ − → [0, ∞] measuring the time at which γ v hits ∂Σ again for all v ∈ ∂ − , and the scattering map s g : ∂ − ∖Γ − → ∂ + associating v ∈ ∂ − ∖Γ − with its exiting vector s g (v).Here Γ − := ℓ −1 g (∞).We say that two metrics g and g ′ on Σ are lens equivalent if ℓ g = l g ′ and s g = s g ′ .For any metric g on Σ, denote by g U the lifted metric on the universal cover Σ.Two metrics g and g ′ on Σ are called marked lens equivalent if the lens data of g U and g ′ U coincide.The lens rigidity (resp.marked lens rigidity) problem asks whether lens equivalent (resp.marked lens equivalent) metrics are isometric via a diffeomorphism fixing ∂Σ.
Together with an argument of Katok [Kat88], we confirm the following extension of Mukhometov-Romanov result [MR78] in the case when hyperbolic trapped sets are allowed.
Corollary B (Marked lens rigidity in a conformal class).-Let ρ : Σ → R + be a smooth function such that the metrics (Σ, g) and (Σ, ρ 2 g) both satisfy the assumptions in Theorem A. Assume that g and ρ 2 g are marked lens equivalent.Then, ρ = 1.
Remark.-Corollary B is related to the boundary rigidity problem, which asks whether one can reconstruct the Riemannian metric g in the interior from knowing the distance d g : ∂Σ × ∂Σ → R between points on the boundary.Michel [Mic81] conjectured that all simple manifolds are boundary rigid, and the surface case was proved by Pestov-Uhlmann [PU05].Partial results in higher dimensions can be found in [SU09], [Var09], [BI10], [BI13], [SUV21], etc.When trapped sets are allowed, the marked lens rigidity is equivalent to marked boundary rigidity, and certain local rigidity results were recently established in [Gui17], [GM18], [Lef19], and [Lef20] in the case when trapped sets are hyperbolic.
Another application is a smooth Livshits theorem for domains with sharp control of regularity of the solution.
Corollary C (Livshits Theorem for domains).-Let (Σ, g) be as in Theorem A and let a C r -smooth (r > 0) function β : T 1 Σ → R be such that its C r -jet vanishes on the boundary ∂T 1 Σ.Assume that for all v ∈ ∂ − \Γ − , Then, there exists u ∈ C r− (T 1 Σ) such that Xu = β and u| ∂(T 1 Σ) = 0, where X is the geodesic spray.
Here r − = r if r is not an integer.If r is an integer then r − = r−1+Lip.Corollary C was also proved in [Gui17,Prop. 5.5], and the proof there applies to u ∈ H s (T 1 Σ) with s > 0. Our proof is more geometric and covers the Hölder regularity.
Remark.-The reason why Livshits theorem is restricted to functions which are flat on the boundary is that, otherwise, the standard bootstrap argument for solution of the cohomological equation [dlLMM86] does not work.However, notice that our condition is not a restriction for the potential application to the deformation lens rigidity (as in [Gui17]) due to a result of Lassas-Sharafutdinov-Uhlmann who recover the jet of the metric from local lens data [LSU03].
Remark.-All our results have low regularity versions in the case when (Σ, g) has finite regularity which exceeds C 3+α for some positive α > 0.
Remark.-The basic example to which Theorem A applies is, of course, when Σ is a strictly convex ball equipped with a simple metric g.In this case the trapped set must be empty since (Σ, g) is assumed to have no conjugate points.When dim Σ = 2 it is easy to make examples which have arbitrary genus, if the genus > 0 then the trapped set is non-empty.It was pointed out to us by one of the referees that examples which satisfy all assumptions of Theorem A and have non-empty trapped set might not exist in dimensions ⩾ 3.While we do not know how to prove that this is the case, we agree that the existence of such example, indeed, seems to be unlikely.We would like to point out that interesting higher dimensional examples with non-empty trapped set exist.While, formally speaking, these examples are not covered by Theorem A, existence of Anosov extension for such examples still holds with some adjustments to the proof of Theorem A.
Let γ be a closed geodesic in a negatively curved manifold M which does not have self-intersections.Then a small neighborhood Σ of the "core" γ in M satisfies all the assumptions of Theorem A except that ∂Σ ≃ S 1 × S n−2 .Note that γ constitutes a non-trivial hyperbolic trapped set for Σ. (Alternatively one can obtain such example by explicitly specifying a negatively curved metric on Σ = S 1 × D n .)We note that Σ already satisfies the conclusion of Theorem A since it is isometrically embedded in M .However, one can deform the metric, for example by creating islands of positive curvature away from ∂Σ and γ, such that existence of Anosov embedding becomes in no way obvious.For this class of examples the proof remains exactly the same up to Section 8, where we take advantage of spherical boundary to glue in out extended domains into a hyperbolic manifold with a large injectivity radius.This argument, with some work, can be adjusted to accommodate the above example.Specifically, the large hyperbolic manifold has to be replaces with a hyperbolic manifold which contains a "large geometric tube" with core γ.Existence of hyperbolic manifolds which contain such "large geometric tubes" was established by Farrell and Jones [FJ93, Cor.3.3] who construct them via a carefully chosen finite cover.
1.1.Outline of the proof of Theorem A. -We construct the extension by hand.Firstly, for each boundary component of the given manifold, we find a metric on a collar that smoothly connects the metric on this boundary to a constant curvature metric.Afterward, we throw away from a compact manifold of a constant sectional curvature (which has the same dimension as the given manifold) finitely many balls (as many as the number of boundary components in the original manifold) that are J.É.P. -M., 2023, tome 10 sufficiently far away (see Lemma 8.3 and the paragraph before it).Finally, in the resulting manifold with boundary and constant negative curvature, we glue in the given manifold with attached collars.The metric on the collar is constructed in several steps.First, we extend the given metric on the neighborhood of the boundary to the negatively curved metric (Section 5).Then, we connect the resulting metric to a rotationally invariant metric in the cylindrical coordinates (Section 6).Finally, we extend the result of the previous extension to a metric of constant curvature (Appendix C).The original metric in the collar is C 1,1 but we smooth it afterward (Section 7).
To guarantee that the constructed compact Riemannian manifold has Anosov geodesic flow we use the criterion by Eberlein (see Theorem 2.3).In particular, we first show that the constructed metric does not have conjugate points.Then, we prove that all nonzero perpendicular Jacobi fields are unbounded.Instead of working directly with the Jacobi field, we estimate the growth rates µ of the logarithm of the square of the norm of nonzero perpendicular Jacobi fields (see (2.4)) using the comparison Lemma 2.8.In particular, the absence of conjugate points means that there is no time interval so that µ tends to infinity as we approach each end of the interval (Proposition 8.10).By Lemma 2.8 and Remark 2.9, we will need to control what are the values of µ as the geodesic enters various regions (the given manifold with boundary and various extension pieces that we construct to obtain the compact manifold with Anosov geodesic flow) so that we have a control from below while it is in the specific region (see Figure 2).To show that all nonzero perpendicular Jacobi fields are unbounded, it is enough to show that the integral of µ over a time ray is unbounded.

1.2.
Organization.-This paper is organized as follows.In Section 2 we set up notation and collect a number of preliminaries from geometry and dynamics.In Section 3 we prove Corollaries B and C using Theorem A. The estimates for Jacobi field within a slightly larger domain containing Σ are carried out in Section 4. The estimates on curvature for certain extension are presented in Sections 5-7.In Section 8 we construct an explicit extension of the metric and prove Theorem A.

Preliminaries
2.1.Geometry of the tangent bundle.-In this section, we formulate some general facts about the tangent bundle.One can find more details in [Ebe73] and [EO80].
Let (M, g) be a C 2+α , α > 0, n-dimensional compact Riemannian manifold with or without a boundary.Denote by T 1 M the unit tangent bundle of M .For any v ∈ T 1 M , let γ v be the unit speed geodesic in (M, g) such that γ ′ v (0) = v.The geodesic flow where R is the Riemann curvature tensor and ′ corresponds to the covariant differentiation along γ v .A Jacobi field is uniquely determined by the values J(0) and J ′ (0).Denote by π : T M → M the canonical projection.For any ξ ∈ T T M , let c(t), for t ∈ (−ε, ε), be a curve on T M with c ′ (0) = ξ.Define the connection map K : The kernel of dπ : T T M → T M , denoted by H, is called the horizontal subbundle, while the kernel V of the connection map K is called the vertical subbundle.The Sasaki metric on T T M is defined via ⟨ξ, η⟩ := g πv (dπξ, dπη) + g πv (Kξ, Kη) for ξ, η ∈ T v T M .We denote by |ξ| := ⟨ξ, ξ⟩ the Sasaki norm of ξ ∈ T T M .
Fact 2.1.-Now vectors in the tangent space T v T 1 M can be identified with Jacobi fields along γ v in the following way: for any ξ ∈ T v (T 1 M ), we define J ξ to be the unique Jacobi field along γ v with J ξ (0) = dπξ and J ′ ξ (0) = Kξ.
The above identification is invariant under the geodesic flow, namely, In particular, if we fix ξ ∈ T v T 1 M then g πv (J ξ (t), γ ′ v (t)) is independent of t.Thus, for any ξ ∈ T v T 1 M , J ξ is perpendicular to γ v if and only if ⟨ξ, X⟩ = 0 where X is the vector field on T M generating the geodesic flow φ t on (M, g).We denote the space of Jacobi fields perpendicular to a geodesic γ by J(γ).
Note that the Sasaki norm of (dφ t )ξ is given by 2.2.Hyperbolicity.-Let φ t : M → M be a smooth flow on a Riemannian manifold and let X be its generating vector field.Recall that an invariant set Λ is λ-hyperbolic (where λ > 0) if there exist C > 0 and a continuous flow-invariant splitting such that for all y ∈ Λ, where ∥•∥ is the norm on T y M induced by the Riemannian metric.Distributions E s and E u are called stable and unstable subbundles on Λ.
If Λ = M then φ t is called an Anosov flow.For Anosov flows the classical Livshits Theorem is stated as follows.Recall that r − = r if r is not an integer and r − = r − 1 + Lip when r is an integer.We will use the following criterion, due to Eberlein, for establishing the Anosov property of geodesic flows.Another proof of this criterion was given Ruggiero [Rug07] following an idea of Mañé.
Theorem 2.3 ( [Ebe73], see also [Rug07]).-Let φ t be a geodesic flow on a closed Riemannian manifold without conjugate points.Then φ t is Anosov if and only if all nonzero perpendicular Jacobi fields are unbounded.
When Λ ̸ = M , the following result lets us extend the hyperbolic structure to a neighborhood of Λ.

Lemma 2.4 ([HPPS70]
). -Let Λ be a λ-hyperbolic set.Then for any ε ∈ (0, λ), there exists an open neighborhood V ε of Λ and extensions E s and E u of the stable and unstable subbundles to V ε with the following properties: (1) Local invariance: if an orbit segment [y, φ t (y)] ⊂ V ε then, dφ t (y)E s (y) = E s (φ t (y)) and dφ t (y)E u (y) = E u (φ t (y)) (2) Hyperbolicity: if an orbit segment [y, φ t (y)] ⊂ V ε then ∥dφ t (y)w∥ ⩽ Ce −(λ−ε)t ∥w∥, ∀w ∈ E s (y) Remark 2.5.-The reference [HPPS70] does not contain an explicit statement about the hyperbolic rate being close to λ (item (2) in Lemma 2.4).However, this rate, indeed can be chosen as close to λ as desired by choosing a sufficiently small neighborhood of Λ.This follows from the fact that the expansion and contraction rates depend continuously on the point.In the case when M is 3-dimensional such extensions of bundles E s and E u can be chosen so that they integrate to locally invariant continuous foliations.In a higher dimension this seems to be unknown.However, for our purposes we will merely need locally invariant bundles which do not necessarily integrate to foliations.

2.3.
The hyperbolic trapped set.-Let (Σ, g) be a smooth n-dimensional compact Riemannian manifold with boundary.Denote by π : T 1 Σ → Σ the canonical projection and T 1 Σ • the interior of T 1 Σ.Let ∂ − and ∂ + be the incoming and outgoing, respectively, subsets of the boundary of T 1 Σ defined by where ν is the unit normal vector field to ∂Σ pointing outwards.For any v ∈ ∂ − , the geodesic γ v starting at v either has an infinite length or exits Σ at a boundary point with tangent vector in ∂ + .We denote by ℓ g (v) ∈ [0, ∞] the length of γ v in Σ.Let For each v ∈ ∂ − ∖ Γ − denote the exit point by s g (v) ∈ ∂ + .Similarly we define the set Γ + ⊂ ∂ + which is trapped in Σ in backwards time.Then the trapped set in the interior of Σ is defined via If Σ has strictly convex boundary, Λ is the set of v ∈ (T 1 Σ) • such that the φ t -orbit of v does not intersect the boundary.Throughout this paper we will always assume that the trapped set is hyperbolic.The stable and unstable bundles of Λ are denoted E s and E u , respectively.It is clear from the discussion in Section 2.1 and (2.3) that both E s and E u are perpendicular to the generating vector field X. [Ebe73] for more details.Now we apply Lemma 2.4 to the trapped set with ε = λ 2 .If v / ∈ Λ then the invariant subbundles along the orbit through v only exist for a finite time and, hence, they do not have to be perpendicular to X. Nevertheless, we can still obtain perpendicular invariant bundles by taking the orthogonal component (which only results in a slightly different constant C in Lemma 2.4).
More specifically, we define the following linear subspaces of the space of Jacobi fields along a geodesic γ v : where J ξ = J ⊥ ξ + J ∥ ξ with J ⊥ ξ being a perpendicular Jacobi vector field and J ∥ ξ being a tangential Jacobi vector field, i.e., J }. Now we have the following variant of Lemma 2.4 near the hyperbolic trapped set Λ of the geodesic flow.
Lemma 2.7.-There exists a neighborhood U of Λ such that U ⊂ V λ/2 and for σ ∈ {s, u}, J.É.P. -M., 2023, tome 10 ; for each w ∈ v ⊥ , there exists a unique vector ξ σ w ∈ E σ ⊥ (v) such that dπξ σ w = w, and the map Proof.-By Lemma 2.4, E σ are continuous and invariant under the flow {φ t } in V λ/2 , so we obtain the first three items of the lemma because the splitting into perpendicular and tangential Jacobi vector fields is invariant under the flow.Since E σ ⊥ (v) = E σ (v) for all v ∈ Λ, there exists a neighborhood U of Λ such that for any v ∈ U for any ξ ∈ E u (v) ∪ E s (v) ∖ {0} we have |⟨ξ, X⟩|/∥ξ∥ ⩽ 1/10.Thus, using Lemma 2. 4 (2), we obtain (4) with C ′ = 2C.By choosing U sufficiently small and using [Ebe73, Prop.2.6], we obtain (5).Finally, (6) follows from Remark 2.6 and (1).□ 2.4.Comparison lemmas of Jacobi fields.-Let J be a nonzero Jacobi field along a unit speed geodesic γ.For any t with J(t) ̸ = 0, define Notice that µ J is invariant under scaling of the Jacobi field J.We will use the following comparison lemma from [Gul75] many times in this paper.
Remark 2.9.-Let u be a solution of u ′′ + f u = 0 where f : R → R is integrable on bounded sets.We define the logarithmic derivative of u by w = u ′ /u.In particular, u(t) = u(0) exp t 0 w(s)ds and w satisfies a first order non-linear equation This equation shows that if f ⩽ 0 then the graph of w crosses the graphs of √ −f and − √ −f horizontally, w monotonically increases between them and decreases while above √ −f and below − √ −f .Thus, we get a good control on w from below (know that it does not drop to −∞ in the considered time) only if w(0) is above − √ −f .In particular, by Lemma 2.8, in that case we get a control on µ J .
If f > 0, then w is monotonically decreasing so there is no good control from below.
Proof.-Because M is compact it admits an upper bound on sectional curvature K 2 and we can assume that K > 1.
We argue by contradiction.Assume that there exists τ 0 > 0 such that for any n ∈ N, there exists v n ∈ T M with γ vn [0, τ 0 ] ⊆ M and perpendicular Jacobi field J n along γ vn with µ Jn (0) > n and µ Jn (s n ) < −n for some Thus, in either cases we have Without loss of generality, we assume that ∥J n ∥ ′ (0) = 1 for all n ∈ N. Thus, On the other hand we have µ J (0) = +∞ and µ J (s) = −∞ thus J(s) = 0.This contradicts to the fact that M has no conjugate points.□ 2.5.The second fundamental form and the shape operator.-In this section, we recall the definitions of the second fundamental form and the shape operator and their connection to sectional curvatures.See [Gro94] for more details.
Let S be an (n − 1)-dimensional smooth manifold.Consider the product (a, b) × S with a Riemannian metric where t ∈ (a, b) and g t is a Riemannian metric on S t := {t} × S. In particular, for any θ ∈ S, we have that γ(t) = (t, θ), where t ∈ (a, b), is a geodesic on (a, b) × S.
Define π s : R×S → R×S by π s (t, θ) = (t+s, θ) for θ ∈ S. The second fundamental form on S t is a quadratic form given by: The shape operator A(t, θ) : In particular, A(t, θ) is diagonalizable and its eigenvalues λ where ⟨• , •⟩ is the inner product corresponding to ds 2 and R is the Riemann curvature tensor.In particular, where [• , •] is the Lie bracket of vector fields.Let T = ∂/∂t.For any vector X ∈ T 1 S t , the sectional curvature of σ X,T = span{X, T } is given by where For any 2-plane σ X,Y = span{X, Y } ⊆ T S t where X, Y ∈ T S t , let K int (σ X,Y ) be the intrinsic sectional curvature of g t at σ X,Y .Then, the relation between K int (σ X,Y ) and K(σ X,Y ) is given by Gauss' equation: We have the following estimate on K(σ) where σ is a 2-plane in T S t .
Lemma 2.11.-Assume S t is strictly convex.Then, for any 2-plane σ ⊆ T (t,θ) S t , Proof.-Let { e i } n−1 i=1 be an orthonormal basis of T (t,θ) S t consisting of principal directions.Then, we have Thus, we have By (2.8), we obtain

Proofs of applications
In this section we give proofs of Corollaries B and C.
Proof of Corollary B. -Denote by µ the normalized Riemannian volume on Σ with respect to g.We can assume that Σ ρ 2 dµ ⩽ 1. (Otherwise we can exchange the roles of g and ρ 2 g so that the conformal factor becomes 1/ρ 2 and proceed in exactly same way.)We begin by applying Theorem A and extend (Σ, g) to a closed Anosov manifold (Σ ext , g ext ).We also extend ρ to ρ ext : Σ ext → R by 1. Denote by µ ext the normalized Riemannian volume on (Σ ext , g ext ).
Assume ρ ext is not 1 everywhere on Σ ext .Then, by Cauchy-Schwartz inequality, we have Now following [Kat88, Th. 2], we apply Birkhoff ergodic theorem and Anosov closing lemma to produce a unit speed geodesic γ which approximates volume measure sufficiently well so that Let c be a connected component of γ ∩ Σ. Denote by c ′ the geodesic segment for ρ 2 g with the same entry and exit point as c.The universal cover Σ equipped with the lift of ρ 2 g does not have conjugate points.Hence the segment c ′ is the global minimizer.Thus where the last equality is due to the lens data assumption.By applying this inequality to each connected component of γ ∩ Σ and noting that ρ ext = 1 outside Σ we obtain which gives a contradiction.Hence ρ ext = 1.□ J.É.P. -M., 2023, tome 10 Remark 3.1.-Using a local argument it is not hard to show that ρ| ∂Σ = 1.However, note that in the above proof we do not need to consider an extension of g ′ and, in principle, ρ is allowed to be discontinuous on the boundary of Σ.
Proof of Corollary C. -We begin by applying our main result to extend X to an Anosov vector field, which we continue denote by X on Σ ext ⊃ Σ.Then we extend β by the zero function.Because C r -jet of β vanishes on the boundary this extension remains C r .For any periodic geodesic γ which intersects boundary of Σ, we have from the assumption of the corollary.Further we also have the following Assuming the lemma we can easily finish the proof by applying the Livshits Theorem 2.2 to β and X to obtain a C r− solution u : Σ ext → R to the cohomological equation Xu = β.To see that u| ∂(T 1 Σ) = 0, pick a dense geodesic which intersect ∂(T 1 Σ) in a dense sequence of points {v n } n∈Z .Because the integral of β from v n to v n+1 vanishes, by Newton's formula we have that u(v n ) = u(v n+1 ) = const.for all n.Hence, after subtracting the constant we indeed have u| ∂(T 1 Σ) = 0. □ To finish the proof of Corollary C, we need to establish the lemma.This lemma is established using a standard shadowing argument.
Proof of Lemma 3.2.-Recall that the trapped set Λ ⊂ Σ consists of all geodesics which are entirely contained in the interior of Σ.In particular, γ ⊂ Λ.Without loss of generality, we may assume that Σ is connected since γ lies in one of the connected components of Σ.
We begin by observing that Λ has a local product structure.Indeed, given a pair of sufficiently close points x, y ∈ Λ the "heteroclinic point" [x, y] = W s (x, ε) ∩ W 0u (y, ε) stays close to the orbit of x in the future and close to the orbit of y in the past and, hence, remains in the interior of Σ as well.
The first step of the proof is show that Λ is nowhere dense.Assume that Λ has non-empty interior int(Λ).Let Λ be the closure of int(Λ).It is easy to see that int(Λ) and Λ still have a local product structure.(Hyperbolic set Λ could be a proper subset of Λ, for example, when Λ has an isolated periodic orbit.)Note that Λ has positive volume.The restriction of the Sasaki volume to Λ is an ergodic measure.Therefore, by ergodicity, there exists a point p ∈ int(Λ) whose forward orbit and backward orbits are both dense in int(Λ) and, hence, are also dense in Λ.Because p is in the interior we have W s (p, ε)∪W u (p, ε) ⊂ Λ for a sufficiently small ε > 0.Then, for any x ∈ Λ, we can pick forward iterates of p which converge to x and, hence, because Λ is closed and W u (p, ε) expands, we have W u (x) ⊂ Λ.In the same way, by considering backwards orbit of p we also have W s (x) ⊂ Λ.Finally, from the local product structure, for sufficiently small ε > 0 we have {x} ̸ In particular, P ε (x) contains a neighborhood of x.Thus, we conclude that x in an interior point of Λ.This gives that the closed set Λ is also open which gives a contradiction because Λ is a proper subset of Σ.Now we can use an approximation argument to show that γ βdt = 0. Let p ∈ γ and let q ∈ γ ′ be a point which is δ-close to p on a periodic geodesic γ ′ which intersects the boundary of Σ. Existence of such a point q follows from density of periodic orbits and the fact that Λ is a closed nowhere dense set.
We now form a pseudo-orbit by pasting γ and γ ′ together and using Anosov closing lemma to produce a periodic orbit α which passes close to [p, q] and first shadows γ and then γ ′ ; see Figure 1.Clearly, such α intersects the boundary of Σ as well and, hence, α βdt = 0. Orbit α can be partitioned into 3 segments: one which shadows γ, one which shadows γ ′ and a short remainder segment which appears due to joint nonintegrability of strong foliations.More precisely, we let α = α 1 ∪ α 2 ∪ α 3 , where α 1 has the same length as γ and relates to γ via unstable-stable holonomy.The segment α 1 is followed by α 2 has the same length as γ ′ and relates to γ ′ via unstable-stable holonomy.Note that if we want the starting point of α 2 to be related to q via unstable-stable holonomy (as indicated on the figure) then we might need to reposition q along γ ′ to achieve that.Finally, the remaining segment α 3 has length < 3δ by application of triangle inequality.(For simplicity, we assume that |α| > |γ| + |γ ′ |; if that is not the case then α 1 and α 2 would overlap and α 3 would the the overlap; the same proof works in this case.)Because the end-points of α 3 are δ-close to p and q we also have Also recall that γ ′ βdt = α βdt = 0. Putting these together we have Taking δ → 0 we obtain γ βdt = 0. □

A Jacobi estimate for geodesics which enter a domain with hyperbolic trapped set
Following the outline of the proof (Section 1.1), we want to control the growth rates of the logarithm of the square of the norm of nonzero perpendicular Jacobi fields for the constructed compact Riemannian manifold.Consider a geodesic γ and let τ M,max (γ) be the length of a maximal time interval so that the geodesic is in the given Riemannian manifold M with boundary.In the presence of a trapped set, τ M,max (γ) can be arbitrarily large as a geodesic can be in the trapped set or accumulate for arbitrarily long time on it.Since the trapped set is hyperbolic, we can show that we have a "good" control on the growth rates of the logarithm of the square of the norm of nonzero perpendicular Jacobi fields in a neighborhood of the trapped set.The precise result is the following proposition.
Proposition 4.1.-Let (M, g) be a manifold with boundary.Assume that (M, g) has no conjugate points and a (possibly empty) hyperbolic trapped set Λ.Then, there exists constants Q M > 0 and C M > 0, which depend only on M , such that for any v ∈ ∂ − and a perpendicular Jacobi field J along γ v with µ J (0) > Q M , J does not vanish as long as γ v lies in M .Moreover, the following properties hold: Then, (1) and (2) remain valid with the same Q M and C M if we replace M with M −δ .
In order to prove Proposition 4.1 we need to analyze the behavior of Jacobi fields J near the hyperbolic trapped set Λ.
Proof.-Notice that for any T ⩾ 0 we have U T is an open set and Λ ⊂ U T .Assume that the conclusion of the lemma does not hold.Then, there exists η 0 > 0 such that for any n ∈ N we have O η0 (Λ) ̸ ⊃ U n .In particular, for any n ∈ N there exists and, by the definition of the trapped set, we have Λ = n∈N U n .
By the compactness of T 1 M , we obtain that there exists x ∈ T 1 M such that In particular, there exists j ∈ N such that x ∈ T 1 M − O η0/2 (U i ) for any i ⩾ j.Thus, we obtain the contradiction to the fact that x n → x as n → +∞ because for any i ⩾ j we have x i ∈ U i , so the distance between x and x i is at least η 0 /2.□ 4.2.Invariant Jacobi fields near Λ. -Let U be the open neighborhood as in Lemma 2.7 with constant C ′ .We pick T 0 satisfying U T0 ⊆ U using Lemma 4.2.For each v ∈ U and w ∈ v ⊥ , let ξ σ w be the vectors defined in Lemma 2.7(5) and denote by J σ w := J ξ σ w .We have (4.1) . By Lemma 2.7(6), there exists L > 0, which is independent of v, such that |U σ v | ⩽ L for all v ∈ U. Together with (2.2) and (4.1) we know that whenever φ t v ∈ U we have Here |•| is the Sasaki norm defined in Section 2.1.Notice that the constants C ′ , L, η depend only on U.

Decomposition of Jacobi fields near
, by Lemma 2.7, we can decompose ξ as ) for σ = s, u.This decomposition can be represented in terms of Jacobi fields as follows: J.É.P. -M., 2023, tome 10 where The following proposition shows that the unstable component of ξ cannot be too small when µ J (0) and ℓ g (v) are sufficiently large.
Proposition 4.3.-Assume the sectional curvature of M is bounded from below by −k 2 .Let Q(T 0 , g) be the constant defined in Corollary 2.10.Then there exists D, ζ > 0 depending on Λ and U such that for any v ∈ ∂ − with ℓ g (v) > 2T 0 + D, and any perpendicular Jacobi field J along γ v with µ J (t) Proof.-We argue by contradiction.Assume that we can find We may assume t n → t, v n → v by passing to a subsequence and it is clear that γ v stays in U T0 for t ⩾ T 0 .In particular, v ∈ Γ − .
By definition of Q(T 0 , g), J n (t) ̸ = 0 for all n and t ∈ [0, T 0 ].Without loss of generality we assume that |ξ n | = 1 for all n thus J n → J for some Jacobi field J along γ v .By Lemma 2.7 the invariant bundles depend continuously on the base vectors, thus the projection to invariant components of Jacobi fields through U is continuous.Hence we have J(t) = J s (t − T 0 ) for t ⩾ T 0 .Since γ v stays in M for t ⩾ T 0 , we also have It is clear that T also depends only on Λ and U. We take with k as in Proposition 4.3 and Q given by Corollary 2.10.First assume that ℓ g (v) ⩾ 2T 0 + T .If µ J (0) ⩾ Q M , by Proposition 4.3 and the parallelogram law, 2) and definition of hyperbolicity we have Hence we finishes the proof of item (1).
Hence when Thus by taking C M := −(2T 0 + T )Q M we finish the proof of (2).The only part left is (3).Recall that all the constant C, L, ζ, T depend on Λ and its neighborhood U.By replacing M with M −δ we still can work on a smaller neighborhood of Λ thus the same argument goes through without any change.Thus we have finished the proof of Proposition 4.1.□

Deformation to negative sectional curvature
In this section we consider a cylinder with a given metric on a neighborhood O of one of the boundaries, and extend it to a metric on the whole cylinder so that the sectional curvatures is arbitrarily negative outside a small neighborhood of O.We provide bounds on both sectional curvatures (see Section 5.1, Proposition 5.3, Lemma 5.4) and the principle curvatures of the equidistant sets.In particular, all the equidistant sets are also strictly convex.See the precise formulation of the main result Proposition 5.2 of this section which is proved using the mentioned curvature bounds.
5.1.The setup and notation.-We use notation from Section 2.5.
Let S be an (n − 1)-dimensional smooth closed manifold.For ε > 0, consider the product (−ε, 0] × S with a Riemannian metric (5.1) where g t is the Riemannian metric on the hypersurface S t = {t} × S. Assume S 0 is strictly convex and recall that h = 2II S0 is the positive definite second fundamental form at t = 0.For any θ ∈ S, since h is symmetric, there exists an orthonormal basis {e i } n−1 i=1 of g 0 such that h(e i , e i ) = 2λ i (0, θ) > 0 where λ i (0, θ) is the i-th principal curvature at (0, θ).Our goal now is to extend the metric in a controlled way for t > 0.
More generally to setup terminology, we can consider a manifold of the form [a, b] × S with coordinates (t, θ) where t ∈ [a, b] and θ ∈ S. We say that a tangent 2-plane σ at (t, θ) is orthogonal to S t if σ contains a normal vector to S t .As a result, Let ρ : R → [0, 1] be a non-increasing C ∞ function such that ρ ≡ 1 on (−∞, 0] and ρ ≡ 0 on [1, ∞).For any ℓ > 0, a function f ℓ : R → R is given by Remark 5.1.-For any metric g ′ on S 0 , we consider its push-forward to a metric (π t ) * g ′ on S t which we still denote by g ′ using a slight abuse of notation.
5.2.Deformation of the metric.-We prove the following result assuming Proposition 5.3 and Lemma 5.4.
Proof.-Recall that h is positive definite.Item (c) will be proved later in Proposition 5.3(1).

Upper bound on orthogonal sectional curvatures
Proposition 5.3 (setting of Proposition 5.2).
(2) Let K ⊥ ℓ,ε (t) be the maximum sectional curvature among planes σ X,T on ([0, 1 + ε] × S, g ℓ,ε ), where X ∈ T 1 S t .Then, for all ℓ > L 1 and all t ∈ [0, 1 + ε], -For any θ ∈ S, let e t i ∈ T (t,θ) S t be defined by e t i = (π t ) * e i , where {e i } is the orthonormal basis in Section 5.1.By construction, {e t i } n−1 i=1 is an orthogonal basis of T (t,θ) S t for t ∈ [0, 1 + ε].Thus, any X ∈ T (t,θ) S t can be written in the coordinates as (X 1 , . . ., X n−1 ) T with respect to {e t i } n−1 i=1 .In particular, For any t ∈ [0, 1 + ε], the second fundamental form on S t is given by Recall that A(t, θ) is the matrix of the shape operator on S t with respect to the basis to {e t i } n−1 i=1 , i.e., the i-th column of A(t, θ) is the image of e t i under the shape operator.Then, by the definition of the shape operator, Therefore, the i-th eigenvalue of A(t, θ) is given by , where η i (t, θ) := ρ(t − ε) 2λ i (0, θ) .
By Lemma 2.11, we have

"Rounding" the metric
In this section we consider a cylinder with given metrics on the boundaries.Then, we use a linear combination of those metrics on each equidistant set to define a metric of the form g = dt 2 + g t on the whole cylinder so that it has the given metrics on the boundary.Then, by choosing an appropriate exponentially growing function f ℓ of the distance to one of the boundaries, we can guarantee that a metric g = dt 2 + f ℓ (t) g t has arbitrarily negative the sectional curvatures (see Propositions 6.3, 6.4).We can guarantee that all the equidistant sets are also strictly convex.See the precise formulation in Proposition 6.1.
Our aim is to glue a given metric on the manifold with boundary with the standard hyperbolic metric.In regards of that, Proposition 6.1 allows us to "round up" the metric through the cylinder meaning have a non-conformal metric on one end of the cylinder and a conformal metric on the other end of it while having arbitrarily negative curvature and strict convexity of the equidistant sets.Proposition 6.1 (Notation of Section 5.1).-Let h and h be Riemannian metrics on S. Consider the manifold [0, 1 + ε] × S with Riemannian metric g ℓ,ε = dt 2 + g t , where Then, for any M 0 > 0 there exists L r = L r (M 0 , ε, h, h, ρ) > 0 such that for any ℓ > L r the following holds: (a) All sectional curvatures of g ℓ,ε are bounded from above by −M 0 .(b) For all t ∈ [0, 1 + ε], S t is strictly convex.Remark 6.2.-We will use Proposition 6.1 in Proposition 7.1 for h = 2II S0 , where S 0 is from Section 5.1, and h being the standard round metric of curvature 1 on a sphere.
Proof.-The proof follows the same general approach as the proof of Proposition 5.2, so we omit some of the details.
For any θ ∈ S, let {e i } n−1 i=1 be an orthonormal basis of h such that h(e i , e j ) = µ i (θ).
For any θ ∈ S, let e t i ∈ T (t,θ) S t be defined by e t i = (π t ) * e i .By the construction, {e t i } n−1 i=1 is an orthogonal basis of T (t,θ) S t for t ∈ [0, 1 + ε].Thus, any X ∈ T (t,θ) S t can be identified with the coordinate vector (X 1 , . . ., X n−1 ) T with respect to {e t i } n−1 i=1 .In particular, For any t ∈ [0, 1 + ε], the i-th eigenvalue of A(t, θ) is given by .
Proof.-For any 2-plane σ ⊆ T (t,θ) S t , we obtain By Proposition 6.3(1), for all ℓ > L 1 and t ∈ [0, ε], we have that S t is strictly convex.Moreover, µ min (S t ) → ∞ uniformly in t ∈ [0, 1 + ε] as ℓ → ∞.Thus, by Lemma 2.11, As a result, for any M 0 > 0 there exists a constant The goal of this section is to construct a C 1,1 -extension to the constant negative curvature of a given metric on the product of infinite ray and a sphere.In the second half of this section we will mollify the C 1,1 metric to obtain a C ∞ metric while still controlling the curvature.
Proof.-Notice that h is a Riemannian metric on S as S 0 is strictly convex.Because f ℓ and ρ are smooth the metric g ℓ,ε is smooth in each component.Via the choice of f ℓ , ρ and κ (in Lemma C.1), it is clear that g ℓ,ε is smooth at t = 1 + ε and C 1,1 at t = 0, 2 + 2ε.Thus we obtain (a).Moreover, Lemma C.1 shows that there exists L 1 = L 1 (M 1 ) such that for any ℓ > L 1 , the associated κ is at least M 1 .Item (c) follows from Proposition 5.2(a), while (e) follows from Lemma B.1.
Notice that the construction on t ∈ [1+ε, 2+2ε] is just a translation reparametrization of the metric in Proposition 6.1.Thus (d) follows from Proposition 5.2(b) and Proposition 6.1(a).Finally we get (b) via Proposition 5.2(c), Proposition 6.1(b) and the assumption of ε above this proposition.□ 7.2.Smoothing of the extension from Section 7.1.-We apply a technique developed in [EK19] to smooth out the C 1,1 metric we obtained in Proposition 7.1.
Step 1: Smoothing near {2 + 2ε} × S. -Notice that for t ∈ [2 + ε, ∞], we can express g ℓ,ε t in the following way: , so is f .The sectional curvature for g on t ⩾ 2 + ε is given by where θ is the angle between the tangent 2-plane σ and T .By Lemma B.1, we have We have that f η is smooth and f η → f in C 1 topology, thus there exists η 1 > 0 such that for all η < η 1 , Hence all S t with t ∈ (2 + 2ε − δ, 2 + 2ε + δ) are strictly convex.
In order to finish the proof of (c) we only have to estimate since f is C 2 on these intervals, we have f η → f in C 2 topology and we can find η 2 such that for any η < η 2 , f ′′ η / f η ⩾ (M 1 − 1) 2 .We finish the proof by taking η < min{η 1 , η 2 }.
Step 2: Smoothing near {0} × S. -We define g η := dt 2 + g η,t on (−δ, δ) × S via convolution It is clear that g η → g in C 1 .Since g is C 1,1 with respect to t and smooth with respect to coordinates on S, d 2 dt 2 g η is bounded by the Lipschitz constant of d dt g, while other second order derivatives of g η converge to those of g.Thus all second derivatives of g η are uniformly bounded on any compact set.Hence there exists η 3 > 0, K 0 > K g such that for any η ∈ (0, η 3 ), the sectional curvatures of g η are bounded above by . We need to establish the bounds on sectional curvature when |t| ∈ [δ/2, δ].Notice that in these domains g is at least C 2 , thus g η → g in C 2 as η → 0 on both [δ/2, δ] × S and [−δ, −δ/2] × S. Hence for any fixed δ, g ℓ,ε η → g in C 2 topology on these domains.Since K 0 > K g and the curvature of g ℓ,ε on (−δ, δ) × S is bounded above by K g J.É.P. -M., 2023, tome 10 by Proposition 7.1(c), there exists η 4 > 0 such that for any η < η 4 , the sectional curvatures of g ℓ,ε η on both [δ/2, δ] × S and [−δ, −δ/2] × S are bounded from above by K 0 .Thus we obtain item (b).Now we prove (d), since g ℓ,ε η → g in C 1 topology as η → 0 and principal curvatures depend merely on g ℓ,ε η and d dt g ℓ,ε η , by Proposition 7.1(b) and the assumption on ε above Proposition 7.1, we know that there exists η 5 > 0 such that for η < η 5 , the principal curvatures has a uniform lower bound λ min (S 0 )/2.

Anosov extension
The goal of this section is to prove the main theorem whose statement we recall.
Theorem 8.1 (Theorem A). -Let (Σ, g) be a compact smooth Riemannian manifold with boundary.Assume that each component of the boundary is a strictly convex sphere.Also assume that (Σ, g) has no conjugate points and the trapped set for the geodesic flow is hyperbolic.Then, there exists a codimension 0 isometric embedding (Σ, g) ⊂ (Σ ext , g ext ) such that (Σ ext , g ext ) is a closed Anosov manifold.
We first describe the main construction where we allow ∂Σ to have several connected components.Afterward, we need to establish the estimates on Jacobi fields, which then allow us to prove the absence of conjugate points and to finish the proof in Section 8.4.For the sake of simpler notation, in this part of the proof we assume that ∂Σ has only one connected component.The argument for the general case is the same.
8.1.Description of the extension.-To describe the extension, we will need the following fact.Lemma 8.2 ([Gui17, Lem.2.3]).-For any sufficiently small δ 0 > 0, there exists an isometrical embedding of (Σ, g) into a smooth Riemannian manifold (Σ δ0 , g δ0 ) with strictly convex boundary which is equidistant to the boundary of Σ, has the same hyperbolic trapped set as (Σ, g), and no conjugate points.Moreover, all hypersurfaces equidistant to the boundary of Σ in Σ δ0 ∖ Σ are strictly convex.
By the lemma we can fix a δ 0 > 0 such that the principal curvatures of all hypersurfaces equidistant to the boundary of Σ in Σ δ0 ∖ Σ are at least λ min (∂Σ)/2 where λ min (∂Σ) is the minimum of principal curvatures of ∂Σ.
We denote by Q 0 := Q Σ δ 0 and C 0 := C Σ δ 0 the constants given by Proposition 4.1 when applied to Σ δ0 .Assume ∂Σ δ0 = ⊔ m j=1 S j with each S j diffeomorphic to a sphere.For any sufficiently small ε ∈ (0, δ 0 ), we can consider normal coordinates in the ε-neighborhoods of each S j .In particular, for each j, the ε-neighborhood of S j is isometric to (−ε, 0] × S j with metric j g = dt 2 + j g t where t ∈ (−ε, 0] parametrizes the (signed) distance to S j and j g t is the Riemannian metric on S j t = {t} × S j .
Remark 8.4.-We want to point out that the resulting constant sectional curvature κ in the extension can be a priori arbitrarily large, and its value depends on ℓ which depends on the given Riemannian manifold with boundary .This can be seen from Lemma C.1.
We introduce notation that we will use in the next sections.Set We decompose Σ ext into three domains where We summarize the properties of the resulting extension that come from Propositions 4.1, Propositions 7.1 and 7.2 with our choice of parameters: (i) We have the conclusion of Proposition 4.1 for (Σ 0 , g ext ) with Q 0 and C 0 .(ii) The sectional curvatures on D − are at most −(Q 0 + 3) 2 .And all maximal geodesic segments within D − have length at least R.
(iii) On C 1 + , the curvature upper bound is K 0 and the principal curvatures for hypersurfaces in C 1 + equidistant to Σ are at least λ min (∂Σ)/4.(iv) On C 2 + , the curvature upper bound is K g and the principal curvatures for hypersurfaces in C 2 + equidistant to Σ are at least λ min (∂Σ)/2.

8.2.
Travel time and Jacobi estimate in the collar.-As we mentioned before, for the sake of simpler notation, we assume that ∂Σ has only one connected component.We denote the boundary of Σ δ0 by S (see Section 8.1) and let S t = {t}×S.We want to estimate the travel time and change of µ J when a geodesic goes through C + .To do that we consider a setting which is (formally) more general than (iii) and (iv) above which we proceed to describe.
Corollary 8.7.-Let c : [0, τ 0 ] → C + be a geodesic in C + and J be a perpendicular Jacobi field along c.
By construction of Σ ext , we know that for all 0 ⩽ k ⩽ n, Firstly, we prove that µ J (a k+1 ) > Q 0 + 2 and (8.5) on which the sectional curvatures are bounded above by −(Q 0 + 3) 2 .By Lemma 2.8, we know that for t In particular, we have (8.5) and (8.6).Together with Corollary 8.7, we have the following two statements: Now we make the estimate on the entire [τ 1 , τ 2 ].Since µ J (τ 1 ) > −Q 0 , by Corollary 8.7(a), we know that µ J (b 0 ) ⩾ −Q 0 − 2. By (8.5) and (8.8), we obtain iii and for any Thus, when n = ∞, τ2 τ1 µ J (τ )dτ = ∞.When τ 2 < ∞, by (8.5) and (8.8), µ J (a n ) > Q 0 + 2, thus µ J (τ 2 ) > Q 0 due to Corollary 8.7 and we obtain i.The only case left is   Proof.-We need to prove that for any geodesic γ v and perpendicular Jacobi field  If a geodesic γ v stays in Σ 0 for all t ∈ R, then v ∈ Λ.Thus any Jacobi field along γ v is unbounded by hyperbolicity.Therefore it remains to consider the case when γ v passes through D − .Let J be a Jacobi field along γ v .By changing the starting time we may assume that the geodesic segment γ v | [−R/2,R/2] lies within D − .We can also assume that J(0) ̸ = 0 and µ J (0) ⩾ 0 (otherwise we can replace v with −v).We will show that ∥J∥(t) → ∞ as t → ∞.
Recall that µ J = ∥J∥ ′ /∥J∥, hence we have only to prove the integral of µ J is unbounded on [0, +∞).As before denote by k ∈ Z being the segments within Σ 0 .

Figure 1 .
Figure 1.Shadowing of γ and γ ′ .Here we use green (resp.red) curves to denote stable (resp.unstable) manifolds.By the standard "exponential slacking" argument which is used in the proof of the Livshits Theorem[Liv71] we have

4. 1 .
Neighborhood of hyperbolic trapped set.-For any T ⩾ 0, let J.É.P. -M., 2023, tome 10 we define orthogonal sectional curvatures of [a, b]×S as curvatures of tangent 2-planes orthogonal to S t for some t ∈ [a, b].