The aim of this note is to establish the correspondence between the twisted localized Pestov identity on the unit tangent bundle of a Riemannian manifold and the Weitzenböck identity for twisted symmetric tensors on the manifold.

On a Riemannian manifold (M,g) with Anosov geodesic flow, the problem of recovering a connection from the knowledge of traces of its holonomies along primitive closed geodesics is known as the holonomy inverse problem. In this paper, we prove Hölder type stability estimates for this inverse problem: 1) locally, near generic connections; 2) globally, for line bundles, and for vector bundles satisfying a certain low-rank assumption over negatively curved base (M,g). The proofs are based on a combination of microlocal analysis along with a new non-Abelian approximate Livšic Theorem in hyperbolic dynamics.

Let (M,g,J) be a closed Kähler manifold with negative sectional curvature and complex dimension m:=dim_C(M)≥2. In this article, we study the unitary frame flow, that is, the restriction of the frame flow to the principal U(m)-bundle F_CM of unitary frames. We show that if m≥6 is even, and m≠28, there exists λ(m)∈(0,1) such that if (M,g,J) has negative λ(m)-pinched holomorphic sectional curvature, then the unitary frame flow is ergodic and mixing. The constants λ(m) satisfy λ(6)=0.9330..., lim_{m→+∞}λ(m)=1112=0.9166..., and m↦λ(m) is decreasing. This extends to the even-dimensional case the results of Brin-Gromov who proved ergodicity of the unitary frame flow on negatively-curved compact Kähler manifolds of odd complex dimension.

We study resonant differential forms at zero for transitive Anosov flows on 3-manifolds. We pay particular attention to the dissipative case, that is, Anosov flows that do not preserve an absolutely continuous measure. Such flows have two distinguished Sinai-Ruelle-Bowen 3-forms, Ω^±_{SRB}, and the cohomology classes [ι_X Ω^±_{SRB}] obtained by contraction (where X is the infinitesimal generator of the flow) play a key role in the determination of the space of resonant 1-forms. When both classes vanish we associate to the flow a helicity that naturally extends the classical notion associated with null-homologous volume preserving flows. We provide a general theory that includes horocyclic invariance of resonant 1-forms and SRB-measures as well as the local geometry of the maps X↦[ι_X Ω^±_{SRB}] near a null-homologous volume preserving flow. Next, we study several relevant classes of examples. Among these are thermostats associated with holomorphic quadratic differentials, giving rise to quasi-Fuchsian flows as introduced by Ghys. For these flows we compute explicitly all resonant 1-forms at zero, we show that [ι_X Ω^±_{SRB}]=0 and give an explicit formula for the helicity. In addition we show that a generic time change of a quasi-Fuchsian flow is semisimple and thus the order of vanishing of the Ruelle zeta function at zero is −χ(M), the same as in the geodesic flow case. In contrast, we show that if (M,g) is a closed surface of negative curvature, the Gaussian thermostat driven by a (small) harmonic 1-form has a Ruelle zeta function whose order of vanishing at zero is −χ(M)−1.

We exhibit a link between the following a priori unrelated three problems: 1) In algebraic geometry: the non-existence of polynomial mappings between spheres; 2) In dynamical systems: the ergodicity of certain partially hyperbolic dynamical systems obtained as isometric extensions of the geodesic flow over negatively-curved Riemannian manifolds; and 3) In spectral theory: Kac's isospectral problem "Can one hear the shape of a drum?" for the Bochner (or connection) Laplacian. We derive two main applications. First, we show that on closed negatively-curved Riemannian manifolds with simple length spectrum, the spectrum of the Bochner Laplacian determines both the isomorphism class of the vector bundle and the connection up to gauge under a low-rank assumption. Second, we show that partially hyperbolic flows arising as extensions of the geodesic flow to certain low-rank frame bundles (over negatively-curved manifolds) are ergodic whenever the bundle admits no holonomy reduction.

We report on some recent progress achieved in [arXiv:2111.14811] on the ergodicity of the frame flow of negatively-curved Riemannian manifolds. We explain the new ideas leading to ergodicity for nearly 0.25-pinched manifolds and give perspectives for future work.

The lens data of a Riemannian manifold with boundary is the collection of lengths of geodesics with endpoints on the boundary together with their incoming and outgoing vectors. We show that negatively-curved Riemannian manifolds with strictly convex boundary are locally lens rigid in the following sense: if g0 is such a metric, then any metric g sufficiently close to g0 and with same lens data is isometric to g0, up to a boundary-preserving diffeomorphism. More generally, we consider the same problem for a wider class of metrics with strictly convex boundary, called metrics of Anosov type. We prove that the same rigidity result holds within that class in dimension 2 and in any dimension, further assuming that the curvature is non-positive.

It is known that the frame flow on a closed n-dimensional Riemannian manifold with negative sectional curvature is ergodic if n is odd and n different from 7. In this paper we study its ergodicity for n ≥ 4 even and n = 7, and we show that: (1) if n ≡ 2 mod 4, or n = 4, the frame flow is ergodic if the manifold is ∼ 0.3-pinched, (2) if n ≡ 0 mod 4, it is ergodic if the manifold is ∼ 0.6-pinched, except in the three dimensions n = 7, 8, 134, where the respective pinching conditions are 0.4962..., 0.6212..., and 0.6716.... In particular, if n = 4 or n ≡ 2 mod 4, this almost solves a long-standing conjecture of Brin asserting that 1/4-pinched even-dimensional manifolds have an ergodic frame flow.

In dimensions ≥3, we prove that the X-ray transform of symmetric tensors of arbitrary degree is generically injective with respect to the metric on closed Anosov manifolds and on manifolds with spherical strictly convex boundary, no conjugate points and a hyperbolic trapped set. Building on earlier work by Guillarmou, Knieper and the second author [arXiv:1806.04218], [arXiv:1909.08666], this solves locally the marked length spectrum rigidity conjecture in a neighborhood of a generic Anosov metric. This is the first work going beyond the negatively-curved assumption or dimension 2. Our method, initiated in [arXiv:2008.09191] and fully developed in the present paper, is based on a perturbative argument of the 0-eigenvalue of elliptic operators via microlocal analysis which turn the analytic problem of injectivity into an algebraic problem of representation theory. When the manifold is equipped with a Hermitian vector bundle together with a unitary connection, we also show that the twisted X-ray transform of symmetric tensors (with values in that bundle) is generically injective with respect to the connection. This property turns out to be crucial when solving the holonomy inverse problem, as studied in a subsequent article [arXiv:2105.06376].

Let (M,g) be a smooth Anosov Riemannian manifold and C♯ the set of its primitive closed geodesics. Given a Hermitian vector bundle E equipped with a unitary connection ∇E, we define T♯(E,∇E) as the sequence of traces of holonomies of ∇E along elements of C♯. This descends to a homomorphism on the additive moduli space A of connections up to gauge T♯:(A,⊕)→ℓ∞(C♯), which we call the primitive trace map. It is the restriction of the well-known Wilson loop operator to primitive closed geodesics. The main theorem of this paper shows that the primitive trace map T♯ is locally injective near generic points of A when dim(M)≥3. We obtain global results in some particular cases: flat bundles, direct sums of line bundles, and general bundles in negative curvature under a spectral assumption which is satisfied in particular for connections with small curvature. As a consequence of the main theorem, we also derive a spectral rigidity result for the connection Laplacian. The proofs are based on two new ingredients: a Livšic-type theorem in hyperbolic dynamical systems showing that the cohomology class of a unitary cocycle is determined by its trace along closed primitive orbits, and a theorem relating the local geometry of A with the Pollicott-Ruelle resonance near zero of a certain natural transport operator.

We show, using semiclassical measures and unstable derivatives, that a smooth vector field X generating a contact Anosov flow on a 3-dimensional manifold M has only finitely many Ruelle resonances in the vertical strips {s∈ℂ | Re(s)∈[−ν_min+ϵ,−1/2ν_max−ϵ]∪[−1/2ν_min+ϵ,0]} for all ϵ>0, where 0<ν_min≤ν_max are the minimal and maximal expansion rates of the flow (the first strip only makes sense if ν_min>ν_max/2). We also show polynomial bounds in s for the resolvent (−X−s)^{−1} as |Im(s)|→∞ in Sobolev spaces, and obtain similar results for cases with a potential. This gives a short microlocal proof of a particular case of the results announced by Faure-Tsujii in \cite{FaTs1}, using that dim E_u = dim E_s = 1.

We show that for a generic conformal metric perturbation of a hyperbolic 3- manifold $\Sigma$, the order of vanishing of the Ruelle zeta function at zero equals $4 − b_1(\Sigma)$, contrary to the hyperbolic case where it is equal to $4 − 2b_1(\Sigma)$. The result is proved by developing a suitable perturbation theory that exploits the natural pairing between resonant and co-resonant differential forms. To obtain a metric conformal perturbation we need to establish the non-vanishing of the pushforward of a certain product of resonant and coresonant states and we achieve this by a suitable regularisation argument. Along the way we describe geometrically all resonant differential forms (at zero) for a closed hyperbolic $3$-manifold and study the semisimplicity of the Lie derivative.

Given a smooth Hermitian vector bundle $\mathcal{E}$ over a closed Riemannian manifold $(M,g)$, we study generic properties of unitary connections $\nabla^{\mathcal{E}}$ on the vector bundle \mathcal{E}. First of all, we show that twisted Conformal Killing Tensors (CKTs) are generically trivial when dim(M) ≥ 3, answering an open question of Guillarmou-Paternain-Salo-Uhlmann. In negative curvature, it is known that the existence of twisted CKTs is the only obstruction to solving exactly the twisted cohomological equations which may appear in various geometric problems such as the study of transparent connections. The main result of this paper says that these equations can be generically solved. As a by-product, we also obtain that the induced connection $\nabla^{\mathrm{End} \mathcal{E}}$ on the endomorphism bundle $\mathrm{End}\mathcal{E}$ has generically trivial CKTs as long as $(M,g)$ has no nontrivial CKTs on its trivial line bundle. Eventually, we show that, under the additional assumption that $(M,g)$ is Anosov (i.e. the geodesic flow is Anosov on the unit tangent bundle), the connections are generically opaque, namely there are no non-trivial subbundles of \mathcal{E} which are generically preserved by parallel transport along geodesics. The proofs rely on the introduction of a new microlocal property for (pseudo)differential operators called operators of uniform divergence type, and on perturbative arguments from spectral theory (especially on the theory of Pollicott-Ruelle resonances in the Anosov case).

We consider Anosov flows on closed 3-manifolds preserving a volume form $\Omega$. Following \cite{DyZw17} we study spaces of invariant distributions with values in the bundle of exterior forms whose wavefront set is contained in the dual of the unstable bundle. Our first result computes the dimension of these spaces in terms of the first Betti number of the manifold, the cohomology class $[\iota_{X}\Omega]$ (where $X$ is the infinitesimal generator of the flow) and the helicity. These dimensions coincide with the Pollicott-Ruelle resonance multiplicities under the assumption of {\it semisimplicity}. We prove various results regarding semisimplicity on 1-forms, including an example showing that it may fail for time changes of hyperbolic geodesic flows. We also study non null-homologous deformations of contact Anosov flows and we show that there is always a splitting Pollicott-Ruelle resonance on 1-forms and that semisimplicity persists in this instance. These results have consequences for the order of vanishing at zero of the Ruelle zeta function. Finally our analysis also incorporates a flat unitary twist in both, the resonant spaces and the Ruelle zeta function.

We study dynamical properties of the billiard flow on convex polyhedra away from a neighbourhood of the non-smooth part of the boundary, called "pockets". We prove there are only finitely many periodic immersed tubes missing the pockets and moreover establish a new quantitative estimate for lengths of such periodic tubes. This extends well-known results in dimension 2. We then apply these dynamical results to prove a quantitative Laplace eigenfunction mass concentration near the pockets of convex polyhedral billiards. As a technical tool for proving our concentration results on irrational polyhedra, we establish a control-theoretic estimate on a product space with an almost-periodic boundary condition. This extends previously known control estimates for periodic boundary conditions, and seems to be of independent interest.

We investigate the Calderón problem for the fractional Schrödinger equation with drift, proving that the unknown drift and potential in a bounded domain can be determined simultaneously and uniquely by an infinite number of exterior measurements. In particular, in contrast to its local analogue, this nonlocal problem does \emph{not} enjoy a gauge invariance. The uniqueness result is complemented by an associated logarithmic stability estimate under suitable apriori assumptions. Also uniqueness under finitely many measurements is discussed. The inverse problem is formulated as a partial data type nonlocal problem; it is considered in any dimensions n≥1.

We give partial answers to the following question: if F is an m by m matrix on R^n satisfying a second order linear elliptic equation, does det F satisfy the strong unique continuation property? We give counterexamples in the case when the operator is a general non-diagonal operator and also for some diagonal operators. Positive results are obtained when n = 1 and any m, when n = 2 for the Laplace-Beltrami operator and also twisted with a Yang-Mills connection. Reductions to special cases when n = 2 are obtained. The last section considers an application to the Calderón problem in 2D based on recent techniques.

We consider the problem of identifying a unitary Yang-Mills connection ∇ on a Hermitian vector bundle from the Dirichlet-to-Neumann (DN) map of the connection Laplacian ∇∗∇ over compact Riemannian manifolds with boundary. We establish uniqueness of the connection up to a gauge equivalence in the case of trivial line bundles in the smooth category and for the higher rank case in the analytic category, by using geometric analysis methods and essentially only one measurement. Moreover, by using a Runge-type approximation argument along curves to re- cover holonomy, we are able to uniquely determine both the bundle structure and the connection, but at the cost of having more measurements. Also, we prove that the DN map is an elliptic pseudodifferential operator of order one on the restriction of the vector bundle to the boundary, whose full symbol determines the complete Taylor series of an arbitrary connection, metric and an associated potential at the boundary.

In this paper we consider the problem of identifying a connection ∇ on a vector bundle up to gauge equivalence from the Dirichlet-to-Neumann map of the connection Laplacian ∇∗∇ over conformally transversally anisotropic (CTA) manifolds. This was proved in [1] for line bundles in the case of the transversal manifold being simple – we generalise this result to the case where the transversal manifold only has an injective ray transform. Moreover, the construction of suitable Gaussian beam solutions on vector bundles is given for the case of the connection Laplacian and a potential, following the works of [2]. Consequently, this enables us to construct the Complex Geometrical Optics (CGO) solutions and prove our main uniqueness result. Finally, we prove the recovery of a flat connection in general from the DN map, up to gauge equivalence, using an argument relating the Cauchy data of the connection Laplacian and the holonomy.