Let be a Riemannian manifold with strictly convex spherical boundary. Assuming absence of conjugate points and that the trapped set is hyperbolic, we show that 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.
Soit une variété riemannienne avec bord sphérique strictement convexe. Lorsque la métrique n’a pas de points conjugués et que l’ensemble capté est hyperbolique, nous montrons que peut être plongée isométriquement dans une variété riemannienne fermée dont le flot géodésique est Anosov. Nous utilisons ce plongement pour établir un lien direct entre le théorème de Livshits classique pour les flots d’Anosov et le théorème de Livshits pour la transformée en rayons X qui apparaît dans le programme de rigidité des bords. Nous donnons également une application pour la rigidité lenticulaire dans une classe conforme.
Accepted:
Published online:
Keywords: Anosov flow, geodesic flow, lens rigidity, Livshits theorem, trapped sets
Mot clés : Flot Anosov, flot géodésique, rigidité lenticulaire, théorème de Livshits, ensemble capté
Dong Chen 1; Alena Erchenko 2; Andrey Gogolev 1
@article{JEP_2023__10__945_0, author = {Dong Chen and Alena Erchenko and Andrey Gogolev}, title = {Riemannian {Anosov} extension and applications}, journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques}, pages = {945--987}, publisher = {\'Ecole polytechnique}, volume = {10}, year = {2023}, doi = {10.5802/jep.237}, language = {en}, url = {https://jep.centre-mersenne.org/articles/10.5802/jep.237/} }
TY - JOUR AU - Dong Chen AU - Alena Erchenko AU - Andrey Gogolev TI - Riemannian Anosov extension and applications JO - Journal de l’École polytechnique — Mathématiques PY - 2023 SP - 945 EP - 987 VL - 10 PB - École polytechnique UR - https://jep.centre-mersenne.org/articles/10.5802/jep.237/ DO - 10.5802/jep.237 LA - en ID - JEP_2023__10__945_0 ER -
%0 Journal Article %A Dong Chen %A Alena Erchenko %A Andrey Gogolev %T Riemannian Anosov extension and applications %J Journal de l’École polytechnique — Mathématiques %D 2023 %P 945-987 %V 10 %I École polytechnique %U https://jep.centre-mersenne.org/articles/10.5802/jep.237/ %R 10.5802/jep.237 %G en %F JEP_2023__10__945_0
Dong Chen; Alena Erchenko; Andrey Gogolev. Riemannian Anosov extension and applications. Journal de l’École polytechnique — Mathématiques, Volume 10 (2023), pp. 945-987. doi : 10.5802/jep.237. https://jep.centre-mersenne.org/articles/10.5802/jep.237/
[BI10] - “Boundary rigidity and filling volume minimality of metrics close to a flat one”, Ann. of Math. (2) 171 (2010) no. 2, p. 1183-1211 | DOI | MR | Zbl
[BI13] - “Area minimizers and boundary rigidity of almost hyperbolic metrics”, Duke Math. J. 162 (2013) no. 7, p. 1205-1248 | DOI | MR | Zbl
[dlLMM86] - “Canonical perturbation theory of Anosov systems and regularity results for the Livsic cohomology equation”, Ann. of Math. (2) 123 (1986) no. 3, p. 537-611 | DOI | MR | Zbl
[DP03] - “Anosov geodesic flows for embedded surfaces”, in Geometric methods in dynamics. II, Astérisque, vol. 287, Société Mathématique de France, Paris, 2003, p. 61-69 | Numdam | Zbl
[DSW21] - “Resonances and weighted zeta functions for obstacle scattering via smooth models”, 2021 | arXiv
[Ebe73] - “When is a geodesic flow of Anosov type? I”, J. Differential Geom. 8 (1973) no. 3, p. 437-463 | MR | Zbl
[EK19] - “Flexibility of entropies for surfaces of negative curvature”, Israel J. Math. 232 (2019) no. 2, p. 631-676 | DOI | MR | Zbl
[EO80] - “Jacobi tensors and Ricci curvature”, Math. Ann. 252 (1980) no. 1, p. 1-26 | DOI | MR | Zbl
[FJ93] - “Nonuniform hyperbolic lattices and exotic smooth structures”, J. Differential Geom. 38 (1993) no. 2, p. 235-261 | MR | Zbl
[GM18] - “Marked boundary rigidity for surfaces”, Ergodic Theory Dynam. Systems 38 (2018) no. 4, p. 1459-1478 | DOI | MR | Zbl
[Gro94] - “Sign and geometric meaning of curvature”, Rend. Sem. Mat. Fis. Milano 61 (1994), p. 9-123 | DOI | MR
[Gui17] - “Lens rigidity for manifolds with hyperbolic trapped sets”, J. Amer. Math. Soc. 30 (2017) no. 2, p. 561-599 | DOI | MR | Zbl
[Gul75] - “On the variety of manifolds without conjugate points”, Trans. Amer. Math. Soc. 210 (1975), p. 185-201 | DOI | MR | Zbl
[HPPS70] - “Neighborhoods of hyperbolic sets”, Invent. Math. 9 (1970), p. 121-134 | DOI | MR | Zbl
[Kat88] - “Four applications of conformal equivalence to geometry and dynamics”, Ergodic Theory Dynam. Systems 8 (1988), p. 139-152 | DOI | MR | Zbl
[Lef19] - “On the s-injectivity of the x-ray transform on manifolds with hyperbolic trapped set”, Nonlinearity 32 (2019) no. 4, p. 1275-1295 | DOI | MR | Zbl
[Lef20] - “Local marked boundary rigidity under hyperbolic trapping assumptions”, J. Geom. Anal. 30 (2020) no. 1, p. 448-465 | DOI | MR | Zbl
[Liv71] - “Certain properties of the homology of -systems”, Mat. Zametki 10 (1971), p. 555-564 | MR
[LSU03] - “Semiglobal boundary rigidity for Riemannian metrics”, Math. Ann. 325 (2003) no. 4, p. 767-793 | DOI | MR | Zbl
[Mal40] - “On isomorphic matrix representations of infinite groups”, Mat. Sb. 8 (50) (1940), p. 405-422 | Zbl
[Mic81] - “Sur la rigidité imposée par la longueur des géodésiques”, Invent. Math. 65 (1981) no. 1, p. 71-83 | DOI | MR | Zbl
[MR78] - “On the problem of finding an isotropic Riemannian metric in an -dimensional space”, Dokl. Akad. Nauk SSSR 243 (1978) no. 1, p. 41-44 | MR
[PU05] - “Two dimensional compact simple Riemannian manifolds are boundary distance rigid”, Ann. of Math. (2) 161 (2005) no. 2, p. 1093-1110 | DOI | MR | Zbl
[Rug07] - Dynamics and global geometry of manifolds without conjugate points, Ensaios Matemáticos, vol. 12, Sociedade Brasileira de Matemática, Rio de Janeiro, 2007
[SU09] - “Local lens rigidity with incomplete data for a class of non-simple Riemannian manifolds”, J. Differential Geom. 82 (2009) no. 2, p. 383-409 | MR | Zbl
[SUV21] - “Local and global boundary rigidity and the geodesic X-ray transform in the normal gauge”, Ann. of Math. (2) 194 (2021) no. 1, p. 1-95 | DOI | MR | Zbl
[Var09] - “A proof of lens rigidity in the category of analytic metrics”, Math. Res. Lett. 16 (2009) no. 6, p. 1057-1069 | DOI | MR | Zbl
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