À l’aide des groupoïdes de Lie, on montre que le rayon d’injectivité d’une variété munie d’une structure de Lie à l’infini est strictement positif. La démonstration s’appuie sur l’intégrabilité de l’algébroïde de Lie correspondant, un résultat bien connu que l’on établit directement en regardant les variétés à coins comme des cas particuliers d’orbifolds.
Using Lie groupoids, we prove that the injectivity radius of a manifold with a Lie structure at infinity is positive. This relies on the integrability of the corresponding Lie algebroid, a well-known result that we prove explicitly by regarding manifolds with corners as particular instances of orbifolds.
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@article{AMBP_2022__29_2_235_0,
author = {Quang-Tu Bui},
title = {Injectivity radius of manifolds with a {Lie} structure at infinity},
journal = {Annales math\'ematiques Blaise Pascal},
pages = {235--246},
publisher = {Universit\'e Clermont Auvergne, Laboratoire de math\'ematiques Blaise Pascal},
volume = {29},
number = {2},
year = {2022},
doi = {10.5802/ambp.412},
language = {en},
url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.412/}
}
TY - JOUR AU - Quang-Tu Bui TI - Injectivity radius of manifolds with a Lie structure at infinity JO - Annales mathématiques Blaise Pascal PY - 2022 SP - 235 EP - 246 VL - 29 IS - 2 PB - Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal UR - https://ambp.centre-mersenne.org/articles/10.5802/ambp.412/ DO - 10.5802/ambp.412 LA - en ID - AMBP_2022__29_2_235_0 ER -
%0 Journal Article %A Quang-Tu Bui %T Injectivity radius of manifolds with a Lie structure at infinity %J Annales mathématiques Blaise Pascal %D 2022 %P 235-246 %V 29 %N 2 %I Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal %U https://ambp.centre-mersenne.org/articles/10.5802/ambp.412/ %R 10.5802/ambp.412 %G en %F AMBP_2022__29_2_235_0
Quang-Tu Bui. Injectivity radius of manifolds with a Lie structure at infinity. Annales mathématiques Blaise Pascal, Tome 29 (2022) no. 2, pp. 235-246. doi : 10.5802/ambp.412. https://ambp.centre-mersenne.org/articles/10.5802/ambp.412/
[1] On the geometry of Riemannian manifolds with a Lie structure at infinity, Int. J. Math., Volume 2004 (2004), pp. 161-193
[2] Pseudo-differential operators on manifolds with a Lie structure at infinity, Ann. Math., Volume 165 (2007) no. 3, pp. 717-747
[3] Polyhomogénéité des métriques compatibles avec une structure de Lie à l’infini le long du flot de Ricci, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 38 (2021) no. 6, pp. 1795-1840 | DOI | MR
[4] Quasi-asymptotically conical Calabi–Yau manifolds, Geom. Topol., Volume 23 (2019) no. 1, pp. 29-100
[5] Integrability of Lie brackets, Ann. Math., Volume 157 (2003) no. 2, pp. 575-620
[6] Holonomy groupoids of singular foliations, J. Differ. Geom., Volume 58 (2001) no. 3, pp. 467-500
[7] Scales, blow-up and quasimode construction, Geometric and computational spectral theory (Contemp. Math.), Volume 700, American Mathematical Society, 2017, pp. 207-266 | Zbl
[8] On manifolds with corners, Advances in geometric analysis (Advanced Lectures in Mathematics), Volume 21, International Press, 2012, pp. 225-258 | MR
[9] Differential analysis on manifolds with corners (available at http://www-math.mit.edu/ rbm/book.html)
[10] Calculus of conormal distributions on manifolds with corners, Int. Math. Res. Not., Volume 1992 (1992) no. 3, pp. 51-61
[11] Pseudodifferential operators on differential groupoids, Pac. J. Math., Volume 189 (1999) no. 1, pp. 117-152
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