2000 character limit reached
Controllability and diffeomorphism groups on manifolds with boundary (2403.12742v2)
Published 19 Mar 2024 in math.DG and math.GR
Abstract: In this article we consider diffeomorphism groups of manifolds with smooth boundary. We show that the diffeomorphism groups of the manifold and its boundary fit into a short exact sequence which admits local sections. In other words, they form an infinite-dimensional fibre bundle. Manifolds with boundary are of interest in numerical analysis and with a view towards applications in machine learning we establish controllability results for families of vector fields. This generalises older results due to Agrachev and Caponigro in the boundary-less case. Our results show in particular that the diffeomorphism group of a manifold with smooth boundary is generated by the image of the exponential map.
- Andrei A Agrachev and Marco Caponigro “Controllability on the group of diffeomorphisms” In Annales de l’Institut Henri Poincaré C, Analyse non linéaire 26.6, 2009, pp. 2503–2509 Elsevier
- Andrei A. Agrachev and Yuri L. Sachkov “Control theory from the geometric viewpoint” Control Theory and Optimization, II 87, Encyclopaedia of Mathematical Sciences Springer-Verlag, Berlin, 2004, pp. xiv+412 DOI: 10.1007/978-3-662-06404-7
- Habib Amiri, Helge Glöckner and Alexander Schmeding “Lie groupoids of mappings taking values in a Lie groupoid.” In Arch. Math., Brno 56.5, 2020, pp. 307–356 DOI: 10.5817/AM2020-5-307
- Sylvain Arguillère “Sub-Riemannian geometry and geodesics in Banach manifolds” In J. Geom. Anal. 30.3, 2020, pp. 2897–2938 DOI: 10.1007/s12220-019-00184-5
- “Sub-Riemannian structures on groups of diffeomorphisms” In J. Inst. Math. Jussieu 16.4, 2017, pp. 745–785 DOI: 10.1017/S1474748015000249
- Augustin Banyaga “The structure of classical diffeomorphism groups” 400, Math. Appl., Dordr. Dordrecht: Kluwer Academic Publishers, 1997
- Martin Bauer, Philipp Harms and Peter W. Michor “Regularity and completeness of half-Lie groups”, 2023 arXiv:2302.01631 [math.DG]
- “Deep neural networks on diffeomorphism groups for optimal shape reparameterization” In BIT Numerical Mathematics 63.50, 2023 DOI: 10.1007/s10543-023-00989-5
- Wei-Liang Chow “Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung” In Math. Ann. 117, 1939, pp. 98–105
- “Groups of diffeomorphisms and the motion of an incompressible fluid” In Ann. Math. (2) 92, 1970, pp. 102–163 DOI: 10.2307/1970699
- “Learning on Manifolds: Universal Approximations Properties using Geometric Controllability Conditions for Neural ODEs”, 2023 arXiv:2305.08849 [math.OC]
- Helge Glöckner “Diffeomorphism groups of convex polytopes” In J. Convex Anal. 30.1, 2023, pp. 343–358 URL: www.heldermann.de/JCA/JCA30/JCA301/jca30019.htm#jca301
- Helge Glöckner, Erlend Grong and Alexander Schmeding “Boundary values of diffeomorphisms of simple polytopes, and controllability” in preparation
- Stefan Haller, Tomasz Rybicki and Josef Teichmann “Smooth perfectness for the group of diffeomorphisms” In J. Geom. Mech. 5.3, 2013, pp. 281–294 DOI: 10.3934/jgm.2013.5.281
- Maximilian Hanusch “A 𝒞ksuperscript𝒞𝑘\mathcal{C}^{k}caligraphic_C start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT-Seeley-extension-theorem for Bastiani’s differential calculus” In Can. J. Math. 75.1, 2023, pp. 170–201 DOI: 10.4153/S0008414X21000596
- Morris W. Hirsch “Differential topology” 33, Grad. Texts Math. Springer, Cham, 1976 DOI: 10.1007/978-1-4684-9449-5
- Boris Khesin, Gerard Misiołek and Alexander Shnirelman “Geometric hydrodynamics in open problems” Id/No 15 In Arch. Ration. Mech. Anal. 247.2, 2023, pp. 43 DOI: 10.1007/s00205-023-01848-x
- Andreas Kriegl and Peter W. Michor “The convenient setting of global analysis” 53, Math. Surv. Monogr. Providence, RI: American Mathematical Society, 1997 URL: www.ams.org/online_bks/surv53/
- John M. Lee “Introduction to smooth manifolds” 218, Grad. Texts Math. New York, NY: Springer, 2013 DOI: 10.1007/978-1-4419-9982-5
- A M Lukatskii “Finite Generation of Groups of Diffeomorphisms” In Russian Mathematical Surveys 33.1, 1978, pp. 207 DOI: 10.1070/RM1978v033n01ABEH002248
- Juan Margalef-Roig and Enrique Outerelo Dominguez “Differential topology. Transl. from the Spanish” 173, North-Holland Math. Stud. Amsterdam etc.: North-Holland, 1992
- Benjamin McKay “Sussmann’s orbit theorem and maps” In Differ. Geom. Appl. 25.3, 2007, pp. 277–280 DOI: 10.1016/j.difgeo.2006.11.005
- Florian Méhats, Yannick Privat and Mario Sigalotti “On the controllability of quantum transport in an electronic nanostructure” In SIAM J. Appl. Math. 74.6, 2014, pp. 1870–1894 DOI: 10.1137/130939328
- Peter W. Michor “Manifolds of differentiable mappings” 3, Shiva Mathematics Series Shiva Publishing Ltd., Nantwich, 1980, pp. iv+158
- Peter W. Michor “Topics in differential geometry” 93, Grad. Stud. Math. Providence, RI: American Mathematical Society (AMS), 2008
- P.K. Rashevskiĭ “On the connectability of two arbitrary points of a totally nonholonomic space by an admissible curve” In Uchen. Zap. Mosk. Ped. Inst. Ser. Fiz.-Mat. Nauk 3.2, 1938, pp. 83–94
- Tomasz Rybicki “Commutators of diffeomorphisms of a manifold with boundary” In Ann. Pol. Math. 68.3, 1998, pp. 199–210 DOI: 10.4064/ap-68-3-199-210
- Alexander Schmeding “An Introduction to Infinite-Dimensional Differential Geometry”, Cambridge Studies in Advanced Mathematics Cambridge University Press, 2022 DOI: 10.1017/9781009091251
- “(Re)constructing Lie groupoids from their bisections and applications to prequantisation” In Differ. Geom. Appl. 49, 2016, pp. 227–276 DOI: 10.1016/j.difgeo.2016.07.009
- R.T. Seeley “Extension of C∞superscript𝐶C^{\infty}italic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT functions defined in a half space” In Proc. Amer. Math. Soc. 15, 1964, pp. 625–626 DOI: 10.2307/2034761
- Héctor J Sussmann “Orbits of families of vector fields and integrability of distributions” In Transactions of the American Mathematical Society 180, 1973, pp. 171–188
- William Thurston “Foliations and groups of diffeomorphisms” In Collected works of William P. Thurston with commentary. Vol. I. Foliations, surfaces and differential geometry Amer. Math. Soc., Providence, RI, 2022, pp. 105–108
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.