Moser's theorem on manifolds with corners
Authors:
Martins Bruveris, Peter W. Michor, Adam Parusiński and Armin Rainer
Journal:
Proc. Amer. Math. Soc. 146 (2018), 4889-4897
MSC (2010):
Primary 53C65, 58A10
DOI:
https://doi.org/10.1090/proc/14130
Published electronically:
August 10, 2018
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: Moser's theorem states that the diffeomorphism group of a compact manifold acts transitively on the space of all smooth positive densities with fixed volume. Here we describe the extension of this result to manifolds with corners. In particular, we obtain Moser's theorem on simplices. The proof is based on Banyaga's paper (1974), where Moser's theorem is proven for manifolds with boundary. A cohomological interpretation of Banyaga's operator is given, which allows a proof of Lefschetz duality using differential forms.
- [1] Augustin Banyaga, Formes-volume sur les variétés à bord, Enseignement Math. (2) 20 (1974), 127–131 (French). MR 0358649
- [2] M. Bruveris, P. W. Michor, and A. Rainer, Determination of all diffeomorphism invariant tensor fields on the space of smooth positive densities on a compact manifold with corners, in preparation, 2018.
- [3] B. Conrad, Stokes' theorem with corners. Math 396 Handout, Stanford University. http://math.stanford.edu/˜conrad/diffgeomPage/handouts/stokescorners.pdf.
- [4] Gyula Csató, Bernard Dacorogna, and Olivier Kneuss, The pullback equation for differential forms, Progress in Nonlinear Differential Equations and their Applications, vol. 83, Birkhäuser/Springer, New York, 2012. MR 2883631
- [5] Bernard Dacorogna and Jürgen Moser, On a partial differential equation involving the Jacobian determinant, Ann. Inst. H. Poincaré Anal. Non Linéaire 7 (1990), no. 1, 1–26 (English, with French summary). MR 1046081, https://doi.org/10.1016/S0294-1449(16)30307-9
- [6] Georges de Rham, La théorie des formes différentielles extérieures et l’homologie des variétés différentiables, Rend. Mat. e Appl. (5) 20 (1961), 105–146 (French). MR 0139102
- [7] A. Douady and L. Hérault. Arrondissement des variétés à coins. Comment. Math. Helv., 48 (1973), 484-491. Appendice à A. Borel and J.-P. Serre: Corners and arithmetic groups.
- [8] R. E. Greene and K. Shiohama, Diffeomorphisms and volume-preserving embeddings of noncompact manifolds, Trans. Amer. Math. Soc. 255 (1979), 403–414. MR 542888, https://doi.org/10.1090/S0002-9947-1979-0542888-3
- [9] S. Lefschetz, Transformations of manifolds with a boundary, Proc. Natl. Acad. Sci. USA, 12 (1926), 737-739.
- [10] R. B. Melrose, Differential Analysis on Manifolds with Corners. 1996. http://www-math.mit.edu/˜rbm/book.html.
- [11] Peter W. Michor, Manifolds of differentiable mappings, Shiva Mathematics Series, vol. 3, Shiva Publishing Ltd., Nantwich, 1980. MR 583436
- [12] Peter W. Michor, Topics in differential geometry, Graduate Studies in Mathematics, vol. 93, American Mathematical Society, Providence, RI, 2008. MR 2428390
- [13] Jürgen Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 286–294. MR 0182927, https://doi.org/10.1090/S0002-9947-1965-0182927-5
- [14] Guillaume Valette, A Lefschetz duality for intersection homology, Geom. Dedicata 169 (2014), 283–299. MR 3175250, https://doi.org/10.1007/s10711-013-9856-z
- [15] Guillaume Valette, Stokes’ formula for stratified forms, Ann. Polon. Math. 114 (2015), no. 3, 197–206. MR 3371768, https://doi.org/10.4064/ap114-3-1
Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 53C65, 58A10
Retrieve articles in all journals with MSC (2010): 53C65, 58A10
Additional Information
Martins Bruveris
Affiliation:
Department of Mathematics, Brunel University London, Uxbridge, UB8 3PH, United Kingdom
Email:
martins.bruveris@brunel.ac.uk
Peter W. Michor
Affiliation:
Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria
Email:
peter.michor@univie.ac.at
Adam Parusiński
Affiliation:
Département de Mathématiques, Université Nice Sophia Antipolis, CNRS, LJAD, UMR 7351, 06108 Nice, France
Email:
adam.parusinski@unice.fr
Armin Rainer
Affiliation:
Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria
Email:
armin.rainer@univie.ac.at
DOI:
https://doi.org/10.1090/proc/14130
Keywords:
Manifolds with corners,
Moser's theorem,
Stokes's theorem
Received by editor(s):
January 3, 2017
Received by editor(s) in revised form:
February 12, 2018
Published electronically:
August 10, 2018
Additional Notes:
This work was supported by the Austrian Science Fund (FWF), Grant P 26735-N25, and by a BRIEF Award from Brunel University London.
Communicated by:
Alexander Braverman
Article copyright:
© Copyright 2018
American Mathematical Society
