Abstract
Let \(F: \mathbb {R}^n\rightarrow \mathbb {R}^n\) be a \(C^{\infty }\) map such that DF(x) is invertible for every \(x\in \mathbb {R}^n\). Although being a local diffeomorphism, F is not necessarily globally injective if \(n\ge 2\). Finding additional assumptions implying the global injectivity of F for \(n\ge 2\) is object of intense study in several areas of mathematics. In this paper, we revisit some assumptions and relations between them in the bi-dimensional case and discuss the natural higher dimensional situation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bass, H., Connell, E.H., Wright, D.: The Jacobian conjecture: reduction of degree and formal expansion of the inverse. Bull. Amer. Math. Soc. 7, 287–330 (1982)
Białynicki-Birula, A., Rosenlicht, M.: Injective morphisms of real algebraic varieties. Proc. Amer. Math. Soc. 1(3), 200–203 (1962)
Braun, F., Oréfice-Okamoto, B.: On polynomial submersions of degree \(4\) and the real Jacobian conjecture in \(\mathbb{R} ^2\). J. Math. Anal. Appl. 443, 688–706 (2016)
Braun, F., dos Santos Filho, J.R.: The real Jacobian conjecture on \(\cal{R} ^2\) is true when one of the components has degree \(3\). Discrete Contin. Dyn. Syst. 2(6), 75–87 (2010)
Braun, F., dos Santos Filho, J.R.: A correction to the paper “Injective mappings and solvable vector fields of Euclidean spaces’’. Topol. Appl. 204, 256–265 (2016)
Camacho, C., Lins Neto, A.: Geometric Theory of Foliations. Translated from the Portuguese by Sue E. Goodman. Birkhäuser Boston, Inc., Boston, MA (1985)
van den Essen, A.: Polynomial automorphisms and the Jacobian conjecture. Progress in Mathematics, vol. 190. Birkhäuser, Basel (2000)
Duistermaat, J.J., Hörmander, L.: Fourier integral operators. II. Acta Math. 128, 183–269 (1972)
Fernandes, A., Gutierrez, C., Rabanal, R.: Global asymptotic stability for differentiable vector fields of \(\cal{R} ^2\). J. Differential Equations 206, 470–482 (2004)
Gutierrez, C.: A solution to the bidimensional global asymptotic stability conjecture. Ann. Inst. H. Poincaré Anal. Non Linéaire 12, 627–671 (1995)
Gutierrez, C., Jarque, X., Llibre, J., Teixeira, M.A.: Global injectivity of \(C^1\) maps of the real plane, inseparable leaves and the Palais–Smale condition. Canad. Math. Bull. 50, 377–389 (2007)
Gutierrez, C., Maquera, C.: Foliations and polynomial diffeomorphims of \(\cal{R} ^3\). Math. Z. 262, 613–626 (2009)
Jelonek, Z.: Geometry of real polynomial mappings. Math. Z. 239, 321–333 (2002)
Krasiński, T., Spodzieja, S.: On linear differential operators related to the \(n\)-dimensional Jacobian conjecture. In: Real Algebraic Geometry (Rennes, 1991), pp. 308–315. Lecture Notes in Math., 1524. Springer, Berlin (1992)
Li, G., Xavier, F.: Non-positive curvature and global invertibility of maps. J. Geom. Anal. 24, 1181–1200 (2014)
Pinchuk, S.: A counterexample to the strong real Jacobian conjecture. Math. Z. 217, 1–4 (1994)
Plastock, R.: Homeomorphisms between Banach spaces. Trans. Amer. Math. Soc. 200, 169–183 (1974)
Stein, Y.: On linear differential operators related to the Jacobian conjecture. J. Pure Appl. Algebra 57, 175–186 (1989)
Stein, Y.: Linear differential operators related to the Jacobian conjecture have a closed image. J. Anal. Math. 54, 237–245 (1990)
Acknowledgements
We thank the referee for important comments that improved the presentation of the paper. The first author was partially supported by FAPESP Grants 10/11323-7, 2014/ 26149-3, and 2017/00136-0. The second and third author were partially supported by FAPESP Grants 07/08231-0 and 07/06896-5, respectively.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Braun, F., dos Santos Filho, J.R. & Teixeira, M.A. Foliations, solvability, and global injectivity. Arch. Math. 119, 649–665 (2022). https://doi.org/10.1007/s00013-022-01789-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-022-01789-z