A note on parallelizable dynamical systems

Josiney A. Souza; Tiago A. Pacifico; Hélio V. M. Tozatti

doi:10.3934/era.2017.24.007

Article Contents

2017, Volume 24: 64-67. Doi: 10.3934/era.2017.24.007

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A note on parallelizable dynamical systems

1.
Departamento de Matemática, Universidade Estadual de Maringá, Maringá-PR, Brasil
2.
Departamento de Matemática, Universidade Tecnolgica Federal do Paraná, Campo Mourõ-PR, Brasil

Received: February 23, 2017

Published: August 2017

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Abstract

Hájek [3] showed that a dynamical system on a Tychonoff space with paracompact orbit space is parallelizable if and only if its corresponding bundle is a locally trivial fiber bundle with fiber $\mathbb{R}$. The present paper provides an enhancement for this classical theorem by omitting all topological hypotheses.

Keywords:

Parallelizability,
principal bundles.

Mathematics Subject Classification: 37C75, 34C27, 34D05, 93D05.

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References

[1]	N. P. Bhatia and G. P. Szegö, Stability Theory of Dynamical Systems, Springer-Verlag, Berlin, 1970.
[2]	J. Dugundji and H. A. Antosiewicz, Parallelizable flows and Lyapunov's second method, Ann. of Math., 73 (1961), 543-555. doi: 10.2307/1970316.
[3]	O. Hájek, Parallelizability revisited, Proc. Amer. Math. Soc., 27 (1971), 77-84. doi: 10.1090/S0002-9939-1971-0271925-7.
[4]	D. Husemoller, Fibre Bundles, Graduate Texts in Mathematics, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4757-2261-1.
[5]	N. Steenrod, The Topology of Fibre Bundles, Princeton University Press, Princeton, 1951.