Singular levels and topological invariants of Morse Bott integrable systems on surfaces

Martínez-Alfaro, José; Meza-Sarmiento, Ingrid S.; Oliveira, Regilene

doi:10.1016/j.jde.2015.09.008

Cited by 8 publications

(3 citation statements)

References 14 publications

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“…About realization of Reeb graphs, there have been a lot of studies, e.g. by Sharko [29], Martínez-Alfaro, Meza-Sarmiento and Oliveira [18,19,20], Masumoto and Saeki [21], Gelbukh [2,3,4,5], Kaluba, Marzantowicz and Silva [12], Michalak [22,23], Michalak and Marzantowicz [17], Kitazawa [13,14,15], etc. Our theorems generalize some of them.…”

Section: Realization IImentioning

confidence: 99%

Reeb spaces of smooth functions on manifolds

Saeki

2020

Preprint

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The Reeb space of a continuous function is the space of connected components of the level sets. In this paper we first prove that the Reeb space of a smooth function on a closed manifold with finitely many critical values has the structure of a finite graph without loops. We also show that an arbitrary finite graph without loops can be realized as the Reeb space of a certain smooth function on a closed manifold with finitely many critical values, where the corresponding level sets can also be preassigned. Finally, we show that a continuous map of a smooth closed connected manifold to a finite connected graph without loops that induces an epimorphism between the fundamental groups is identified with the natural quotient map to the Reeb space of a certain smooth function with finitely many critical values, up to homotopy.

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Section: Realization IImentioning

confidence: 99%

Reeb spaces of smooth functions on manifolds

Saeki

2020

Preprint

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“…In the study of integrable systems on the plane, it is possible to relate the topological equivalence of such systems to the topological equivalence of their first integrals. For instance, J. Martínez-Alfaro et al [9,10] gave complete topological invariants for the integrable Morse-Bott systems and for their first integrals, the so called Morse-Bott functions. In a similar line, K.R.…”

Section: Introductionmentioning

confidence: 99%

Topological equivalence of submersion functions and topological equivalence of their foliations on the plane: the linear-like case

Braun¹,

Meza-Sarmiento²

2022

Preprint

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Let f, g : R 2 → R be two submersion functions and F (f ) and F (g) be the regular foliations of R 2 whose leaves are the connected components of the levels sets of f and g, respectively. The topological equivalence of f and g implies the topological equivalence of F (f ) and F (g), but the converse is not true, in general. In this paper, we introduce the class of linear-like submersion functions, which is wide enough in order to contain non-trivial behaviors, and provide conditions for the validity of the converse implication for functions inside this class. Our results lead us to a complete topological invariant for topological equivalence in a certain subclass of linear-like submersion functions.

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“…These foliations, in particular, foliations by level sets of Morse-Bott functions, play an important role in Hamiltonian dynamics and Poisson geometry, see e.g. [18], [19], [20], [21], [22].…”

Section: Introductionmentioning

confidence: 99%

Diffeomorphisms preserving Morse–Bott functions

Khohliyk

Maksymenko

2020

Indagationes Mathematicae

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Let f : M → R be a Morse-Bott function on a closed manifold M , so the set Σ f of its critical points is a closed submanifold whose connected components may have distinct dimensions. Denote by S(f ) = {h ∈ D(M ) | f • h = f } the group of diffeomorphisms of M preserving f and let D(Σ f ) be the group of diffeomorphisms of Σ f . We prove that the "restriction to Σ f " map ρ : S(f ) → D(Σ f ), ρ(h) = h| Σ f , is a locally trivial fibration over its image ρ(S(f )).2010 Mathematics Subject Classification. 57R30, 57R45, 53B15,

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Singular levels and topological invariants of Morse Bott integrable systems on surfaces

Cited by 8 publications

References 14 publications

Reeb spaces of smooth functions on manifolds

Reeb spaces of smooth functions on manifolds

Topological equivalence of submersion functions and topological equivalence of their foliations on the plane: the linear-like case

Diffeomorphisms preserving Morse–Bott functions

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