Morse-Bott functions with two critical values on a surface

Gelbukh, Irina

doi:10.21136/cmj.2021.0125-20

Cited by 8 publications

(3 citation statements)

References 15 publications

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“…[1] and [2] are related studies where Morse-Bott functions on closed surfaces and general closed manifolds are considered. Studies respecting preimages of regular values are new important problems and regarded as variants of Main Problem.…”

Section: Proofs Of Main Theorems and Remarksmentioning

confidence: 99%

Realization problems of graphs as Reeb graphs of Morse functions with prescribed preimages

Kitazawa¹

2021

Preprint

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The present paper presents some new results on so-called realization problems of graphs as Reeb graphs of Morse functions having nice structures. More precisely, we assign (types of) closed and smooth manifolds of suitable classes to edges and construct good functions whose preimages are as prescribed.The Reeb graph of a smooth function is a graph which is the quotient space of the manifold of the domain obtained by the following equivalence relation; two points in the manifold is equivalent if and only if they are in a same connected component of a same preimage. Note that in considerable cases such as cases where there exists finitely many singular values.Reeb graphs represent the manifolds of the domains compactly in general. They are not only fundamental and important tools in geometry using differentiable functions and maps as fundamental tools, but also in applied mathematics or applications of mathematics such as visualizations. Realization problems have been also fundamental and important.

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Section: Proofs Of Main Theorems and Remarksmentioning

confidence: 99%

Realization problems of graphs as Reeb graphs of Morse functions with prescribed preimages

Kitazawa¹

2021

Preprint

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show abstract

“…In particular topological classification of Morse-Bott functions on oriented surfaces were studied by E. B. Batista et al [2] and by J. Martínez-Alfaro et al [16]. I. Gelbukh classified Morse-Bott functions with the only 2 nondegenerate critical points on manifolds [4]. Topological properties foliations with Morse-Bott singularities of codimension-1 were investigated by B. Scárdua and J. Seade [19,20], and homotopy properties of diffeomorphisms preserving Morse-Bott foliations on lens spaces were studied by S. Maksymenko [14,15].…”

Section: Introductionmentioning

confidence: 99%

“…We can also describe functions listed above via so-called Kronrod-Reeb graphs, see for example [2,4]. A Kronrod-Reeb graph of a function from ( 1)-( 4) is a path graph, and a circle graph in the case (5).…”

Section: Introductionmentioning

confidence: 99%