On Reeb graphs induced from smooth functions on closed or open manifolds

Kitazawa, Naoki

doi:10.48550/arxiv.1908.04340

Cited by 6 publications

(5 citation statements)

References 10 publications

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“…It is also remarkable that we construct smooth functions which are not real algebraic or real analytic functions in several scenes in the study. Important functions in [9,26] show explicit cases. Remark 2.…”

Section: As Important Tools We Consider the Partial Derivatives Of Th...mentioning

confidence: 99%

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Real algebraic maps of non-positive codimensions of certain general types

Kitazawa¹

2023

Preprint

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We present new real algebraic maps of non-positive codimensions with images of some general types. We also investigate the compositions of the maps with canonical projections.This gives a new answer to construction of explicit real algebraic maps of negative codimensions with explicit information on preimages. Thanks to some celebrated theory of Nash, smooth closed manifolds are smooth real algebraic manifolds and the zero sets of some real polynomials. In considerable cases, we can approximate smooth functions or more generally, maps, by real algebraic ones. It is in general difficult to have explicit examples. We have constructed some previously and we have ones with new topological structures.

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Section: As Important Tools We Consider the Partial Derivatives Of Th...mentioning

confidence: 99%

“…It asks whether we can respect singularities of the functions and preimages in addition. For related studies, see [8,9] as pioneering studies by the author for example. [26] is also a related study based on our informal discussions on [8].…”

Section: As Important Tools We Consider the Partial Derivatives Of Th...mentioning

confidence: 99%

Real algebraic maps of non-positive codimensions of certain general types

Kitazawa¹

2023

Preprint

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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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“…These study cases for smooth functions on closed surfaces and Morse functions such that each connected component of each preimage containing no singular points is always a sphere for example. In [5,6,9,10], the author has studied cases where such connected components are general manifolds with mild conditions on singularities of the functions for example. [21] appears as a paper motivated by [5] and through related informal discussions by us.…”

Section: Introductionmentioning

confidence: 99%

Real algebraic functions on closed manifolds whose Reeb spaces are given graphs

Kitazawa¹

2023

Preprint

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In our paper, we construct a real-algebraic function on some closed manifold whose Reeb (Kronrod-Reeb) graph is a graph respecting some algebraic domain: a graph for this is called a Poincaré-Reeb graph.The Reeb graph of a smooth function is defined as a natural graph which is the quotient space of the manifold of the domain under a natural equivalence relation for some wide and nice class of smooth functions. The vertex set is defined as the set of all connected components containing some singular points of the function: a singular point of a smooth function is a point where the differential vanishes. Morse-Bott functions give very specific cases. The relation is to contract each connected component of each preimage to a point.Sharko has asked a natural and important problem: can we construct a nice smooth function whose Reeb graph is a given graph? Explicit answers have been given first by Masumoto-Saeki in a generalized manner for closed surfaces. After that various answers have been presented by various researchers and most of them are essentially for functions on closed surfaces and Morse functions such that connected components of preimages containing no singular points are spheres. Recently the author has also considered questions and solved in cases the preimages are general manifolds.

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On Reeb graphs induced from smooth functions on closed or open manifolds

Cited by 6 publications

References 10 publications

Real algebraic maps of non-positive codimensions of certain general types

Real algebraic maps of non-positive codimensions of certain general types

Reeb spaces of smooth functions on manifolds

Real algebraic functions on closed manifolds whose Reeb spaces are given graphs

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