arXiv subjects

Naoki Kitazawa

Publications and source records attributed to Naoki Kitazawa.

At least 19 recordsLinked to original sources

arXiv2026

On Reeb spaces of non-proper smooth functions which are $1$-dimensional CW complexes

We discuss Reeb spaces of non-proper smooth functions. (Non-)proper functions are maps between topological spaces the preimages of compact sets by which are compact (resp. may not be compact). Their Reeb spaces are the spaces of connected components of their level sets and with the natural quotient topologies. In proper cases, explicit theory are developing since the 1950s. This is closely related to Morse(-Bott) function theory and in such cases we have graphs naturally. Recently, related general topological studies with combinatorial ones are actively developing, mainly due to Gelbukh and Saeki, and they are at most $1$-dimensional in tame cases. We extend a theorem by Saeki in the 2020s: the Reeb space of a smooth function on a closed manifold is naturally a graph if and only if its critical value set is finite. The author has been a pioneer of the non-proper case and previously investigated examples.

math.GM↗

arXiv2026

Reeb spaces of 1st derivatives of proper submersions of certain classes

We study the (canonical) 1st derivatives of {\it proper} submersions represented as height functions and belonging to a certain class: a proper map means a map the preimage of a compact set by which is always compact. We investigate their {\it Reeb spaces}. They are the quotient spaces defined by the equivalence relations on the manifolds of the domains where we identify two points in a same connected component of a same level set of them. They have been important since the establishment of theory of Morse functions, in the 20th century, They are in certain tame situations $0$- or $1$-dimensional and graphs naturally. Related facts have been shown by Gelbukh and Saeki in the 2020s for certain proper smooth real-valued functions. In non-proper cases, even related explicit theory has been difficult, except some previously given case of the author. Our study is on a new related case.

math.DG↗

arXiv2026

Height functions on products of spheres and associated Reeb digraphs and level sets

Height functions are fundamental and important objects and tools in mathematics, especially in geometry such as differential topology and differential geometry and some related singularity theory of differentiable maps. Our interest, especially interest of the author, lies in obtaining explicit lists of such functions. Recently, as information on so-called higher degrees, he is also interested in their naturally defined 1st derivatives. We consider natural maps on products of spheres which are variants of specific cases of so-called moment maps on toric symplectic manifolds. We also generalize cases of the canonical projections of the unit spheres. This is a further result on related previous study of the author. We use Reeb graphs, graphs being natural quotient spaces of manifolds of the domains of nice functions such as Morse-Bott functions, and consisting of connected components of level sets. They are fundamental tools and objects since the last century.

math.DG↗

arXiv2026

Regions represented as foliated forms and natural smooth maps onto them

The author is interested in regions surrounded by hypersurfaces and natural smooth maps onto them respecting the canonical projections of the unit spheres and so-called special generic maps and moment maps, more generally. We consider situations where these regions are foliated via 1-dimensional families of functions and their zero sets (smoothly). We including the author are also interested in explicit and nice functions obtained by composing the canonical projections and their topological or combinatorial properties. This is of singularity theory of differentiable maps and applications to differential topology and various real geometry. Explicitly, here, as a new challenge, we discuss the 1st derivative of such a function and critical sets of this.

math.AG↗

arXiv2026

Compactifying real analytic functions and resulting Reeb spaces

We formulate compactifications of continuous maps naturally. We consider real analytic functions mainly. We are interested in topological properties and combinatorial ones of explicit resulting maps. For understanding them, we use their Reeb spaces, being quotient spaces of the spaces of the domains of the functions and defined by the equivalence relation identifying two points in same components of their level sets. They are known to be $0$- or $1$-dimensional (metrizable) cell-complexes, in our situations or more general certain tame cases. Reeb spaces have been important in understanding topological properties and combinatorial ones of functions and spaces roughly, since the last century. These compactifications have been explicitly studied by the author previously and recently. We have obtained real algebraic functions whose Reeb spaces are not so complicated and which seem to be of most natural and simplest. We present new discussions and examples.

math.AG↗

arXiv2026

Representations of Reeb spaces via simplified graphs and examples

Reeb spaces of continuous real-valued functions on topological spaces are fundamental and strong tools in investigating the spaces. The Reeb space is the natural quotient space of the space of the domain represented by connected components of its level sets. They have appeared in theory of Morse functions in the last century and as important topological objects, they are shown to be graphs for tame functions on (compact) manifolds such as Morse(-Bott) functions and naturally generalized ones. Related general theory develops actively, recently, mainly by Gelbukh and Saeki. For nice Haudorff spaces and continuous functions there, they are "$1$-dimensional". We concentrate on Reeb spaces which are not CW complexes and study their representations by graphs and nice examples. Reconstructing nice smooth functions with given Reeb graphs is of related studies and pioneered by Sharko and followed by Masumoto, Michalak, Saeki, and so on. The author has also contributed to it.

math.AG↗

arXiv2026

Morse functions with regular level sets consisting of $2$-dimensional spheres, $2$-dimensional tori, or Klein Bottles

In this paper, we study Morse functions with regular level sets consisting of spheres, tori, or Klein Bottles on $3$-dimensional closed manifolds. We characterize $3$-dimensional manifolds represented by connected sums each of whose summands is the product $S^1 \times S^2$ of the circle $S^1$ and the sphere $S^2$, lens spaces, or non-orientable closed and connected manifolds of genus $1$ by a certain subclass of such Morse functions. This is a kind of extensions of the orientable case, by Saeki, in 2006. This is a variant of its extension by the author for $3$-dimensional orientable manifolds represented by connected sums each of whose summands is the product $S^1 \times S^2$, lens spaces, or torus bundles over $S^1$ by a certain class of Morse-Bott functions. We also classify Morse functions with given regular level sets consisting of $S^2$, $S^1 \times S^1$, or Klein Bottles in a certain sense, generalizing some previous work by the author.

math.GT↗

arXiv2026

A note on asymptotic behaviors and topological properties of two smooth real-valued functions and several graphs associated to them

This is a note on the graphs of two smooth real-valued functions in the plane with no intersection and the natural map onto the region surrounded by them with the canonical projection to the line composed, yielding its Reeb space. The Reeb space of a real-valued function on a topological space is the set of all connected components of all level sets and topologized naturally. Such spaces have been fundamental and strong tools in theory of Morse functions and its generalization and variants, since the former half of the 20th century. They are graphs for tame functions such as Morse(-Bott) functions. The author has launched and has been studying this problem since 2020s, interested in Reeb spaces of smooth or non-analytic non-proper functions. For smooth closed manifolds and nice compact spaces, topological properties and combinatorial ones on Reeb spaces have been investigated by Gelbukh, Saeki, and so on.

math.GN↗

arXiv2026

Reeb spaces of smooth functions associated to globally similar graphs of smooth functions

Previously, we have investigated a natural smooth map onto the region surrounded by the graphs of two smooth real-valued functions in the plane converging to a same value or diverges to $+\infty$ or $-\infty$ simultaneously, at each infinity, and topological properties and combinatorial ones of its composition with the canonical projection. Here, we consider smooth functions with congruent or globally similar graphs instead. Here, the Reeb space of a smooth function on a manifold with no boundary is fundamental and important. This is the naturally topologized quotient space of the manifold, consisting of all connected components (contours) of the function and is a graph under a certain nice situation. Related studies also related to the present study were started due to interest of the author in theory of Reeb spaces of non-proper functions. For proper functions, in 2020s related studies have developed mainly due to Gelbukh and Saeki.

math.GN↗

arXiv2026

Fundamental examples of Reeb spaces of smooth functions defined from two graphs of smooth functions with same asymptotic behaviors

Reeb spaces of (continuous) real-valued functions on (nice) topological spaces are the spaces whose underlying sets consist of all connected components (contours) of their level sets and seen naturally as quotient spaces of the spaces. They are "$1$-dimensional" spaces in various nice cases. They are graphs or graphs with ends for smooth function cases with nice singularities and behaviors. Reeb spaces have been fundamental and important in theory of Morse functions and more general smooth functions and applications to geometry, since the 20th century. We present Reeb spaces homeomorphic to infinite graphs (with ends) for functions on non-compact manifolds with no boundary. This paper is a note on cases previously obtained by the author. More explicitly, we consider a natural smooth map onto the region surrounded by the graphs of two smooth real-valued functions in the plane and its composition with the canonical projection.

math.GN↗

arXiv2026

On real algebraic realization of round fold maps of codimension $-1$

The canonical projections of the unit spheres are generalized to special generic maps and round fold maps, for example. They are generalizations from the viewpoint of singularity theory of differentiable maps and these maps restrict the topologies and the differentiable structures of the manifolds. We are concerned with round fold maps, defined as smooth maps locally represented as the product map of a Morse function and the identity map on a smooth manifold, and maps with singular value sets being concentric spheres. A bit different from differential topology, we are concerned with real algebraic geometric aspects of these maps. We discuss real algebraic realization of round fold maps of codimension $-1$ as our new work. Real algebraic realization of these maps is of fundamental and important studies in real algebraic geometry and a new study recently developing mainly due to the author.

math.AG↗

arXiv2026

Regions surrounded by parabolas in the plane and trees representing their shapes respecting their natural projection to the line

The author has been interested in regions surrounded by real algebraic curves of degree $1$ or $2$ in the plane. The author is mainly interested in their shapes and combinatorics. This is a fundamental and natural problem in mathematics being also elementary and connected to various fields. The shapes are understood via graphs the regions collapsing to respecting the canonical projection onto the 1st component. Our main result is the following: each tree is realized by regions surrounded by parabolas of two types, here. Related studies are elementary and interesting and surprisingly, this explicit field is started very recently, by Bodin, Popescu-Pampu and Sorea in the 2020s. After that, this is developing, due to the author. The author also investigates this motivated by studies on explicit construction of real algebraic maps onto the regions locally so-called moment maps: this comes from singularity theory of differentiable maps and real algebraic geometry.

math.AG↗

arXiv2026

Reeb spaces of functions being analytic on dense subsets and their graph structures

Reeb spaces of real-valued functions on manifolds are the spaces of all connected components (contours) of level sets and endowed with the natural quotient topology. They have been fundamental and strong tools in investigating manifolds via smooth functions with mild critical points since the birth of fundamental theory of Morse functions in the 20th century. We are concerned with topologies and combinatorics of them. Following an explicit note on explicit Reeb spaces of explicit functions which are real analytic (on dense sets) and seem to be simplest and most fundamental, edited by the author himself. We investigate other construction of examples of such functions and their Reeb spaces. Reeb spaces are naturally graphs in considerable cases and as another work, we also discuss natural definitions of vertices for them.

math.GN↗

arXiv2026

A note on Reeb spaces of some explicit real analytic functions

Reeb spaces of smooth functions are fundamental and strong tools in understanding manifolds via smooth functions with mild critical points. They are defined as the natural spaces of all connected components of level sets. They are also important objects in related studies. Realization of graphs as Reeb spaces of smooth functions of certain nice classes is of such studies. In this paper, we present Reeb spaces of explicit real analytic functions which are not finite graphs. Related problems were started by Sharko, in 2006, who has studied smooth functions with critical points represented by certain elementary polynomials, and followed by a study of Masumoto and Saeki, which is on smooth functions on closed surfaces under an extended situation, and a study of Michalak, which is on Morse functions on closed manifolds. The author has contributed to this by respecting topologies of level sets, and real algebraic construction.

math.GM↗

arXiv2026

Regions surrounded by cylinders of real algebraic manifolds and natural decompositions

The author has been interested in regions surrounded by cylinders of real algebraic hypersurfaces and their shapes and polynomials associated to them. Here, we formulate and investigate natural decompositions into such cylinders of real algebraic hypersurfaces. Especially, intersections of these cylinders of real algebraic hypersurfaces, which give important information on regions, are investigated via singularity theory. This is a kind of natural problems on real geometry. This also comes from construction of explicit real algebraic maps onto explicit regions in real affine spaces on real algebraic manifolds. More generally, we are interested in difficulty in explicit construction of real algebraic objects, where existence and approximation has been well-known, since pioneering studies by Nash and Tognoli, in the latter half of 20th century. This also comes from interest in singularity theory of differentiable, smooth or real algebraic functions and maps, especially, explicit construction.

math.AG↗

arXiv2025

Regions surrounded by cylinders of circles of fixed radii and exposition of their shapes by natural graphs

We investigate regions formed by cylinders of circles of fixed radii. We investigate graphs obtained by collapsing each level set of the functions represented by the natural projections of them to the $1$-dimensional line. Some specific trees obtained in simple ways from so-called balanced trees are shown to be realized as such graphs. Related studies on regions in the Euclidean plane surrounded by real algebraic curves are presented by several researchers. One of pioneering studies is presented by Bodin, Popescu-Pampu and Sorea in 2022--3 as an elementary and surprisingly new study. The author has been interested in related studies and also in constructing natural and explicit real algebraic maps onto such regions, generalizing the canonical projections of the unit spheres. Such studies in real algebraic geometry, different from theory of existence in the last century, mainly studied by Nash and Tognoli, are remarked.

math.AG↗

arXiv2025

Regions surrounded by circles whose Poincar\'e-Reeb graphs are trees

Regions in the Euclidean plane surrounded by circles are fundamental geometric and combinatorial objects. Related studies have been done and we cannot explain them precisely, or roughly, well. We study such regions whose Poincar\'e-Reeb graphs are trees and investigate the trees obtained by a certain inductive rule from a disk in the plane. The Poincar\'e-Reeb graph of such a region is a graph whose underlying set is the set of all components of level sets of the restriction of the canonical projection to the closure and whose vertices are points corresponding to the components containing {\it singular} points. Related studies were started by the author, motivated by importance and difficulty of explicit construction of a real algebraic map onto a prescribed closed region in the plane.

math.AG↗

arXiv2025

Smooth functions which are Morse on preimages of values not being local extrema and constructing natural functions of the class on connected sums of manifolds admitting these functions

We discuss smooth functions which are Morse on preimages of values not being local extrema. We call such a function internally Morse or I-Morse. The Reeb graph of a smooth function is the space of all connected components of preimages of single points of it topologized with the natural quotient topology of the manifolds and a vertex of it is a point corresponding to a preimage with critical points. A smooth function is neat with respect to the Reeb graph or N-Reeb if the preimages of the vertices are the closed subsets in the manifolds of the domains with interiors being empty. We discuss I-Morse and N-Reeb functions, IN-Morse-Reeb functions. Our main result presents an IN-Morse-Reeb function respecting two such functions, on a connected sum of these given manifolds.

math.GN↗