arXiv subjects

Naoki Kitazawa

Publications and source records attributed to Naoki Kitazawa.

At least 37 records · Page 2Linked to original sources

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.

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arXiv2025

Planar graphs embedded in generic ways and realizing them as Reeb graphs of real algebraic functions

This paper is concerned with long-time interest of us, especially, the author, in realizing graphs as Reeb graphs of real algebraic functions of certain nice classes. The Reeb graph of a differentiable function is the set consisting of all components of preimages of all single points and endowed with the quotient topology canonically. In tame cases, such objects are graphs. The Reeb graph of the natural height of the unit sphere of dimension at least $2$ is a graph with exactly one edge and homeomorphic to a closed interval. These graphs have been fundamental and strong tools in geometry since theory of Morse functions has been established in the former half of the last century. We present a new answer to the problem, saying that generically embedded planar graphs are homeomorphic to the Reeb graphs of real algebraic functions obtained by elementary polynomials and elementary procedures.

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arXiv2025

Graphs with tree decompositions of small graphs and realizing them as the Reeb graphs of real algebraic functions

We have been interested in graphs and realizing them as Reeb graphs of explicit real algebraic functions. The Reeb graph of a differentiable function is the quotient space of the manifold of the domain, regarded as the space consisting of all components of preimages of all single points. Reeb graphs have been fundamental and strong tools in geometry of manifolds since the birth of theory of Morse functions, in the former half of the 20th century. We can easily see that the Reeb graph of the natural height of the unit sphere whose dimension is at least $2$ is a graph with exactly one edge and two edges. We are concerned with realizations of graphs decomposed into trees nicely, each vertex of which corresponds to a graph with exactly one edge and two edges or a graph with exactly two edges homeomorphic to a circle.

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arXiv2025

Reconstruction of real algebraic functions into curves with prescribed Reeb graphs

We discuss reconstructing smooth real algebraic maps onto curves whose Reeb graph is as prescribed. This can be contributed to real algebraic geometry, especially in explicit examples in real algebraic geometry in a new way. The Reeb graph of a smooth function is the space of all connected components of preimages of all single points and a natural quotient space of the manifold with the vertex set being all connected components containing some singular points of it. This gives a strong tool in geometry of manifolds and appeared already in 1950 with Morse functions. The Reeb graph of the natural height of the unit sphere of dimension at least 2 is a graph with exactly two vertices and one edge. We reconstruct functions, from general finite graphs, conversely. In the differentiable situations, Sharko pioneered this in 2006, followed by Masumoto-Saeki and Michalak, mainly. Related real algebraic situations have been launched and studied by the author. The curve-valued case is first considered here.

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arXiv2025

On reconstructing Morse-Bott functions with prescribed preimages on $3$-dimensional manifolds and conditions for the reconstruction

We present conditions for reconstruction of Morse-Bott functions with prescribed preimages on $3$-dimensional manifolds. The present work strengthens a previous result for the Morse function case by the author and present a related example as another result. This shows a new result on reconstruction of nice smooth functions such that preimages are as prescribed. Such a study has been fundamental, natural, and surprisingly, founded recently, in 2006, by Sharko. Reconstruction of nice smooth functions on closed surfaces has been followed by Masumoto-Saeki, for example, and later, Gelbukh, Marzantowicz, Michalak, and so on, are studying Morse function cases further. The author has started explicit studies for $3$-dimensional cases respecting topologies of preimages of single points and obtained several results. We add another result on this.

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arXiv2025

Moment-like maps and real algebraic functions with prescribed preimages

We discuss a problem on singularity theory of differentiable (smooth) or real algebraic maps which is different from knowing existence and has been difficult: constructing explcit real algebraic functions. We discuss construction of real algebraic functions with exactly one singular value, the singular points being of definite type, and prescribed preimages of single points. We discuss generalizations of the canonical projection of the unit sphere around the pole and the Morse-Bott functions around the boundaries of the images with preimages diffeomorphic to the torus. This has been discussed in the differentiable (smooth) category since the pioneering study of Sharko in 2006, followed by Masumoto-Saeki, Michalak and so on: the author has first considered the cases respecting the topologies of the preimages of the points where these studies had not done this essentially. Related real algebraic studies have been started by the author essentially in 2020's and the studies are developing, mainly due to the author.

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arXiv2025

Refined algebraic domains with finite sets in the boundaries respecting differential geometry

We are interested in shapes of real algebraic curves in the plane and regions surrounded by them: they are named refined algebraic domains by the author. As characteristic finite sets, we consider points contained in two curves and the sets of singular points of the restrictions of the projections to the lines to the curves. As a new case, we respect differential geometry and consider inflection points and points of some double tangent lines of a single connected curve. We prove fundamental properties and investigate some examples. We have also previously considered the cases where the curves are straight lines, circles, or boundaries of ellipsoids for example. Such simple cases are trivial in our new consideration.

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arXiv2025

Some remark on real algebraic maps which are topologically special generic maps and generalize the canonical projections of the unit spheres

Morse functions with exactly two singular points on homotopy spheres and canonical projections of spheres are generalized as special generic maps. A special generic map is, roughly, a smooth map represented as the composition of a smooth surjection onto a manifold whose preimages are diffeomorphic to a unit sphere in the interior of the manifold and single point sets on the boundary with a smooth immersion of codimension $0$. This paper constructs real algebraic maps topologically special generic maps whose images are smoothly embedded manifolds. We are also interested in construction of explicit and meaningful smooth maps in differential topology and recently ones in real algebraic geometry. This has been an important and difficult problem. In such stories, we have previously constructed real algebraic maps topologically regarded as special generic maps. This paper is a kind of additional short remark on such maps.

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arXiv2025

Refined algebraic domains with finite sets in the boundaries

Refined algebraic domains are regions in the plane surrounded by finitely many non-singular real algebraic curves which may intersect with normal crossing. We are interested in shapes of such regions with surrounding real algebraic curves. Poincar'e-Reeb Graphs of them are graphs the regions naturally collapse to respecting the projection to a straight line. Such graphs were first formulated by Sorea, for example, around 2020, and regions surrounded by mutually disjoint non-singular real algebraic curves were mainly considered. The author has generalized the studies to several general situations. We find classes of such objects defined inductively by adding curves. We respect characteristic finite sets in the curves. We consider regions surrounded by the curves and of a new type. We investigate geometric properties and combinatorial ones of them and discuss important examples. We also previously studied explicit classes defined inductively in this way and review them.

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arXiv2025

Arrangements of circles supported by small chords and compatible with natural real algebraic functions

We have previously proposed a study of arrangements of small circles which also surround regions in the plane realized as the images of natural real algebraic maps yielding Morse-Bott functions by projections. Among studies of arrangements, families of smooth regular submanifolds in smooth manifolds, this study is fundamental, explicit, and new, surprisingly. We have obtained a complete list of local changes of the graphs the regions naturally collapse to in adding a (generic) small circle to an existing arrangement of the proposed class. Here, we propose a similar and essentially different class of arrangements of circles. The present study also yields real algebraic maps and nice real algebraic functions similarly and we present a similar study. We are interested in topological properties and combinatorics among such arrangements and regions and applications to constructing such real algebraic maps and manifolds explicitly and understanding their global structures.

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arXiv2025

Realizations of planar graphs as Poincar'e-Reeb graphs of refined algebraic domains

Algebraic domains are regions in the plane surrounded by mutually disjoint non-singular real algebraic curves. Poincar'e-Reeb Graphs of them are graphs they naturally collapse: such graphs are formally formulated by Sorea, for example, around 2020. Their studies found that nicely embedded planar graphs are Poincar'e-Reeb graphs of some algebraic domains. These graphs are generic with respect to the projection to the horizontal axis. Problems, methods and results are elementary and natural and they apply natural approximations nicely for example. We present our new approach to extension of the result to a non-generic case and an answer. We first formulate generalized algebraic domains, surrounded by non-singular real algebraic curves which may intersect with normal crossings. Such domains and certain classes of them appear in related studies of graphs and regions surrounded by algebraic curves explicitly.

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arXiv2025

Arrangements of circles, the regions surrounded by them and labeled Poincaré-Reeb graphs

We are interested in arrangements of circles and the regions surrounded by them. {\it Poincaré-Reeb graphs} have been fundamental and strong tools in studying shapes of regions surrounded by real algebraic curves, since around 2020. They are natural graphs the regions naturally collapse to and were first formulated by Sorea with several researchers. Studying shapes of such regions is one of fundamental studies in real algebraic geometry and combinatorics for example. This is surprisingly new and recently developing. Our study introduces labels on vertices and edges of such graphs encoding information of the circles where we concentrate on regions surrounded by circles. The author studied local changes of Poincaré-Reeb graphs by addition of circles under certain rules before and we discuss changes of new types. The author has started related studies motivated by singularity theory of real algebraic maps and found first that our regions are the images of natural real algebraic maps, generalizing natural projections of spheres.

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arXiv2025

Arrangements of small circles for Morse-Bott functions

As a topic of mathematics, "arrangements", systems of hyperplanes, circles, and general (regular) submanifolds, attract us strongly. We present a natural elementary study of arrangements of circles. It is also a kind of new studies. Our study is closely related to geometry and singularity theory of Morse(-Bott) functions. Regions surrounded by circles are regarded as images of real algebraic maps and composing them with projections gives Morse-Bott functions: this observation is natural, and surprisingly, recently presented first, by the author. We present a systematic way of constructing such arrangements by choosing small circles centered at existing circles inductively. We are interested in graphs the regions surrounded by the circles naturally collapse. We have studied local changes of the graphs in adding these circles. These graphs are essentially so-called {\it Reeb graphs} of the previous Morse-Bott functions: they are spaces of all components of preimages of single points for the functions.

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arXiv2024

Characterizing $3$-dimensional manifolds represented as connected sums of Lens spaces, $S^2 \times S^1$, and torus bundles over the circle by certain Morse-Bott functions

We characterize $3$-dimensional manifolds represented as connected sums of Lens spaces, copies of $S^2 \times S^1$, and torus bundles over the circle by certain Morse-Bott functions. This adds to our previous result around 2024, classifying Morse functions whose preimages containing no singular points are disjoint unions of spheres and tori on $3$-dimensional manifolds represented as connected sums of connected sums of Lens spaces and copies of $S^2 \times S^1$: we have strengthened and explicitized Saeki's result, characterizing the manifolds via such functions, in 2006. We apply similar arguments. However we discuss in a self-contained way essentially.

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arXiv2024

On a classification of Morse functions on $3$-dimensional manifolds represented as connected sums of manifolds of Heegaard genus one

Morse functions are important objects and tools in understanding topologies of manifolds since the 20th century. Their classification has been natural and difficult problems, and surprisingly, this is recently developing. Since the 2010's, results for cases of surfaces have been presented by Gelbukh, Marzantowicz and Michalak for example. We have also longed for higher dimensional cases. We present a classification of Morse functions on $3$-dimensional manifolds represented as connected sums of manifolds of Heegaard genus one. We concentrate on Morse functions such that preimages of single points containing no singular points are disjoint unions of spheres and tori. Existence of such functions implies that the $3$-dimensional closed and connected manifolds are of such manifolds. This has been shown by Saeki in 2006 and we further study structures of these functions.

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arXiv2024

Reconstructing real algebraic maps locally like moment-maps with prescribed images and compositions with the canonical projections to the $1$-dimensional real affine space

We present new real algebraic maps of non-positive codimensions with prescribed images whose boundaries consist of explicit non-singular real algebraic hypersurfaces satisfying so-called "transversality" as follows. Explicit information on important real polynomials is given. Preimages are one-point sets or products of spheres. They are locally like so-called moment maps. Celebrated theory of Nash and Tognoli says that smooth closed manifolds are {\it non-singular} real algebraic manifolds and the zero sets of some real polynomial maps. In general, 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 maps of a specific class of the present class containing the canonical projections of the unit spheres previously where preimages are one-point sets or spheres. We also present explicit families of functions represented as compositions of such maps with the canonical projections.

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arXiv2024

Construction of real algebraic functions with prescribed preimages

Nash and Tognoli show that smooth closed manifolds can be the zero sets of some real polynomial maps and non-singular. The canonical projections of spheres naturally embedded in the $1$-dimensional higher Euclidean spaces and some natural functions on projective spaces, Lie groups and their quotient spaces are important examples of real algebraic functions being also Morse. In general, it is difficult to construct such examples of maps and the structures of the manifolds. In addition the maps are hard to understand globally. We construct examples by answering to a problem from singularity theory and differential topology. It asks whether we can reconstruct nice smooth functions with prescribed preimages. We have previously given an answer with real algebraic functions. This previous result is one of our key ingredients.

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arXiv2023

Proofs of the non-existence of special generic maps on the $3$-dimensional complex projective space

We prove the non-existence of special generic maps on $3$-dimensional complex projective space as our new result and a corollary by several methods. Special generic maps are generalizations of Morse functions with exactly two singular points on spheres and canonical projections of unit spheres are special generic. Our paper focuses on such maps on closed and simply-connected manifolds of classes containing the $3$-dimensional complex projective space. The differentiable structures of spheres admitting special generic maps are known to be restricted strongly. Special generic maps on closed and simply-connected manifolds and projective spaces have been studied by various people including the author. The (non-)existence and construction are main problems. Studies on such maps on closed and simply-connected manifolds whose dimensions are greater than $5$ have been difficult.

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