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SAJI KentaroGraduate School of Science / Division of MathematicsProfessor
Research activity information
■ Paper- Union Matematica Argentina, Aug. 2024, Revista de la Unión Matemática ArgentinaScientific journal
- Springer Science and Business Media LLC, Jun. 2024, Arnold Mathematical JournalScientific journal
- 2023, TOHOKU MATHEMATICAL JOURNAL, 75(1) (1), 131 - 141, EnglishScientific journal
- Abstract Consider an oriented curve $$\Gamma $$ in a domain D in the plane $${\varvec{R } }^2$$. Thinking of D as a piece of paper, one can make a curved folding in the Euclidean space $${\varvec{R } }^3$$. This can be expressed as the image of an “origami map” $$\Phi :D\rightarrow {\varvec{R } }^3$$ such that $$\Gamma $$ is the singular set of $$\Phi $$, the word “origami” coming from the Japanese term for paper folding. We call the singular set image $$C:=\Phi (\Gamma )$$ the crease of $$\Phi $$ and the singular set $$\Gamma $$ the crease pattern of $$\Phi $$. We are interested in the number of origami maps whose creases and crease patterns are C and $$\Gamma $$, respectively. Two such possibilities have been known. In the authors’ previous work, two other new possibilities and an explicit example with four such non-congruent distinct curved foldings were established. In this paper, we determine the possible values for the number N of congruence classes of curved foldings with the same crease and crease pattern. As a consequence, if C is a non-closed simple arc, then $$N=4$$ if and only if both $$\Gamma $$ and C do not admit any symmetries. On the other hand, when C is a closed curve, there are infinitely many distinct possibilities for curved foldings with the same crease and crease pattern, in general.Springer Science and Business Media LLC, Dec. 2022, Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, 63(4) (4), 723 - 761Scientific journal
- 2022, JOURNAL OF SINGULARITIES, 25, 299 - 324, EnglishScientific journal
- Springer Science and Business Media LLC, Dec. 2021, Journal of Geometry, 112(3) (3)Scientific journal
- Springer, Nov. 2021, Monatshefte fur Mathematik, 196(3) (3), 553 - 575, EnglishScientific journal
- Institute of Mathematics, University of Tsukuba, Jul. 2021, Tsukuba Journal of Mathematics, 45(1) (1)Scientific journal
- 2021, Beitrage zur Algebra und GeometrieScientific journal
- Obtaining complete information about the shape of an object by looking at it from a single direction is impossible in general. In this paper, we theoretically study obtaining differential geometric information of an object from orthogonal projections in a number of directions. We discuss relations between (1) a space curve and the projected curves from several distinct directions, and (2) a surface and the apparent contours of projections from several distinct directions, in terms of differential geometry and singularity theory. In particular, formulae for recovering certain information on the original curves or surfaces from their projected images are given.World Scientific Pub Co Pte Lt, Sep. 2020, International Journal of Mathematics for Industry, 2050004 - 2050004Scientific journal
- Worldwide Center of Mathematics, 2020, Journal of Singularities, 22, 40 - 58, EnglishScientific journal
- 2020, Mediterranean Journal of Mathematics, 17(2) (2), 20 pp., English, International magazine[Refereed]Scientific journal
- Curved foldings with common creases and crease patternsConsider a curve $\Gamma$ in a domain $D$ in the plane $\boldsymbol R^2$. Thinking of $D$ as a piece of paper, one can make a curved folding $P$ in the Euclidean space $\boldsymbol R^3$. The singular set $C$ of $P$ as a space curve is called the crease of $P$ and the initially given plane curve $\Gamma$ is called the crease pattern of $P$. In this paper, we show that in general there are four distinct non-congruent curved foldings with a given pair consisting of a crease and crease pattern. Two of these possibilities were already known, but it seems that the other two possibilities (i.e. four possibilities in total) are presented here for the first time.Oct. 2019
- Oct. 2019, Kodai Mathematical Journal, 42(3) (3), 496 - 525, English, International magazine[Refereed]Scientific journal
- 2019, Anais da Academia Brasileira de Ciências, 91(3) (3), English[Refereed]Scientific journal
- 2019, São Paulo Journal of Mathematical Sciences, 13(2) (2), 663 - 677, English[Refereed]Scientific journal
- 2019, Hokkaido Math. J., 48, 281 - 308, EnglishCharacterizing singularities of a surface in Lie sphere geometry[Refereed]Scientific journal
- A mixed type surface is a connected regular surface in a LorentzianElsevier BV, Nov. 2018, Advances in Mathematics (2020), 365, 107036 - 107036
3-manifold with non-empty spacelike and timelike point sets. The induced metric
of a mixed type surface is a signature-changing metric, and their lightlike
points may be regarded as singular points of such metrics. In this paper, we
investigate the behavior of Gaussian curvature at a non-degenerate lightlike
point of a mixed type surface. To characterize the boundedness of Gaussian
curvature at a non-degenerate lightlike points, we introduce several
fundamental invariants along non-degenerate lightlike points, such as the
lightlike singular curvature and the lightlike normal curvature. Moreover,
using the results by Pelletier and Steller, we obtain the Gauss-Bonnet type
formula for mixed type surfaces with bounded Gaussian curvature.[Refereed]Scientific journal - Elsevier B.V., Feb. 2018, Topology and its Applications, 234, 209 - 219, English[Refereed]Scientific journal
- 2018, Rev. Mat. Complut, 31(no.3) (no.3), 627 - 650, EnglishOn the geometry of folded cuspidal edges[Refereed]Scientific journal
- 2018, Adv. Stud. Pure Math., 78, 411 - 429, EnglishOn pairs of geometric foliations on a cuspidal edge. Singularities in generic geometry[Refereed]Scientific journal
- 2018, Internat. J. Math, 29(no.7) (no.7), 1850046,17 pp, EnglishNormal form of the swallowtail and its applications[Refereed]Scientific journal
- 2018, Hokkaido Math. J, 47(no.2) (no.2), 245 - 267, EnglishFold singularities on spacelike CMC surfaces in Lorentz-Minkowski space[Refereed]Scientific journal
- University of California, Berkeley, 2018, Pacific Journal of Mathematics, 294(2) (2), 505 - 509, English[Refereed]Scientific journal
- Worldwide Center of Mathematics, 2017, Journal of Singularities, 16, 73 - 100, English[Refereed]Scientific journal
- Jan. 2017, JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN, 69(1) (1), 417 - 457, English[Refereed]Scientific journal
- Apr. 2016, CANADIAN JOURNAL OF MATHEMATICS-JOURNAL CANADIEN DE MATHEMATIQUES, 68(2) (2), 445 - 462, English[Refereed]Scientific journal
- 2016, Journal of Gökova Geometry Topology, 10, EnglishExtensions of Koenderink’s formula[Refereed]Scientific journal
- 2016, RIMS Kôkyûroku Bessatsu, B55, 203 - -222., EnglishCriteria for Morin singularities into higher dimensions[Refereed]Scientific journal
- 2016, REAL AND COMPLEX SINGULARITIES, 675, 315 - 336, English[Refereed]International conference proceedings
- 2016, GEOMETRY AND TOPOLOGY OF MANIFOLDS, 154, 247 - 281, English[Refereed]International conference proceedings
- Jun. 2015, INTERNATIONAL JOURNAL OF MATHEMATICS, 26(6) (6), English[Refereed]Scientific journal
- Mar. 2015, INTERNATIONAL JOURNAL OF MATHEMATICS, 26(4) (4), English[Refereed]Scientific journal
- 2015, ANNALES POLONICI MATHEMATICI, 115(3) (3), 241 - 261, English[Refereed]Scientific journal
- Sep. 2014, SCIENCE CHINA-MATHEMATICS, 57(9) (9), 1841 - 1866, English[Refereed]Scientific journal
- Worldwide Center of Mathematics, 2014, Journal of Singularities, 10, 264 - 270, English[Refereed]Scientific journal
- Apr. 2012, JOURNAL OF GEOMETRIC ANALYSIS, 22(2) (2), 383 - 409, English[Refereed]Scientific journal
- Oct. 2011, KODAI MATHEMATICAL JOURNAL, 34(3) (3), 390 - 409, EnglishA(2)-SINGULARITIES OF HYPERSURFACES WITH NON-NEGATIVE SECTIONAL CURVATURE IN EUCLIDEAN SPACE[Refereed]Scientific journal
- Jun. 2011, DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS, 29(3) (3), 409 - 425, English[Refereed]Scientific journal
- Recently we discovered a new geometry on submanifolds in hyperbolic n-space which is called horospherical geometry. Unfortunately this geometry is not invariant under the hyperbolic motions (it is invariant under the canonical action of SO(n)), but it has quite interesting features. For example, the flatness in this geometry is a hyperbolic invariant and the total curvatures are topological invariants. In this paper, we investigate the horospherical flat surfaces (flat surfaces in the sense of horospherical geometry) in hyperbolic 3-space. Especially, we give a generic classification of singularities of such surfaces. As a consequence, we can say that such a class of surfaces has quite a rich geometric structure.The Mathematical Society of Japan, Jul. 2010, Journal of Mathematical Society of Japan, 62(3) (3), 789 - 849, English[Refereed]Scientific journal
- Jun. 2010, COMPTES RENDUS MATHEMATIQUE, 348(11-12) (11-12), 665 - 668, English[Refereed]Scientific journal
- 2010, J. Singul., 2, 92 - 127, EnglishThe mandala of Legendrian dualities for pseudo-spheres in Lorentz-Minkowski space and ``flat'' spacelike surfaces[Refereed]Scientific journal
- We give useful criteria for S_1 singularities in the Mond classification table, and cuspidal S_k singularities. As applications, we give a simple proof of a result given by Mond and a characterization of cuspidal S_k singularities for the composition of a cuspidal edge and a fold map indicated by Arnol'd for the case k=1.2010, J. Gökova Geom. Topol., English[Refereed]Scientific journal
- We give a definition of `front bundles', which is an intrinsic formulation of wave fronts. In our definition of front bundles, the first fundamental forms and the third fundamental forms are considered as induced metrics of certain homomorphisms between vector bundles. They satisfy the completely same conditions, and so can reverse roles with each other, which is the meaning of the intrinsic duality in this paper. For a given wave front of a 2-manifold, there are two Gauss-Bonnet formulas. By exchanging the roles of the fundamental forms, we get two new additional Gauss-Bonnet formulas for ...Oct. 2009
- We give useful and simple criteria for determining $D_4^\pm$ singularities of wave fronts. As an application, we investigate behaviors of singular curvatures of cuspidal edges near $D_4^+$ singularities.Jun. 2009, Tohoku Math. J., English[Refereed]Scientific journal
- May 2009, MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY, 146(3) (3), 731 - 746, English[Refereed]Scientific journal
- We shall give useful criteria of lips, beaks and swallowtail singularities of smooth map from the plane into the plane. As an application of criteria, we will discuss the singularities of Cauchy problem of single conservation law.May 2009, Hiroshima Math. J., English[Refereed]Scientific journal
- Apr. 2009, JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN, 61(2) (2), 559 - 578, English[Refereed]Scientific journal
- 2009, Ann. of Math., 169, 491 - 529
- The lightlike hypersurfaces in Lorentz-Minkowski space are of special interest in Relativity Theory. In particular, the singularities of these hypersurfaces provide good models for the Study of different horizon types. We introduce the notion of flatness for these hypersurfaces and study their singularities. The classification result asserts that a generic classification of flat lightlike hypersurfaces is quite different from that of generic lightlike hypersurfaces.Elsevier B.V., 2009, Journal of Geometry and Physics, 59(11) (11), 1528 - 1546, English[Refereed]Scientific journal
- Aug. 2008, MATHEMATISCHE ZEITSCHRIFT, 259(4) (4), 827 - 848, English[Refereed]Scientific journal
- Aug. 2008, 数理解析研究所講究録, 1610, 114 - 120, EnglishKoenderink type theorems for fronts
- Let M2 be an oriented 2-manifold and f : M2→R3 a C∞-map. A point p ∈M2 is called a singular point if f is not an immersion at p. The map f is called a front (or wave front), if there exists a unit C∞-vector field ν such that the image of each tangent vector df (X) (X ∈ TM2) is perpendicular to ν, and the pair (f, ν) gives an immersion into R3 × S2. In a previous paper, we gave an intrinsic formulation of wave fronts in R3. In this paper, we investigate the behavior of cuspidal edges near corank-one singular points and establish Gauss-Bonnet-type formulas under the intrinsic formulation.Kyushu University, 2008, Kyusyu Journal of Mathematics, 62(1) (1), 259 - 280, English
- 2007, PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS, 137, 415 - 430, EnglishSingularities of smooth mappings with patterns[Refereed]Scientific journal
- 2005, Pacific J. Math., 221(2) (2), 303 - 351Singularities of flat fronts in hyperbolic space
- Mar. 2004, JOURNAL OF KNOT THEORY AND ITS RAMIFICATIONS, 13(2) (2), 249 - 257, EnglishBifurcations of Voronoi diagrams and its application to braid theory[Refereed]Scientific journal
- 2003, Proceedings of the Royal Society of Edinburgh, 133A, 1341 - 1359, EnglishSingularities of line congruences[Refereed]Scientific journal
- On the existence of four or more curved foldings with common creases and crease patternsConsider a curve $\Gamma$ in a domain $D$ in the plane $\boldsymbol{R}^2$. Thinking of $D$ as a piece of paper, one can make a curved folding in the Euclidean space $\boldsymbol{R}^3$. This can be expressed as the image of an `origami map' $\varphi:D\to \boldsymbol{R}^3$ such that $\Gamma$ is the singular set of $\varphi$, the word `origami' coming from the Japanese term for paper folding. We call the singular set image $C:=\varphi(\Gamma)$ the crease of $\varphi$ and the singular set $\Gamma$ the crease pattern of $\varphi$. We are interested in the number of origami maps whose creases and crease patterns are $C$ and $\Gamma$ respectively. Two such possibilities have been known. In the previous authors' work, two other new possibilities and an explicit example with four such non-congruent distinct curved foldings were established. In this paper, we show that the case of four mutually non-congruent curved foldings with the same crease and crease pattern occurs if and only if $\Gamma$ and $C$ do not admit any symmetries. Moreover, when $C$ is a closed curve, we show that there are infinitely many distinct possibilities for curved foldings with the same crease and crease pattern, in general.17 Nov. 2019
- Duality on generalized cuspidal edges preserving singular set images and first fundamental formsIn the second, fourth and fifth authors' previous work, a duality on generic06 Jun. 2019
real analytic cuspidal edges in the Euclidean 3-space $\boldsymbol R^3$
preserving their singular set images and first fundamental forms, was given. In
this paper, we show that this duality extends to generalized cuspidal edges in
$\boldsymbol R^3$, including cuspidal cross caps, and 5/2-cuspidal edges. When
the singular set image has no symmetries and does not lie in a plane, the dual
generalized cuspidal edge is not congruent to the original one. We construct
concrete examples of these duals. Moreover, we give several new geometric
insights on this duality.Technical report - Kyoto University, Apr. 2016, RIMS Kokyuroku Bessatsu, 55, 205 - 224, EnglishCriteria for Morin singularities into higher dimensions (Theory of singularities of smooth mappings and around it)
- Jun. 2010, Osaka Journal of Mathematics, 47(2) (2), 591 - 607, EnglishThe duality between singular points and inflection points on wave fronts
- Hiroshima University, Jul. 2002, Hiroshima mathematical journal, 32(2) (2), 309 - 323, EnglishSingularities of non-degenerate 2-ruled hypersurfaces in 4-space
■ Lectures, oral presentations, etc.
- 日本数学会トポロジー分科会, Mar. 2019, Japanese, 東京工業大学, Domestic conference特異点をもつ曲面とカスプ辺との接触Oral presentation
- 幾何学的視点からの形状形成, Sep. 2018, English, 九州大学伊都キャンパス・ウエスト1 号館, International conferenceSingularities of wave fronts and curvatures[Invited]Invited oral presentation
- 15th edition of the International Workshop on Real and Complex Singularities, Instituto de Ciências Matemáticas e de Computacão, Jul. 2018, English, Universidade de São Paulo, Campus São Carlos, Brazil, International conferenceSO(3) normal forms and differential geometric invariants of singular surfaces[Invited]Invited oral presentation
- 微分幾何学・微分式系・特異点論の応用, Jun. 2018, Japanese, 広島工業大学広島校舎２０１号室, Domestic conferenceSingular surfaces with prescribed unbounded mean curvatureOral presentation
- 幾何学と特異点２０１８, Mar. 2018, Japanese, 東京学芸大学自然科学１号館, Domestic conference与えられた非有界平均曲率をもつ特異点付き回転面Oral presentation
- 第７回神戸特異点研究会, Mar. 2018, English, 兵庫教育大学ハーバーランドキャンパス, International conferenceSingular surfaces of revolution with prescribed unbounded mean curvatureOral presentation
- 特異点論とその応用, Feb. 2018, English, 北海道大学学術交流会館, 理学部, International conferenceベクトル場と関数の接触とその応用Oral presentation
- Geometric and Algebraic Singularity Theory, Sep. 2017, English, The Mathematical Research and Conference Center Bedlewo, Poland, International conferenceNormal form of swallowtail and its applicationsOral presentation
- 山口佳三先生退職記念研究集会, Jul. 2017, Japanese, 神戸大学理学部, Domestic conferenceランフォイドカスプの不変量とその応用,Oral presentation
- 微分幾何学と特異点論の応用, Jun. 2017, Japanese, 岩手医科大学創立６０周年記念館, Domestic conferenceスワローテイルのノーマルフォームとその応用Oral presentation
- 微分幾何学と可積分系,, Mar. 2017, English, Osaka city University, Osaka city University, International conferenceGeometric foliations of fronts[Invited]Invited oral presentation
- 14th International Workshop on Real and Complex Singularities, Jul. 2016, English, ICMC, Sao Carlos - SP, Brazil, International conferenceInvariants of cuspidal edges and flat surfacesOral presentation
- Encounter with Mathematics, Mar. 2016, Japanese, 中央大学理工学部5号館,, Domestic conference誰にでもできる特異点の判定法[Invited]Invited oral presentation
- ４次元トポロジー, Nov. 2015, Japanese, 大阪市立大学理学部E棟4階E408, Domestic conferenceファイバー型モラン特異点の判定法とその応用Oral presentation
- Geometric Singularity Theory, Sep. 2015, English, Banach Center, Warsaw, International conferenceOn pairs of geometric foliations on a cuspidal edgeOral presentation
- トポロジーシンポジウム, Aug. 2015, Japanese, 名古屋工業大学51号館1階5111, Domestic conference可微分写像の特異点の判定法とその応用[Invited]Invited oral presentation
- 「Transformations \& singularities」, Sep. 2014, English, DA09 E10 Konferenzraum,Technische Universit{"a}t Wien, International conferenceIsotopy of map-germs and Morin singularities[Invited]Invited oral presentation
- Transformations & singularities, Sep. 2014, English, DA09 E10 Konferenzraum,Technische Universität Wien, International conferenceIsotopy of map-germs and Morin singularities[Invited]Invited oral presentation
- 「The 2nd Franco-Japanese-Vietnamese symposium on singularities」, Aug. 2014, English, 北海道大学理学部7-310, International conferenceGeometry of singularities of fronts[Invited]Invited oral presentation
- The 2nd Franco-Japanese-Vietnamese symposium on singularities, Aug. 2014, English, 北海道大学理学部7-310, International conferenceGeometry of singularities of fronts[Invited]Invited oral presentation
- 「13th International Workshopon Real and Complex Singularities」, Jul. 2014, English, Society of Kobe University, Instituto de Ci{\^e}nciasMate{\'a}ticas e de Computa\c{c},{\~a}oUniversidade de S{\~a}o Paulo, Campus S{\~a}o Carlos., International conferenceCriteria and geometries of Morin singularitiesOral presentation
- 13th International Workshop on Real and Complex Singularities, Jul. 2014, English, Society of Kobe University, Instituto de Ciências Mateáticas e de Computação Universidade de São Paulo, Campus São Carlos, International conferenceCriteria and geometries of Morin singularitiesOral presentation
- 「写像の特異点論及び関連する科学の諸問題 」, Jun. 2014, Japanese, 主催者名１, 主催者名２, 都城工業高等専門学校多目的ホール, Domestic conference異次元モラン特異点の判定法とその応用Oral presentation
- 写像の特異点論及び関連する科学の諸問題, Jun. 2014, Japanese, 主催者名１, 主催者名２, 都城工業高等専門学校多目的ホール, Domestic conference異次元モラン特異点の判定法とその応用Oral presentation
- 日本数学会春季年会トポロジー分科会, Mar. 2014, Japanese, 日本数学会, 学習院大学目白キャンパス, Domestic conferenceモラン特異点のイソトピーOral presentation
- 「日本数学会春季年会トポロジー分科会」, Mar. 2014, Japanese, 明治大学 駿河台キャンパス リバティタワー, Domestic conferenceカスプ辺の幾何的模様Oral presentation
- 日本数学会春季年会トポロジー分科会, Mar. 2014, Japanese, 明治大学 駿河台キャンパス リバティタワー, Domestic conferenceカスプ辺の幾何的模様Oral presentation
- The second Japanese-Spanish workshop on Differential Geometry, Feb. 2014, English, 京都大学数理解析研究所, 京都大学数理解析研究所, Domestic conferenceGeometric invariants of non-degenerate singular pointson wave fronts[Invited]Invited oral presentation
- 大阪市立大学幾何学セミナー, Dec. 2013, Japanese, 大阪市立大学数学研究所, 大阪市立大学数学研究所, Domestic conferenceカスプ辺の幾何的不変量Oral presentation
- Theory ofsingularities of smooth mappings and around it, Nov. 2013, English, 京都大学数理解析研究所, 京都大学数理解析研究所, Domestic conferenceGeometric invariants of cuspidal edgesOral presentation
- Japanese-Brazilian Singularity Days, Sep. 2013, English, Sao Paulo 大学, Sao Carlos キャンパス,Brazil, International conferenceIsotopy of Morin singularitieOral presentation
- Geometric Singularity Theory Polish-Japanese Singularity Theory Working Days, Aug. 2013, English, Warsaw Center of Mathematics andComputer Science, Warsaw, Poland, International conferenceGeometric invariants of cuspidal edgesOral presentation
- 位相幾何・微分幾何と特異点論の関わり, May 2013, Japanese, 弘前大学創立51周年記念会館 岩木ホール, Domestic conferenceモラン写像芽のトポロジーOral presentation
- 「第7回札幌・福岡幾何学セミナー」, Feb. 2013, Japanese, 北海道大学理学部, Domestic conference可微分写像の特異点判定法とその応用I, II[Invited]Invited oral presentation
- 「12th International Workshop on Real and Complex singularities」, Jul. 2012, English, Sao Paulo 大学 ,Sao Carlos キャンパス,, International conferenceSingularities of surfaces swept by constant curvature curvesOral presentation
- 「特異点論と幾何構造」, Jun. 2012, Japanese, 長野市生涯学習センター, Domestic conference双曲空間内の定曲率線織面とその特異点Oral presentation
- 日本学術振興会, 科学研究費助成事業, 基盤研究(C), 神戸大学, 01 Apr. 2023 - 31 Mar. 2028Global behavior of discrete surfaces via integrabilityThe purpose of this research is to develop the connection between complex analytic methods in surface theory with the integrable systems methods in transformation theory, and produce new results with this.
The former methods involve primarily the use of Weierstrass and DPW type representations to construct surfaces with particular curvature properties, the first example of this being minimal surfaces in Euclidean space, but including many other classes of surfaces in a variety of spaceforms. The latter methods include transformations of surfaces, such as Baecklund and Darboux transformations, together with permutability properties, in a Moebius geometric context. - Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Fund for the Promotion of Joint International Research (Fostering Joint International Research (B)), Kobe University, 07 Oct. 2022 - 31 Mar. 2028Cross-disciplinary fusion of singular phenomena by singularity theory波面の特異点の微分幾何学的研究に関して、標準形を与えることができたD4+および-特異点に対して係数の内在性を検討した。2次のジェットで定まる係数は両者ひとつだけであるが、どちらの特異点ともこの係数としての微分幾何的不変量が内在的であることを示した。これは第一基本形式の情報のみで表現できる座標を用いてこの係数を第一基本形式の係数のみを用いて書いた。この式は示唆的であり、今後の研究にも役立つと思われる。さらに、特異点曲線の角度を用いて幾何学的特徴づけを与えることができた。 また、回転面の平均曲率の研究を発展させて、螺旋曲面の研究を行った。螺旋曲面では曲率が１変数関数にならないため、与えられた曲率を持つ螺旋曲面の構成は容易でない。そのために特異点をもつ平面曲線を螺旋運動させてできる曲面の特異点と曲率の性質を検討した。特異点についてはそのような曲面は螺旋の平行移動方向に直交する平面の切り口に表れる曲線のサスペンション曲面となることがわかり、切り口の特異点を検討して、特異点をもつ曲線の曲率を用いて特異点の特徴づけを行った。 回転面のうち与えられた法曲率をもつものの構成問題の検討を行い、解くべき微分方程式はわかったが、積分する方法がまだ未解明であり、今後の研究課題として残っている。特異点をもつ場合のHeinzの定理の検討を行った。特異点をもつ場合は平均曲率はほとんどの場合発散するため、定理の定式化自体が自明でなく、これも今後の研究課題として残っている。 特異点をもつ場合のd'Ocagneの定理に関して検討を行った。ツバメの尾特異点等の特異点の標準形ができているものに関しては標準形の係数を不変量の代わりに使えるため、d'Ocagneの定理を検討しうることがわかった。さらに特異点を射影した際に得られる平面間の写像への理解も深まっており、これはすぐに着手可能な研究課題となった。
- Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), Kobe University, 01 Apr. 2022 - 31 Mar. 2026Characterization of singular points and study of surface singularities
- Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (B), Institute of Science Tokyo, 01 Apr. 2021 - 31 Mar. 2026Differential geometrey of singularities and its applications1. 一昨年度に取り組んだ藤森氏・川上氏・國分氏・Rossman氏・山田氏との「実解析写像の像の解析的拡張の不可能性」の判定に大きな進展があり，11月の数理解析研究所の研究集会において研究発表を行った．これに平行して，東工大の山田氏，横浜国大の本田氏との共同研究で，上記の理論が完成したという仮定のもとで，「de Sitter 時空内の平均曲率が１のカテノイド型の空間的曲面」の解析的な拡張後の曲面がすべて，これ以上の解析的拡張を持たないことを示した． 2. 埼玉大学の福井氏，東工大の卒業生である木下氏，中国のDonghe Pei氏とHaiou Yu氏らと，３次元Minkowski時空の一般化されたカスプ辺の研究について，昨年度に取り組めなかった次数が４の場合を研究し，論文にして投稿した． 3. 熊本大学の安藤直也氏との共同研究として，３次元時空の空間的曲面と時間的曲面それぞれについて，与えたれた滑らかな関数のヘッセ行列から誘導される行列の固有ベクトル場が曲率線となる座標系を見いだし，応用として，これらの時空の曲面の臍点の指数に関する新たな知見を論文にまとめ，年度内に国際誌に掲載された． 4. 研究分担者の山田氏・佐治氏との共同研究として昨年度に行った「３次元の定曲率空間のカスプ辺の写像芽」に関する研究の発展として，一般化されたツバメの尾を定義し，これらについても同様の結論を引き出すことに成功した． 5. 昨年度に引き続き，協力者の北川氏・榎本氏と，３次元球面内において「はめ込まれた平坦トーラスの直径予想」に取り組み，最終的な解決に「あと一歩」という状況に到達した．また，指導していた修士学生の高田君と平面三角形の諸心の他の空間形への変形について，筆者の実解析写像の研究の応用という位置づけで共同研究を行い，成果があがりつつある．
- Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (B), Tokyo Institute of Technology, 01 Apr. 2021 - 31 Mar. 2026Differential geometrey of singularities and its applications１．分担者の山田氏，佐治氏，協力者の本田氏，直川氏との共同研究として，波面について知られるZakalyukinの補題をフロンタル特異点にまで拡張し応用を与えた．特に以前，筆者等が与えたツバメの尾の判定条件の証明が簡明になった．関連して，フロンタルではないが，曲面に現れる主要な特異点である「交叉帽子特異点」についても同様の研究を行い，Zakalyukin 型の定理を証明することに成功した．これらを２つの論文にまとめた． ２．3次元de Sitter空間の平均曲率1の空間的な曲面として，「不動点をもつ１係数等長変換群」で不変な曲面が複数存在し，G-catenoid とよばれる．分担者の山田氏・ラスマン氏，協力者の藤森氏，國分氏，川上氏，Yang氏らと，それらの多くが，非自明な解析的な拡張をもち，２次元Double-Cone manifoldの構造をもつことを示した．さらに，この事実から，拡張後のG-catenoidが，これ以上の解析的拡張をもたないことが示せた．この成果は論文として現在投稿中である． ３．ローレンツ・ミンコフスキー時空における超曲面は，すべての点で誘導計量が退化するとき「光的」であるという．分担者の山田氏，協力者の赤嶺氏，本田氏との共同研究により，光的な超曲面で対応する光的ベクトル場が完備となるものは，ユークリッド空間の超曲面の平行曲面族から構成されるものに限ることを示し，いくつかの重要な応用を与えた．この成果は，論文として現在投稿中である． ４．「三次元球面に，はめ込まれた平坦トーラスの直径が，球面の直径に一致するだろう」という予想に長年，筆者は取り組み成果を積んできたが，協力者の北川氏，榎本氏らと共同研究を行い，１つの新たな知見を得ることができた．これを基に今後も研究を継続していく所存である．
- 学術研究助成基金助成金／基盤研究(C), Apr. 2018 - Mar. 2022, Principal investigatorCompetitive research funding
- 学術研究助成基金助成金／基盤研究(C), Apr. 2015 - Mar. 2020Competitive research funding
- Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (B), Hokkaido University, 01 Apr. 2015 - 31 Mar. 2019Developments and applications of geometric singularity theoryIn a space with symmetry, we study the geometric solutions of naturally appeared equations, regarding the characteristic singularities associated to the geometric structure. Under geometric motivations, we analyse and classify singularities and clarify their geometric meanings. Thus we present new methods in geometry and apply them. By this project, we have completed the generic classification of singularities in tangent surfaces under a far general framework. Moreover we have established the classification of tangent surfaces to null curves in any 3-dimensional Lorentz manifold, relating Engel pseudo-product structures, projective contact structures and the geometric theory of third order OED. Furthermore we have succeeded the clarification of the general framework on the G2 duality of (2,3,5)-distributions.
- Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Challenging Exploratory Research, Hokkaido University, 01 Apr. 2015 - 31 Mar. 2018Algebro-geometric method for singularity criteria in kinematicsWe developed effective criteria of singularities of differentiable maps in the following three topics. 1. We have newly developed a local theory originated by Darborx and Wilcynski in projective differential geometry of surfaces in relation with applications to computer vision. 2. Combining Geometric Algebra and classification theory of map-germs, we studied singularities arising in differential line geometry (ruled surfaces, line congruences, etc), that suggests a new method in Applied Geometry. 3. Using classification of map-germs and universal polynomials of singularities, we developed classical enumerative geometry of curves and surfaces, that proposes a new algebro-geometric approach to computer vision. 4. By means of `singularity criteria', we studied A-tangent spaces and logarithmic tangents to discriminants (free divisors).
- Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (B), Hokkaido University, 01 Apr. 2014 - 31 Mar. 2018Geometric studies on singularity of non-linear phenomenaIn this research project we have shown that Lagrangian equivalence among Lagrangian submanifold germs and a certain equivalence relation among the corresponding graph-like wave fronts are the same. Applications of this result include the classical differential geometry, the geometry of the space-time, the differential geometry of singular surfaces and mappings, and so on. On the other hand, a new application of singularity theory to quantum mechanics is also discovered in this research project.
- 学術研究助成基金助成金／基盤研究(C), Apr. 2014 - Mar. 2018, Principal investigatorCompetitive research funding
- Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (A), 01 Apr. 2010 - 31 Mar. 2015Geometry of curves, surfaces and hypersurfaces with singularitiesUsing the concept of coherent tangent bundles, we (the head investigator and the research group) found four new Gauss-Bonnet type formulas for closed surfaces with singularities in Euclidean 3-space. Using the fact that a spacelike maximal surface with fold singularities has an analytic extension across those singularities, we showed that the analytic extensions of the triply-periodic Schwarz D type maximal surfaces are all embedded. In a joint work with Yoshihisa Kitagawa, we proved that the Clifford torus is rigid in the class of immersed flat tori whose mean curvature functions do not change sign. Moreover, we obtained some interesting results for plane curves. For example, a simplification of the proof of Bol's conjecture on sextactic points was discovered.
- 科学研究費補助金／若手研究(B), Apr. 2011 - Mar. 2015, Principal investigatorCompetitive research funding
- Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Grant-in-Aid for Young Scientists (B), Gifu University, 2008 - 2010Differential geometric study of surfaces with singularitiesI gave criteria for cuspidal lips, cuspidal beaks, cuspidal butterfly, umbilics of singularities fronts. Furthermore, I gave criteria for Mond singularity and higher dimensional singularities of fronts. As applications of them, I clarified relationships between singularities and differential geometric properties of surfaces. Furthermore, I gave an intrinsic formulation of fronts. Using this, I gave formulas about numbers of singularities.
- 日本学術振興会, 科学研究費助成事業, 若手研究(スタートアップ), 北海道大学, 2007 - 2008特異点理論とその曲面論、低次元トポロジーへの応用この研究により得られた結果は以下の通り。 1,平面間の写像に関して余次元1の特異点の認識問題を研究し、有用な判定法を得た。この判定法を応用して、微分方程式の特性曲面の特異性を調べ、特異点の分類や微分方程式の形作用素と呼ぶべき物を発見した。三次元空間内の曲面の平面への射影についても研究し、射影する方向と射影に現れる特異点との関係を明らかにした。 2,三次元空間内の波面のA型特異点の有用な判定法を作り、超幾何微分方程式のシュワルツ写像の特異点を佐々木武氏、吉田正章氏と共同で研究した。シュワルツ写像から作られる曲面には三個のスワローテイルが一つになる現象が起こるが、これがA5特異点であることを示した。 3,梅原雅顕氏と山田光太郎氏と共同で高次元空間内の波面のA型特異点の有用な判定法を作り、波面の大域的な不変量であるジグザグ数と曲率写像との関係を明らかにした。 4,双曲空間内の曲面に関して、泉屋周一氏と共同で擬球面内の曲面の双対曲面を様々な場合に研究した。特異点相互の関係や微分幾何学的特徴と特異点の性質を明らかにした。 5,光錐内の曲面に関して、泉屋氏、カルメン・ロメロフスター氏と共同で研究し、ジェネリックに現れる特異点の分類を得た。締活線を定義し、柱面的な光錐内の曲面を定義する方法が解った。
- 日本学術振興会, 科学研究費助成事業, 特別研究員奨励費, 2004 - 2006特異点理論による幾何学の新展開この研究により得られた結果は以下の通り。 1,波面の幾何学に関して昨年、梅原雅顕氏と山田光太郎氏と共同で、波面の特異点に関して曲率を導入し、その性質を調べた。その後、ルジャンドル多様体への写像として定式化されていた波面の概念を内在的な概念に定式化し、ピークという非常に弱い条件の特異点のみを許した内在的な波面に対して、重要な特異曲線の上組・下組の概念を発見し、それによりガウス・ボンネの定理を証明した。また、高次元の波面のA型と呼ばれる特異点に関して、使いやすい判定法を証明した。 2,ユークリッド空間内の波面の曲率と、波面を平面へ射影したときに出来る特異点の像の曲率との関係を明らかにした。これにより平面に射影して出来る特異点の数を数えることによって波面の曲がり具合をあらわす大域的な不変量を計算できるようになる可能性が高い。 3,双曲空間内の曲面に関して、泉屋周一氏と高橋雅朋氏と共同でホロ球との近さを量る幾何学を研究した。そのような意味での曲率が消える曲面を定義し、それをユークリッド空間の線織面の研究と対比させて締活線の存在や特異点について論じた。また、波面のくちばし、くちびると呼ばれる特異点に関して使いやすい判定法を証明した。双曲空間と光錐内の曲面間の双対性を論じ、特異点を持つ点は対応しており、その性質が対になっていることを発見した。 4,波面のA4分岐と呼ばれる特異点に関して使いやすい判定法を証明した。同次元間の写像のモラン型と呼ばれる特異点に関して使いやすい判定法を証明した。また、他の特異点(モンド型)等に関しては判定法は得られなかったが、判定法を得るための足がかりをいくつか得た。これらの判定法は様々な場面で有用であり、今後はこれらの応用を考えていきたい。