Researcher Profile and Settings

Master

Affiliation (Master)

• Faculty of Science　Mathematics　Mathematics

Affiliation (Master)

• Faculty of Science　Mathematics　Mathematics

researchmap

Profile and Settings

Affiliation

• Hokkaido University, Faculty of Science Department of Mathematics, Professor

Profile and Settings

• Name (Japanese)

  Inoguchi

• Name (Kana)

  Junichi

• Name

  200901040204658633

Affiliation

• Hokkaido University, Faculty of Science Department of Mathematics, Professor

Achievement

Research Interests

• Integrable Systems   loop group   harmonic map   minimal surface   CMC surface   DPW-method   contact structure   spin structure   magnetic harmonic map   Differential Geometry   Discrete Differential Geometry   

Research Areas

• Natural sciences / Geometry

Research Experience

• 2022/10 - Today Hokkaido University Graduate School of Science
• 2015/04 - 2022/09 University of Tsukuba

Published Papers

• Parallel and totally umbilical hypersurfaces of the four‐dimensional Thurston geometry $\text{Sol}^4_0$

  Marie D'haene, Jun‐ichi Inoguchi, Joeri Van der Veken

  Mathematische Nachrichten 297 (5) 1879 - 1891 0025-584X 2024/05 [Refereed]
   

  Abstract We study hypersurfaces of the four‐dimensional Thurston geometry , which is a Riemannian homogeneous space and a solvable Lie group. In particular, we give a full classification of hypersurfaces whose second fundamental form is a Codazzi tensor—including totally geodesic hypersurfaces and hypersurfaces with parallel second fundamental form—and of totally umbilical hypersurfaces of . We also give a closed expression for the Riemann curvature tensor of , using two integrable complex structures.
• Homogeneous geodesics of 4-dimensional solvable Lie groups

  Jun-ichi Inoguchı

  International Electronic Journal of Geometry 17 (1) 106 - 136 2024/04/23 [Refereed][Invited]
   

  We study homogeneous geodesics in $4$-dimensional solvable Lie groups $\mathrm{Sol}_0^4$, $\mathrm{Sol}_1^4$, $\mathrm{Sol}_{m,n}$ and $\mathrm{Nil}_4$.

• Geodesics and magnetic curves in the 4-dim almost Kähler model space F⁴

  Zlatko Erjavec, Jun-ichi Inoguchi

  Complex Manifolds 11 (1) 2024/04/18 [Refereed]
   

  Abstract We study geodesics and magnetic trajectories in the model space F4{ { \rm{F } } }^{4}. The space F4{ { \rm{F } } }^{4} is isometric to the 4-dim simply connected Riemannian 3-symmetric space due to Kowalski. We describe the solvable Lie group model of F4{ { \rm{F } } }^{4} and investigate its curvature properties. We introduce the symplectic pair of two Kähler forms on F4{ { \rm{F } } }^{4}. Those symplectic forms induce invariant Kähler structure and invariant strictly almost Kähler structure on F4{ { \rm{F } } }^{4}. We explore some typical submanifolds of F4{ { \rm{F } } }^{4}. Next, we explore the general properties of magnetic trajectories in an almost Kähler 4-manifold and characterize Kähler magnetic curves with respect to the symplectic pair of Kähler forms. Finally, we study homogeneous geodesics and homogeneous magnetic curves in F4{ { \rm{F } } }^{4}.
• Homogeneity of magnetic trajectories in the real special linear group

  Jun-ichi Inoguchi, Marian Ioan Munteanu

  Proceedings of the American Mathematical Society 0002-9939 2023/12/18 [Refereed]
   

  We prove the homogeneity of contact magnetic curves in the real special linear group of degree . Every contact magnetic trajectory is a product of a homogeneous geodesic and a charged Reeb flow.
• On the $\eta$-parallelism in almost Kenmotsu $3$-manifolds

  Jun-ichi Inoguchi, Ji-Eun Lee

  Journal of the Korean Mathematical Society 60 (6) 1303 - 1336 2023/11 [Refereed]
  
• J-trajectories in 4-dimensional solvable Lie Group $\mathrm{Sol}_1^4$

  Zlatko Erjavec, Jun-ichi Inoguchi

  Journal of Nonlinear Science 33 (6) 0938-8974 2023/09/25 [Refereed]
  
• Characteristic Jacobi operator on almost Kenmotsu $3$-manifolds

  Jun-ichi Inoguchi

  International Electronic Journal of Geometry 16 (2) 464 - 525 2023/09/22 [Refereed]
   

  The Ricci tensor field, $\varphi$-Ricci tensor field and the characteristic Jacobi operator on almost Kenmotsu $3$-manifolds are investigated. We give a classification of locally symmetric almost Kenmotsu $3$-manifolds.

• Log-aesthetic curves: Similarity geometry, integrable discretization and variational principles

  Jun-ichi Inoguchi, Yoshiki Jikumaru, Kenji Kajiwara, Kenjiro T. Miura, Wolfgang K. Schief

  Computer Aided Geometric Design 105 102233 - 102233 0167-8396 2023/09 [Refereed]
  
• Minimal submanifolds in $\mathrm{Sol}_1^4$

  Zlatko Erjavec, Jun-ichi Inoguchi

  Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 117 (4) 1578-7303 2023/08/22 [Refereed]
  
• Contact 3-manifolds with pseudo-parallel characteristic Jacobi operator

  Jun-ichi Inoguchi, Ji-Eun Lee

  Mediterranean Journal of Mathematics 20 (5) 1660-5446 2023/08/05 [Refereed]
  
• Minimal submanifolds in $\mathrm{Sol}_0^4$

  Zlatko Erjavec, Jun-ichi Inoguchi

  The Journal of Geometric Analysis 33 (9) 1050-6926 2023/06/16 [Refereed]
  
• Killing magnetic curves in $\mathbb{H}^{3}$

  Zlatko Erjavec, Jun-ichi Inoguchi

  International Electronic Journal of Geometry 2023/04/16 [Refereed]
   

  We consider magnetic curves corresponding to the Killing magnetic fields in hyperbolic 3-space.

• Pseudo-symmetric almost cosymplectic 3-manifolds

  Jun-ichi Inoguchi, Ji-Eun Lee

  International Journal of Geometric Methods in Modern Physics 0219-8878 2023/04/07 [Refereed]
  
• Killing submersions and magnetic curves

  Jun-ichi Inoguchi, Marian Ioan Munteanu

  Journal of Mathematical Analysis and Applications 520 (2) 126889 - 126889 0022-247X 2023/04 [Refereed]
  
• Magnetic unit vector fields

  Jun-ichi Inoguchi, Marian Ioan Munteanu

  Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas 117 (2) 1578-7303 2023/02/20 [Refereed]
  
• Magnetic Jacobi fields in Sasakian space forms

  Jun-ichi Inoguchi, Marian Ioan Munteanu

  Mediterranean Journal of Mathematics 20 (1) 1660-5446 2022/12/11 [Refereed]
  
• Minimal surfaces with non-trivial geometry in the three-dimensional Heisenberg group

  Josef F. Dorfmeister, Jun-ichi Inoguchi, Shimpei Kobayashi

  Complex Manifolds 9 (1) 285 - 336 2022/11/15 [Refereed]
   

  Abstract We study symmetric minimal surfaces in the three-dimensional Heisenberg group Nil₃ using the generalized Weierstrass type representation, the so-called loop group method. In particular, we will present a general scheme for how to construct minimal surfaces in Nil₃ with non-trivial geometry. Special emphasis will be put on equivariant minimal surfaces. Moreover, we will classify equivariant minimal surfaces given by one-parameter subgroups of the isometry group Iso_◦(Nil₃) of Nil₃.
• On some curves in 3-dimensional hyperbolic geometry and solvgeometry

  Inoguchi, Jun-ichi

  Journal of Geometry SPRINGER BASEL AG 113 0047-2468 2022/06/28 [Refereed]
   

  We study curve geometry in para-Sasakian 3-manifolds, especially in the hyperbolic 3-space and the space Sol3 of solvgeometry. Para- metric expression for φ-trajectories in the hyperbolic 3-space is given.
• Almost cosymplectic 3-manifolds with pseudo-parallel characteristic Jacobi operator

  Inoguchi, Jun-ichi, Lee, ji-Eun

  International Journal of Geometric Methods in Modern Physics WORLD SCIENTIFIC PUBL CO PTE LTD 19 (8) 0219-8878 2022/06 [Refereed]
   

  In this paper, we classify almost cosymplectic 3-manifolds with pseudo-parallel char- acteristic Jacobi operator. The only simply connected and complete non-cosymplectic almost cosymplectic 3-manifold with pseudo parallel characteristic Jacobi operator is the Minkowski motion group.
• Biharmonic curves in f-Kenmotsu 3-manifolds

  Inoguchi, Jun-ichi, Lee, ji-Eun

  Journal of Mathematical Analysis and Applications 509 (1) 2022/05 [Refereed]
   

  It is known that there exist no proper biharmonic helices in Kenmotsu 3-manifolds. In this paper we show the existence of proper biharmonic helices in certain f-Kenmotsu 3-manifolds.
• アフィン接続と接触構造に関する話題から

  井ノ口, 順一

  Geometry and Analysis Fukuoka 11 - 34 2022/03 [Not refereed]
  
• J-trajectories in 4-dimensional solvable Lie group Sol_0^4

  Erjavec, Zlatko, Inoguchi, Jun-ichi

  Mathematical Physics, Analysis and Geometry 25 2022/03 [Refereed]
  
• Magnetic Jacobi fields in 3-dimensional Sasakian space forms

  Inoguchi, Jun-ichi, Munteanu, Marian Ioan

  The Journal of Geometric Analysis SPRINGER 32 (3) 1050-6926 2022/03 [Refereed]
   

  Representative examples of uniform magnetic fields are furnished by Miller magnetic fields. From this point of view, magnetic Jacobi fields on surfaces or Kahler manifolds were investigated by Adachi and Gouda. On the contrary, Sasakian manifolds have non-uniform magnetic fields. We obtain all magnetic Jacobi fields along contact magnetic curves in 3-dimensional Sasakian space forms.
• Gridshell structures with discrete curvature lines :Modeling technique and evaluation of mechanical performance

  Yokosuka, Yohei, Inoguchi, Jun-ichi, Ohsaki, Makoto, Honma, Toshio

  Proceedings of IASS Annual Symposia, IASS 2020/21 Surrey Symposium: Conceptual design International Association for Shell and Spatial Structures (IASS) 821 - 833 2021/06 [Refereed]
  
• 𝜖𝜅 -Curves: controlled local curvature extrema

  Miura, Kenjiro T, Gobithaasan, R. U, Salvi, Péter, Wang, Dan, Sekine, Tadatoshi, Usuki, Shin, Inoguchi, Jun-ichi, Kajiwara, Kenji

  The Visual Computer Springer 38 (8) 2723 - 2738 0178-2789 2021/05 [Refereed]
   

  The kappa-curve is a recently published interpolating spline which consists of quadratic Bezier segments passing through input points at the loci of local curvature extrema. We extend this representation to control the magnitudes of local maximum curvature in a new scheme called extended- or is an element of kappa-curves. kappa-curves have been implemented as the curvature tool in Adobe Illustrator (R) and Photoshop (R) and are highly valued by professional designers. However, because of the limited degrees of freedom of quadratic Bezier curves, it provides no control over the curvature distribution. We propose new methods that enable the modification of local curvature at the interpolation points by degree elevation of the Bernstein basis as well as application of generalized trigonometric basis functions. By using is an element of kappa-curves, designers acquire much more ability to produce a variety of expressions, as illustrated by our examples.
• 魅力的な曲線たちー拡がりゆく可積分幾何・差分幾何ー

  井ノ口 順一

  SUGAKU 社団法人 日本数学会 73 (1) 88 - 103 0039-470X 2021/01 [Refereed]
  
• A characterization of the alpha-connections on the statistical manifold of normal distributions

  Furuhata, Hitoshi, Inoguchi, Jun-ichi, Kobayashi, Shimpei

  Information Geometry SPRINGER 4 177 - 188 2511-2481 2020/10 [Refereed]
  
• The Gauss maps of Demoulin surfaces with conformal coordinates

  Inoguchi, Jun-ichi, Kobayashi, Shimpei

  Science China Mathematics SPRINGER 64 (7) 1479 - 1492 1674-7283 2020/10 [Refereed]
   

  Demoulin surfaces in the real projective 3-space are investigated. Our result enables us to establish a generalized Weierstrass type representation for definite Demoulin surfaces by virtue of primitive maps into a certain semi-Riemannian 6-symmetric space.
• Magnetic curves in the real special linear group

  Inoguchi, Jun-ichi, Munteanu, Marian Ioan

  Advances in Theoretical and Mathematical Physics International Press 23 (8) 2161 - 2205 1095-0761 2020/05 [Refereed][Not invited]
   

  We investigate contact magnetic curves in the real special linear group of degree 2. They are geodesics of the Hopf tubes over the projection curve. We prove that periodic contact magnetic curves in SL2R can be quantized in the set of rational numbers. Finally, we study contact homogeneous magnetic trajectories in SL2R and show that they project to horocycles in H-2(-4).
• Generation of Discrete Log-aesthetic Curves based on Similarity Geometry and Euclidean Geometry

  Miura,Kenjiro T, Kajiwara,Kenji, Inoguchi,Jun-ichi

  Proceedings of JSPE Semestrial Meeting 公益社団法人 精密工学会 2019 872 - 873 2019/09 [Not refereed][Not invited]
   

  近年の研究により，対数型美的曲線(log-aesthetic curve)は相似幾何により適切に定式化・解析できることが明らかとなった．本研究では，その離散化である離散対数型美的曲線（discrete log-aesthetic curve: dLAC）を相似幾何およびユークリッド幾何に基づいて生成する手法を提案する．
• Generation of Discrete Log-aesthetic Curves based on Similarity Geometry and Euclidean Geometry

  Miura,Kenjiro T, Kajiwara,Kenji, Inoguchi,Jun-ichi

  Proceedings of JSPE Semestrial Meeting 公益社団法人 精密工学会 2019 872 - 873 2019/09 [Not refereed][Not invited]
   

  近年の研究により，対数型美的曲線(log-aesthetic curve)は相似幾何により適切に定式化・解析できることが明らかとなった．本研究では，その離散化である離散対数型美的曲線（discrete log-aesthetic curve: dLAC）を相似幾何およびユークリッド幾何に基づいて生成する手法を提案する．
• Discrete local induction equation

  Inoguchi, Jun-ichi, Kajiwara, Kenji, Matsuura, Nozomu, Ohta, Yasuhiro

  Journal of Integrable Systems Oxford University Press 4 (1) 2019/06 [Refereed][Not invited]
  
• Grassmann geometry on the 3-dimensional non-unimodular Lie groups

  Inoguchi, Jun-ichi, Naitoh, Hiroo

  Hokkaido Mathematical Journal HOKKAIDO UNIV, DEPT MATHEMATICS 48 (2) 385 - 406 0385-4035 2019/06 [Refereed][Not invited]
   

  We study the Grassmann geometry of surfaces when the ambient space is a 3-dimensional non-unimodular Lie group with left invariant metric. This work together with our previous papers yield a complete classification of Grassmann geometry of orbit type in all 3-dimensional homogeneous spaces.
• Generalization of log-aesthetic curves via similarity geometry

  Inoguchi, Jun-ichi, Ziatdinov, Rushan, Miura, Kenjiro T

  Japan Journal of Industrial and Applied Mathematics Springer Japan 36 (1) 239 - 259 0916-7005 2019/01 [Refereed][Not invited]
   

  The class of log-aesthetic curves includes the logarithmic spiral, clothoid, and involute of a circle. Although most of these curves are expressed only by an integral form of the tangent vector, it is possible to interactively generate and deform them, thereby presenting many applications in industrial and graphic design. The use of the log-aesthetic curves in practical design, however, is still limited. Therefore, we should extend its formula to obtain curves that solve various practical design problems such as 𝐺𝑛 G^n Hermite interpolation, deformation, smoothing, data-point fitting, and blending plural curves. In this paper, we present a systematic approach to representing log-aesthetic curves via similarity geometry. In turn, this research provides a unified framework for various studies on log-aesthetic curves, particularly of log-aesthetic curve formulation.
• Affine spheres and finite gap solutions of Tzitzèica equation

  Inoguchi, Jun-ichi, Seiichi, Udagawa

  Journal of Physics Communications IOP Publishing home 2 (11) 2399-6528 2018/11 [Refereed][Not invited]
   

  The purpose of the present paper is to give an explicit form of the finite gap solutions to the Tzitzeica equation (2D Toda equation of type A_2^2) in terms of Riemann theta function. We give explicit expressions of proper affiene spheres derived from finite gap solutions to the Tzitzeica equation.
• Log-Aesthetic Curves: Similarity Geometry, Integrable Discretization and Variational Principles

  Jun-ichi Inoguchi, Kenji Kajiwara, Kenjiro T. Miura, Hyeongki Park, Wolfgang K. Schief

  arXiv:1808.03104 2018/08 [Not refereed][Not invited]
   

  In this paper, we consider a class of plane curves called log-aesthetic curves and their generalization which is used in CAGD. We consider these curves in the context of similarity geometry and characterize them in terms of a ``stationary'' integrable flow on plane curves which is governed by the Burgers equation. We propose a variational principle for these curves, leading to the stationary Burgers equation as the Euler-Lagrange equation. As an application of the formalism developed here, we propose a discretization of both the curves and the associated variational principle which preserves the underlying integrable structure. We finally present an algorithm for the generation of discrete log-aesthetic curves for given ${\rm G}^1$ data. The computation time to generate discrete log-aesthetic curves is much shorter than that for numerical discretizations of log-aesthetic curves due to the avoidance of fine numerical integration to calculate their shapes. Instead, only coarse summation is required.
• The hidden symmetry of chiral fields and the Riemann-Hilbert problem, revisited

  井ノ口, 順一

  京都大学数理解析研究所講究録 京都大学数理解析研究所 2071 1 - 16 2018/04 [Not refereed][Not invited]
   

  We generalize the Ueno-Nakamura theory and the Uhlenbeck-Segal theory for harmonic maps of Riemann surfaces into compact semi-simple Lie groups to those of (affine) harmonic maps into general Lie groups with torsion free bi-invariant connection in terms of loop groups
• Fairness metric of plane curves defined with similarity geometry invariants

  Kenjiro T. Miura, Sho Suzuki, R. U. Gobithaasan, Shin Usuki, Jun-ichi Inoguchi, Masayuki Sato, Kenji Kajiwara, Yasuhiro Shimizu

  Computer-Aided Design and Applications 15 (2) 256 - 263 1686-4360 2018/03/04 [Refereed][Not invited]
   

  A curve is considered fair if it consists of continuous and few monotonic curvature segments. Polynomial curves such as Bézier and B-spline curves have complex curvature function, hence the curvature profile may oscillate easily with a little tweak of control points. Thus, bending energy and shear deformation energy are common fairness metrics used to produce curves with monotonic curvature profiles. The fairness metrics are used not just to evaluate the quality of curves, but it also aids in reaching to the final design. In this paper, we propose two types of fairness metric functionals to fair plane curves defined by the similarity geometry invariants, i.e. similarity curvature and its reciprocal to extend a variety of aesthetic fairing metrics. We illustrate numerical examples to show how log-aesthetic curves change depending on σ and G1 constraints. We extend LAC by modifying the integrand of the functionals and obtain quasi aesthetic curves. We also propose σ-curve to introduce symmetry concept for the log-aesthetic curve.
• Log-aesthetic curves as similarity geometric analogue of Euler's elasticae

  Jun-ichi Inoguchi, Kenji Kajiwara, Kenjiro T. Miura, Masayuki Sato, Wolfgang K. Schief, Yasuhiro Shimizu

  Computer Aided Geometric Design 61 1 - 5 0167-8396 2018/03/01 [Refereed][Not invited]
   

  In this paper we consider the log-aesthetic curves and their generalization which are used in CAGD. We consider those curves under similarity geometry and characterize them as stationary integrable flow on plane curves which is governed by the Burgers equation. We propose a variational formulation of those curves whose Euler–Lagrange equation yields the stationary Burgers equation. Our result suggests that the log-aesthetic curves and their generalization can be regarded as the similarity geometric analogue of Euler's elasticae.
• 対数型美的曲線の相似幾何学的定式化

  井ノ口, 順一

  2018年度精密工学会春季大会シンポジウム資料集 54 - 57 2018/03 [Not refereed][Not invited]
  
• Elasticae in similarity geometry and their discretization.

  井ノ口, 順一, 梶原健司, 三浦憲二郎, 朴炯基, Schief, Wolfgang

  Reports of RIAM Symposium No.29AO-S7 New Trends in Nonlinear Waves - Theory and Applications - 九州大学応用力学研究所 29AO-S7 61 - 68 2018/03 [Refereed][Not invited]
   

  弾性エネルギーの臨界点である平面曲線は弾性曲線とよばれる．弾性曲線はmKdV 方程式と深く関連し，実際，平面曲線の等周変形を記述するmKdV 方程式の進行波解から定まる曲線が弾性曲線である．本稿では相似幾何学の枠組みを用いて工業意匠設計で用いられている対数型美的曲線（LAC）とその一般化を考察し，それらが平面曲線の等角変形を記述するBurgers 方程式の定常解として特徴付けられること，および適当なエネルギーの臨界点として定式化できることを報告する．この結果は，LAC が弾性曲線の相似幾何類似であることを示唆する．以上の理論的枠組みに基づき，可積分離散化の手法を応用したLAC の離散化を提案する．さらに，それらを離散変分問題の解として定式化する．
• 19世紀の微分幾何

  井ノ口, 順一

  津田塾大学 数学・計算機科学研究所報 津田塾大学 38 (38) 68 - 80 2017/03 [Not refereed][Not invited]
  
• Periodic magnetic curves in Berger spheres

  Jun-Ichi Inoguchi, Marian Ioan Munteanu

  Tohoku Mathematical Journal 69 (1) 113 - 128 0040-8735 2017/03/01 [Refereed][Not invited]
   

  It is an interesting question whether a given equation of motion has a periodic solution or not, and in the positive case to describe it. We investigate periodic magnetic curves in elliptic Sasakian space forms and we obtain a quantization principle for periodic magnetic flowlines on Berger spheres. We give a criterion for periodicity of magnetic curves on the unit sphere S3.
• Finite gap solutions for horizontal minimal surfaces of finite type in 5-sphere

  Inoguchi, Jun-ichi, Taniguchi, Tetsuya, Seiichi, Udagawa

  Journal of Integrable Systems Oxford University Press 1 (1) 2016/12 [Refereed][Not invited]
  
• A loop group method for affine harmonic maps into Lie groups

  Josef F. Dorfraeister, Jun-ichi Inoguchi, Shimpei Kobayashi

  ADVANCES IN MATHEMATICS 298 207 - 253 0001-8708 2016/08 [Refereed][Not invited]
   

  We generalize the UhlenbeckSegal theory for harmonic maps into compact semi-simple Lie groups to general Lie groups equipped with torsion free bi-invariant connection. (C) 2016 Elsevier Inc. All rights reserved.
• Magnetic curves in cosymplectic manifolds

  Simona-Luiza Druţă-Romaniuc, Jun-ichi Inoguchi, Marian Ioan Munteanu, Ana Irina Nistor

  Reports on Mathematical Physics 78 (1) 33 - 48 0034-4877 2016/08 [Refereed]
  
• dNLS Flow on Discrete Space Curves

  Sampei Hirose, Jun-ichi Inoguchi, Kenji Kajiwara, Nozomu Matsuura, Yasuhiro Ohta

  Mathematical Progress in Expressive Image Synthesis III, Mathematics for Industry 24 137 - 149 2016/06 [Refereed][Not invited]
   

  The local induction equation, or the binormal flow on space curves is a well-known model of deformation of space curves as it describes the dynamics of vortex filaments, and the complex curvature is governed by the nonlinear Schr\"odinger equation (NLS). In this paper, we present its discrete analogue, namely, a model of deformation of discrete space curves by the discrete nonlinear Schr\"odinger equation (dNLS). We also present explicit formulas for both NLS and dNLS flows in terms of the $\tau$ function of the 2-component KP hierarchy.
• On the Bernstein problem in the three-dimensional Heisenberg group

  Josef F. Dorfmeister, Jun-ichi Inoguchi, Shimpei Kobayashi

  Canadian Mathematical Bulletin 59 (01) 50 - 61 0008-4395 2016/03 [Refereed]
   

  Abstract In this note we present a simple alternative proof for the Bernstein problem in the threedimensional Heisenberg group Nil₃ by using the loop group technique. We clarify the geometric meaning of the two-parameter ambiguity of entire minimal graphs with prescribed Abresch- Rosenberg diòerential.
• A loop group method for minimal surfaces in the three-dimensional Heisenberg group

  Josef F. Dorfmeister, Jun-Ichi Inoguchi, Shimpei Kobayashi

  Asian Journal of Mathematics 20 (3) 409 - 448 1093-6106 2016
• dNLS Flow on Discrete Space Curves

  Sampei Hirose, Jun-ichi Inoguchi, Kenji Kajiwara, Nozomu Matsuura, Yasuhiro Ohta

  MI Lecture Note 64 93 - 102 2015/09 [Refereed][Not invited]
   

  The local induction equation, or the binormal flow on space curves is a well-known model of deformation of space curves as it describes the dynamics of vortex filaments, and the complex curvature is governed by the nonlinear Schödinger equation (NLS). In this paper, we present its discrete analogue, namely, a model of deformation of discrete space curves by the discrete nonlinear Schrödinger equation (dNLS). We also present explicit formulas for both NLS and dNLS flows in terms of the tau function of the 2-component KP hierarchy.
• Attractive plane curves in Differential Geometry

  Inoguchi, Jun-ichi

  MI Lecture Note Kyushu University 64 121 - 124 2188-1200 2015/09 [Not refereed][Not invited]
  
• Harmonic maps in almost contact geometry

  Inoguchi,Jun-ichi

  SUT Journal of Mathematics 50 (2) 353 - 382 0916-5746 2014/12 [Refereed][Not invited]
   

  We study harmonicity and pluriharmonicity of holomorphic maps in almost contact geometry.
• Constant Gaussian curvature surfaces in the 3-sphere via loop groups

  David Brander, Jun-ichi Inoguchi, Shimpei Kobayashi

  Pacific Journal of Mathematics 269 (2) 281 - 303 0030-8730 2014/07/26 [Refereed]
  
• Discrete mKdV and discrete sine-Gordon flows on discrete space curves

  Jun-ichi Inoguchi, Kenji Kajiwara, Nozomu Matsuura, Yasuhiro Ohta

  JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL 47 (23) 235202  1751-8113 2014/06 [Refereed][Not invited]
   

  In this paper, we consider the discrete deformation of the discrete space curves with constant torsion described by the discrete mKdV or the discrete sine-Gordon equations, and show that it is formulated as the torsion-preserving equidistant deformation on the osculating plane which satisfies the isoperimetric condition. The curve is reconstructed from the deformation data by using the Sym-Tafel formula. The isoperimetric equidistant deformation of the space curves does not preserve the torsion in general. However, it is possible to construct the torsion-preserving deformation by tuning the deformation parameters. Further, it is also possible to make an arbitrary choice of the deformation described by the discrete mKdV equation or by the discrete sine-Gordon equation at each step. We finally show that the discrete deformation of discrete space curves yields the discrete K-surfaces.
• Constant mean curvature surfaces in hyperbolic 3-space via loop groups

  Josef F. Dorfmeister, Jun-ichi Inoguchi, Shimpei Kobayashi

  JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK 686 1 - 36 0075-4102 2014/01 [Refereed][Not invited]
   

  In hyperbolic 3-space H-3 surfaces of constant mean curvature H come in three types, corresponding to the cases 0 <= H < 1, H = 1, H > 1. Via the Lawson correspondence the latter two cases correspond to constant mean curvature surfaces in Euclidean 3-space E-3 with H = 0 and H not equal 0, respectively. These surface classes have been investigated intensively in the literature. For the case 0 <= H < 1 there is no Lawson correspondence in Euclidean space and there are relatively few publications. Examples have been difficult to construct. In this paper we present a generalized Weierstrass type representation for surfaces of constant mean curvature in H-3 with particular emphasis on the case of mean curvature 0 <= H < 1. In particular, the generalized Weierstrass type representation presented in this paper enables us to construct simultaneously minimal surfaces (H = 0) and non-minimal constant mean curvature surfaces (0 < H < 1).
• Magnetic maps

  Jun-Ichi Inoguchi, Marian Ioan Munteanu

  International Journal of Geometric Methods in Modern Physics 11 (6) 1450058  0219-8878 2014 [Refereed][Not invited]
   

  In this paper, we introduce the notion of magnetic maps between Riemannian manifolds. They are generalizations of both magnetic curves and harmonic maps. We provide some fundamental examples of magnetic maps. Furthermore, we study some classes of magnetic surfaces in Euclidean 3-space. © 2014 World Scientific Publishing Company.
• Gauss maps of constant mean curvature surfaces in three-dimensional homogeneous spaces

  Jun-ichi Inoguchi, Joeri Van der Veken

  Kobe Journal of Mathematics 31 (1-2) 45 - 62 2014 [Refereed][Not invited]
  
• Contact metric hypersurfaces in complex space forms

  Jong Taek CHO, Jun-ichi INOGUCHI

  Differential Geometry of Submanifolds and its Related Topics 2013/10/29 [Refereed][Invited]
  
• Integrable discretizations of the Dym equation

  Bao-Feng Feng, Jun-ichi Inoguchi, Kenji Kajiwara, Ken-ichi Maruno, Yasuhiro Ohta

  FRONTIERS OF MATHEMATICS IN CHINA 8 (5) 1017 - 1029 1673-3452 2013/10 [Refereed][Not invited]
   

  Integrable discretizations of the complex and real Dym equations are proposed. N-soliton solutions for both semi-discrete and fully discrete analogues of the complex and real Dym equations are also presented.
• Semi-discrete analogues of the elastic beam equation and the short pulse equation

  K. Maruno, B.F. Feng, J. Inoguchi, K. Kajiwara, Y. Ohta

  Proceedings of 2013 International Symposium on Nonlinear Theory and its Applications 278 - 281 2013/09 [Refereed][Not invited]
   

  Two integrable nonlinear differential- difference systems, semi-discrete analogues of the Wadati-Konno-Ichikawa elastic beam equation and the short pulse equation, are constructed by using a geometric approach.
• Biminimal curves in $2$-dimensional space forms

  Jun-Ichi Inoguchi, Ji-Eun Lee

  Communications of the Korean Mathematical Society 27 (4) 771 - 780 1225-1763 2012/10/31 [Refereed]
  
• MOTION AND BACKLUND TRANSFORMATIONS OF DISCRETE PLANE CURVES

  Jun-ichi Inoguchi, Kenji Kajiwara, Nozomu Matsuura, Yasuhiro Ohta

  KYUSHU JOURNAL OF MATHEMATICS Faculty of Mathematics, Kyushu University,九州大学大学院数理学研究院 66 (2) 303 - 324 1340-6116 2012/09 [Refereed][Not invited]
   

  We construct explicit solutions to the discrete motion of discrete plane curves that has been introduced by one of the authors recently. Explicit formulas in terms of the tau function are presented. Transformation theory of the motions of both smooth and discrete curves is developed simultaneously.
• Affine biharmonic curves in 3-dimensional homogeneous geometries

  Jun-ichi Inoguchi, Ji-Eun Lee

  Mediterranean Journal of Mathematics 10 (1) 571 - 592 1660-5446 2012/04/20 [Refereed]
  
• Minimal translation surfaces in the Heisenberg group $\mathrm{Nil}_3$

  Jun-ichi Inoguchi, Rafael López, Marian-Ioan Munteanu

  Geometriae Dedicata 161 (1) 221 - 231 0046-5755 2012/02/25 [Refereed]
  
• Explicit solutions to the semi-discrete modified KdV equation and motion of discrete plane curves

  Jun-ichi Inoguchi, Kenji Kajiwara, Nozomu Matsuura, Yasuhiro Ohta

  JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL 45 (4) 045206  1751-8113 2012/02 [Refereed][Not invited]
   

  We construct explicit solutions to continuous motion of discrete plane curves described by a semi-discrete potential modified KdV equation. Explicit formulas in terms of the tau function are presented. Backlund transformations of the discrete curves are also discussed. We finally consider the continuous limit of discrete motion of discrete plane curves described by the discrete potential modified KdV equation to motion of smooth plane curves characterized by the potential modified KdV equation.
• Discrete integrable systems and hodograph transformations arising from motions of discrete plane curves

  Bao-Feng Feng, Jun-ichi Inoguchi, Kenji Kajiwara, Ken-ichi Maruno, Yasuhiro Ohta

  JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL 44 (39) 395201  1751-8113 2011/09 [Refereed][Not invited]
   

  We consider integrable discretizations of some soliton equations associated with the motions of plane curves: the Wadati-Konno-Ichikawa elastic beam equation, the complex Dym equation and the short pulse equation. They are related to the modified KdV or the sine-Gordon equations by the hodograph transformations. Based on the observation that the hodograph transformations are regarded as the Euler-Lagrange transformations of the curve motions, we construct the discrete analogues of the hodograph transformations, which yield integrable discretizations of those soliton equations.
• semi-discrete modified KdV方程式と平面離散曲線の時間発展

  井ノ口順一, 梶原健司, 松浦望, 太田泰広

  九州大学応用力学研究所研究集会報告 22AO-S8 75 - 81 2011/03 [Refereed][Not invited]
  
• Grassmann geometry on the 3-dimensional unimodular lie groups II

  Jun-Ichi Inoguchi, Hiroo Naitoh

  Hokkaido Mathematical Journal 40 (3) 411 - 429 0385-4035 2011 [Refereed][Not invited]
   

  We study the Grassmann geometry of surfaces in the special real linear group SL(2, R).
• On $\varphi$-Einstein contact Riemannian manifolds

  Jong Taek Cho, Jun-ichi Inoguchi

  Mediterranean Journal of Mathematics 7 (2) 143 - 167 1660-5446 2010/04/22 [Refereed]
  
• Grassmann geometry on the 3-dimensional unimodular Lie groups I

  Jun-ichi Inoguchi, Hiroo Naitoh

  HOKKAIDO MATHEMATICAL JOURNAL 38 (3) 427 - 496 0385-4035 2009/08 [Refereed][Not invited]
   

  We study the Grassmann geometry of surfaces when the ambient space is a 3-dimensional unimodular Lie group with left invariant metric, that is, it is one of the 3-dimensional commutative Lie group, the 3-dimensional Heisenberg group, the groups of rigid motions on the Euclidean or the Minkowski planes, the special unitary group SU(2), and the special real linear group SL(2, R).
• LIGHTLIKE SURFACES IN MINKOWSKI 3-SPACE

  Jun-Ichi Inoguchi, Sungwook Lee

  INTERNATIONAL JOURNAL OF GEOMETRIC METHODS IN MODERN PHYSICS 6 (2) 267 - 283 0219-8878 2009/03 [Refereed][Not invited]
   

  We study lightlike surfaces in Minkowski 3-space.
• A complete classification of parallel surfaces in three-dimensional homogeneous spaces

  Jun-ichi Inoguchi, Joeri Van der Veken

  GEOMETRIAE DEDICATA 131 (1) 159 - 172 0046-5755 2008/02 [Refereed][Not invited]
   

  We complete the classification of surfaces with parallel second fundamental form in all three-dimensional homogeneous spaces.
• A Weierstrass type representation for minimal surfaces in Sol

  Jun-Ichi Inoguchi, Sungwook Lee

  PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 136 (6) 2209 - 2216 0002-9939 2008 [Refereed][Not invited]
   

  The normal Gauss map of a minimal surface in the model space Sol of solvegeometry is a harmonic map with respect to a certain singular Riemannian metric on the extended complex plane.
• Parallel surfaces in the motion groups E(1,1) and E(2)

  Inoguchi, Jun-ichi, Van der Veken, Joeri

  Bulletin of the Belgian Mathematical Society - Simon Stevin Belgian Mathematical Society 14 (2) 321 - 332 2007/06 [Refereed][Not invited]
   

  We give a classification of parallel surfaces in the groups of rigid motions of Minkowski plane and Euclidean plane, equipped with a general left-invariant metric. Our result completes the classification of parallel surfaces in the eight three-dimensional model geometries of Thurston and in three-dimensional unimodular Lie groups with maximal isometry group.
• Biminimal submanifolds in contact 3-manifolds

  Jun-ichi Inoguchi

  BALKAN JOURNAL OF GEOMETRY AND ITS APPLICATIONS 12 (1) 56 - 67 1224-2780 2007 [Refereed][Not invited]
   

  We study biminimal submanifolds in contact 3-manifolds. In particular, biminimal curves in homogeneous contact Riemannian 3-manifolds and biminimal Hopf cylinders in Sasakian 3-space forms are investigated.
• On slant curves in Sasakian 3-manifolds

  Jong Taek Cho, Jun-ichi Inoguchi, Ji-Eun Lee

  BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY 74 (3) 359 - 367 0004-9727 2006/12 [Refereed][Not invited]
   

  A classical theorem by Lancret says that a curve in Euclidean 3-space is of constant slope if and only if its ratio of curvature and torsion is constant. In this paper we study Lancret type problems for curves in Sasakian 3-manifolds.
• Characterizations of Bianchi-Backlund transformations of constant mean curvature surfaces

  S Kobayashi, J Inoguchi

  INTERNATIONAL JOURNAL OF MATHEMATICS 16 (2) 101 - 110 0129-167X 2005/02 [Refereed][Not invited]
   

  We show that Bianchi-Backlund transformation of a constant mean curvature surface is equivalent to the Darboux transformation and the simple type dressing.
• Timelike minimal surfaces via loop groups

  J. Inoguchi, M. Toda

  Acta Applicandae Mathematicae 83 (3) 313 - 355 0167-8019 2004/09 [Refereed]
  
• Schrodinger flows, binormal motion for curves and the second AKNS-hierarchies

  Q Ding, J Inoguchi

  CHAOS SOLITONS & FRACTALS 21 (3) 669 - 677 0960-0779 2004/07 [Refereed][Not invited]
   

  In this paper, we present a unified geometric interpretation of the second AKNS-hierarchies via the geometric concept of Schrodinger flows in the category of symplectic manifolds and binormal motion for curves in the Minkowski 3-space. (C) 2004 Elsevier Ltd. All rights reserved.
• Invariant minimal surfaces in the real special linear group of degree 2

  Jun-ichi Inoguchi

  Italian Journal of Pure and Applied Mathematics 16 61 - 80 2004 [Refereed]
  
• Minimal surfaces in 3-dimensional solvable Lie groups

  J Inoguchi

  CHINESE ANNALS OF MATHEMATICS SERIES B 24 (1) 73 - 84 0252-9599 2003/01 [Refereed][Not invited]
   

  The author studies minimal surfaces in 3-dimensional solvable Lie, groups with left invariant Riemannian metrics. A Weierstrass type integral representation formula for minimal surfaces is obtained.
• Timelike Bonnet surfaces in Lorentzian space forms

  A Fujioka, J Inoguchi

  DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS 18 (1) 103 - 111 0926-2245 2003/01 [Refereed][Not invited]
   

  We study timelike surfaces in Lorentzian space forms which admit a one-parameter family of isometric deformations preserving the mean curvature. (C) 2002 Elsevier Science B.V. All rights reserved.
• On time-like surfaces of positive constant Gaussian curvature and imaginary principal curvatures

  C.H. Gu, H.S. Hu, Jun-Ichi Inoguchi

  Journal of Geometry and Physics 41 (4) 296 - 311 0393-0440 2002/04
• Surfaces with parallel second fundamental form in Bianchi-Cartan-Vranceanu spaces

  Mohamed Belkhelfa, Franki Dillen, Jun-ichi Inoguchi

  PDEs, Submanifolds and Affine Differential Geometry 67 - 87 2002 [Refereed]
  
• Darboux transformations on timelike constant mean curvature surfaces

  J Inoguchi

  JOURNAL OF GEOMETRY AND PHYSICS 32 (1) 57 - 78 0393-0440 1999/11 [Refereed][Not invited]
   

  We give loop group theoretic reformulated Backlund transformations on constant mean curvature timelike surfaces in Minkowski 3-space. Further we present 1-soliton surfaces explicitly. (C) 1999 Elsevier Science B.V. All rights reserved.
• On some generalisations of constant mean curvature surfaces

  Atsushi Fujioka, Jun-ichi Inoguchi

  Lobachevskii Journal of Mathematics 3 73 - 95 1999
• Timelike surfaces of constant mean curvature in Minkowski 3-space

  Jun-ichi INOGUCHI

  Tokyo Journal of Mathematics 21 (1) 0387-3870 1998/06/01 [Refereed]
  
• Bonnet surfaces with constant curvature

  Atsushi Fujioka, Jun-ichi Inoguchi

  Results in Mathematics 33 (3-4) 288 - 293 0378-6218 1998/05 [Refereed]
  

MISC

• 曲面と解析力学（特集 曲線と曲面を考える）

  井ノ口順一  数理科学  62-  (2)  54  -61  2024/02
• Sine-Gordon方程式の解法とその離散化

  宇田川 誠一, 井ノ口 順一, 梶原 健司  日本大学医学部一般教育研究紀要 / 日本大学医学部一般教育 編  (50)  9  -26  2022/12
• ベクトル解析から曲面論へ (特集 ベクトル解析の力 : 多彩な問題を通じてその姿に迫る)

  井ノ口 順一  数理科学  52-  (8)  15  -20  2014/08
• Deformation of Space Discrete Curves by Discrete mKdV and Discrete Sine-Gordon Equations (Novel Development of Nonlinear Discrete Integrable Systems)

  INOGUCHI Jun-ichi, KAJIWARA Kenji, MATSUURA Nozomu, OHTA Yasuhiro  RIMS Kokyuroku Bessatsu  47-  1  -21  2014/06  

  "Novel Development of Nonlinear Discrete Integrable Systems". September 2～4, 2013. edited by Junta Matsukidaira. The papers presented in this volume of RIMS Kokyuroku Bessatsu are in final form and refereed.
• Submanifold geometry of contact 3-manifolds (Development in Differential Geometry of Submanifolds)

  Inoguchi Jun-ichi  RIMS Kokyuroku  1880-  72  -99  2014/04
• 自己適合移動格子スキームとミンコフスキー平面上の離散曲線の運動について

  丸野健一, 梶原健司, 井ノ口順一, 太田泰広, FENG Baofeng  日本応用数理学会年会講演予稿集(CD-ROM)  2014-  2014
• Discrete differential geometry of surfaces (Progress in Mathematics of Integrable Systems)

  Inoguchi Jun-ichi  RIMS Kokyuroku Bessatsu  30-  77  -99  2012/04
• 離散平面曲線の時間発展に現れる離散可積分系と離散ホドグラフ変換

  FENG Baofeng, 井ノ口順一, 梶原健司, 丸野健一, 太田泰広  日本応用数理学会年会講演予稿集  2011-  2011
• A note on almost contact Riemannian 3-manifolds

  Inoguchi Jun-ichi  Bulletin of the Yamagata University. Natural science  17-  (1)  1  -6  2010/02/15  

  We investigate curvatures of normal almost contact Riemannian 3-manifolds. In particular, we show that Kenmotsu 3-manifolds of constant scalar curvature are of constant curvature
• On Homogeneous Contact 3-Manifolds

  Inoguchi Jun-ichi  Bulletin of the Faculty of Education, Utsunomiya University  59-  1  -12  2009/03/10
• 戸田方程式と微分幾何

  井ノ口順一  戸田格子40周年 非線形波動研究の歩みと展望  47  -62  2008
• Geometry, it's a secret ingredient that makes integrable system theory interesing for us (Perspective and Application of Integrable Systems)

  Inoguchi Jun-ichi  RIMS Kokyuroku  1422-  134  -153  2005/04
• Real hypersurfaces of complex space forms with symmetric Ricci *-tensor

  Hamada Tatsuyoshi, Inoguchi Jun-ichi  Memoirs of the Faculty of Science and Engineering, Shimane University. Series B, Mathematical science  38-  1–5  2005/03
• Integrable systems in unfashionable geometries (Theory of integrable systems and related topics : State of arts and perspectives)

  Inoguchi Jun-ichi  RIMS Kokyuroku  1400-  127  -144  2004/11
• A QUICK INTRODUCTION TO DISCRETISED PROJECTIVE DIFFERENTIAL GEOMETRY (Development in Discrete Integrable Systems : Ultra-Discretization, Quantization)

  Inoguchi Jun-ichi  RIMS Kokyuroku  1221-  112  -124  2001/07
• INTEGRABLE SURFACES AND THEIR DISCRETISATION (Recent Topics on Discrete Integrable Systems)

  Inoguchi Jun-Ichi  RIMS Kokyuroku  1170-  9  -22  2000/09
• Bonnet Surfaces with Constant Curvature (Homogeneous Structures and Theory of Submanifolds)

  Fujioka Atsushi, Inoguchi Jun-ichi  RIMS Kokyuroku  1069-  73  -82  1998/11
• SURFACES IN MINKOWSI 3-SPACE AND HARMONIC MAPS

  INOGUCHI JUN-ICHI  RIMS Kokyuroku  995-  58  -69  1997/05

Books etc

• Textbook: Analytic Geometry and Linear Algebra

  Inoguchi Jun-ichi (Single work)
  現代数学社 2024/05 (ISBN: 9784768706350) 360
• 1+3 dimensional world: From Surfcaes to Manifolds and Spacetimes

  Inoguchi, Jun-ichi (Single work)
  Gendai Sugakusha 2023/04 (ISBN: 9784768706046) 268
• Contact Geometry of Slant Submanifolds

  Inoguchi, Jun-ichi, Munteanu, Marian Ioan (ContributorSlant Curves and Magnetic Curves)
  Springer Nature Singapore Pte Ltd. 2022/06 (ISBN: 9789811600166) 

  This chapter treats slant curves and magnetic curves in almost contact metric manifolds. Special attention is paid to magnetic curves in Sasakian manifolds. We describe magnetic slant curves in Sasakian space forms.
• 1+2 dimensional world: Curves and Surfaces in Minkowski Space

  Inoguchi, Jun-ichi (Single work)
  Gendai Sugakusha 2022/02 (ISBN: 9784768705766) 204
• 1+1 dimensional world: Geometry of Minkowski Plane

  Inoguchi, Jun-ichi (Single work)
  現代数学社 2021/12 189
• A First Course to Vector Analysis

  Inoguchi, Jun-ichi (Single work)
  現代数学社 2020/12 (ISBN: 9784768705476) 396
• A First Course to Partial differentiation

  Inoguchi, Jun-ichi (Single work)
  現代数学社 2019/09 (ISBN: 9784768705162) 222
• 解析学百科II 可積分系の数理

  Inoguchi, Jun-ichi (Contributor幾何学と可積分系)
  朝倉書店 2018/03
• A First Course to Lie Algebras

  Inoguchi, Jun-ichi (Single work)
  現代数学社 2018/02 (ISBN: 9784768704714) 280
• A First Course to Lie Groups

  Inoguchi, Jun-ichi (Single work)
  現代数学社 2017/07 (ISBN: 9784768704707) 272
• Surface Geometry and Integrable Systems

  Inoguchi,Jun-ichi (Single work)
  Asakura Shoten 2015/10 (ISBN: 9784254117684) vi, 212p
• 応用数理ハンドブック

  Inoguchi, Jun-ichi (Contributor幾何学と可積分系)
  朝倉書店 2013/11
• 負定曲率曲面とサイン・ゴルドン方程式

  Inoguchi, Jun-ichi (Single work)
  Saitama University 2012/04
• 離散可積分系・離散微分幾何チュートリアル2012

  Inoguchi, Jun-ichi (Contributor可積分幾何入門)
  Kyushu University 2012/03
• リッカチのひ・み・つ

  Inoguchi, Jun-ichi (Single work)
  日本評論社 2010/09
• どこにでも居る幾何. アサガオから宇宙まで

  Inoguchi, Jun-ichi (Single work)
  日本評論社 2010/09 (ISBN: 9784535786110)
• Plane curves and Solitons

  Inoguchi, Jun-ichi (Single work)
  朝倉書店 2010/03 (ISBN: 9784254117349)
• いろいろな幾何と曲線の時間発展

  Inoguchi, Jun-ichi (Single work)
  Hokkaido University 2008/09
• 幾何学いろいろ

  Inoguchi, Jun-ichi (Single work)
  日本評論社 2007/11 (ISBN: 9784535784628)
• 曲面の微分幾何学とソリトン方程式 : 可積分幾何入門

  Inoguchi, Jun-ichi (Contributor負定曲率曲面とサイン・ゴルドン方程式)
  立教大学 2005/10

Presentations

• 3次元接触多様体の磁場軌道

  井ノ口順一

  接触構造、特異点、微分方程式及びその周辺  2024/01  金沢大学サテライト・プラザ
• Differential Geometry of Industrial Shape Design  [Invited]

  Jun-ichi Inoguchi

  第22回水戸幾何セミナー  2023/11
• Contact geometry and magnetic trajectories  [Invited]

  Jun-ichi Inoguchi

  横国大幾何トポロジーセミナー  2023/10
• Discrete Differential Geometry. Developments and Perspectives  [Invited]

  Jun-ichi Inoguchi

  日本建築学会大会（近畿）構造部門（シェル・空間構造）パネルディスカッション  2023/09
• Submanifold Geometry of LCK surfaces  [Invited]

  Jun-ichi Inoguchi

  The 20th Mito Geometry Seminar  2023/02
• Lie sphere geometry: Is it future promising?  [Invited]

  Jun-ichi Inoguchi

  Mini-Workshop "Differential Geometry, Integrable Systems, and Shape Generation"  2023/02
• アフィン接続と接触構造に関する話題から  [Invited]

  井ノ口, 順一

  福岡大学 微分幾何研究会  2021/11  福岡大学（ハイブリッド）
• Similarity geometry revisited: Differential geometry and CAGD  [Invited]

  井ノ口, 順一

  8th European Congress of Mathematics (8ECM) Minisymposium Differential Geometry: Old and New  2021/06  スロベニア Portoroz  European Mathematical Society
  
• 「離散微分幾何と有限要素法の融合，建築とCGへの応用」  [Not invited]

  井ノ口, 順一

  AIMaP集会「離散微分幾何と有限要素法の融合，建築とCGへの応用」  2020/12  九州大学（ハイブリッド）  筑波大学数理科学研究コア
  
• 3次元球面内の曲線に関する話題  [Invited]

  井ノ口, 順一

  北川義久教授ご退職記念研究集会  2020/11  東京工業大学（オンライン）
• Tzitzeica方程式をめぐって  [Invited]

  井ノ口, 順一

  リーマン面に関連する 位相幾何学  2020/08  東京大学（オンライン）
• Slant Curves in contact geometry  [Invited]

  井ノ口, 順一

  International Workshop on Geometry of Submanifolds, 2019  2019/11  トルコ Istanbul center for mathematical Science
• 3次元等質空間内の曲面のグラスマン幾何  [Invited]

  井ノ口, 順一

  北九州幾何学研究集会2019  2019/07  九州工業大学
• Harmonic map into Lie groups, revisited  [Invited]

  井ノ口, 順一

  The Joint International Meeting of the Chinese mathematical Society and American Mathematical Society  2018/06  中華人民共和国 復旦大学
• Curve flows, integrable systems and industrial design  [Invited]

  井ノ口, 順一

  Integrable Geometry at Bayrischzell  2018/05  ドイツ Bayrischzell Gasthof zur Post
• 対数型美的曲線の相似幾何学的定式化  [Invited]

  井ノ口, 順一

  AIMaP数学応用シンポジウム：精密工学と幾何学の新たな出会い  2018/03  中央大学  公益社団法人 精密工学会
  
• Elasticae in similarity geometry and their discretization.  [Not invited]

  井ノ口, 順一, 梶原健司, 三浦憲二郎, 朴炯基, Schief, Wolfgang

  非線形波動研究の新潮流 .理論とその応用  2017/11  九州大学応用力学研究所
• Grassmann geometry of surfaces in 3-dimensional homogeneous spaces  [Invited]

  井ノ口, 順一

  INTERNATIONAL CONFERENCE ON APPLIED AND PURE MATHEMATICS (ICAPM 2017)  2017/11  ルーマニア "Gheorghe Asachi" Technical University, Iaşi
• 相似幾何不変量による平面曲線 の Fairness 測度  [Not invited]

  三浦憲二郎, 鈴木晶, 臼杵深, Gobithaasan, Rudrusamy, 井ノ口, 順一, 佐藤雅之, 梶原健司, 清水保弘

  日本応用数理学会2017年度年会  2017/09  武蔵野大学
• 対数型美的曲線の相似幾何における平面曲線に対する変分原理による 定式化  [Not invited]

  井ノ口, 順一, 梶原健司, 三浦憲二郎, Schief, Wolfgang

  日本応用数理学会2017年度年会  2017/09  武蔵野大学
• 平面曲線と意匠設計  [Invited]

  井ノ口, 順一

  第63回幾何学シンポジウム  2016/08  岡山大学
• Grasmann geometry of 3-dimensional homogeneous spaces  [Invited]

  井ノ口,順一

  内藤博夫先生退職記念研究集会  2016/03  山口大学
• 魅力的な曲線たち  [Invited]

  井ノ口,順一

  日本数学会北海道支部会  2015/12  北海道大学  日本数学会北海道支部会
  
• Attractive plane curves in Differential Geometry  [Invited]

  Inoguchi,Jun-ichi

  Mathematical Progress in Expressive Image Synthesis 2015  2015/09  日本 九州大学  九州大学
  
• New examples of biharmonic hypersurfaces  [Not invited]

  井ノ口,順一

  International Workshop on Finite Type Submanifolds, 2014  2014/09  トルコ イスタンブール工科大学

Association Memberships

• 日本数学会   THE SOCIETY FOR SCIENCE ON FORM, JAPAN   The Japan Society for Industrial and Applied Mathematics   

Research Projects

• Construction of harmonic maps into hyperbolic space and applications to surface theory in homogeneous spaces

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2019/04 -2023/03 

  Author : 井ノ口順一

   

  本研究の主要課題である「調和写像の構成と等質空間内の曲面論への応用」に関し、等質空間の幾何学の観点から研究を遂行し以下の研究成果を得た。 （１）前年度に得た、双曲平面Hに値をもつ「1径数変換群の作用で同変的な調和写像」を用いた３次元ハイゼンベルグ群内の「対称性を備えた極小曲面」の構成法（Dorfmeister氏、小林氏との共同研究）に関し、具体例の詳細な記述を得ることに成功した。（３）Hと数直線の直積空間HXRの軌道型グラスマン幾何に関する前年度の研究成果と調和写像の関連を深めるために新たな研究視点と手法を導入した。Hを複素部分多様体として含む４次元等質空間（サーストン幾何の４次元類似）である2種の空間Sol40およびSol41の曲線論と曲面論を創始した（部分多様体論は未開であった）。調和写像の伝統的構成法である「極小部分多様体の構成」に着手した。極小部分多様体を複素構造の観点から構成し、いくつかの設定下で分類した（Erjavec氏との共著論文投稿中）。さらにJ-軌道（磁場軌道に相当）を分類した。（３）リー球面幾何学の建築構造設計への応用に関する研究成果を国際会議論文として発表した（横須賀氏、大崎氏、本間氏との共著）（４）(1)から(3)の研究過程において、情報幾何学への予期せぬ応用が発見された。正規分布のなす統計多様体に指定される甘利-Chentsov接続は数理統計学に由来するものであり、微分幾何学的な意味、必然性は未解明であった。正規分布のなす統計多様体を統計リー群として実現することによりある種の対称性をもつ唯一の線型接続であることを証明した（古畑氏、小林氏との共著論文を発表）（５）前年度に行った3次元佐々木空間形における磁場軌道の分類を論文発表した（Munteanu氏との共著）。Munteanu氏との検討を継続し、一般の奇数次元への拡張に成功した（共著論文を投稿中）
• Theoretical Analysis on Trimmed Surface Connection and Generation of high-quality Trimmed Surface from Measured Point Data

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2019/04 -2022/03 

  Author : Miura Kenjiro T.

   

  Free-form surfaces used in automobile exterior design are required to be smooth, beautiful and of high quality. Although there is a strong need in practice, little theoretical research has been done on trimmed surfaces. So, first, as a reverse engineer for the exterior of automobiles using a trim curved surface. A least squares approximation was performed, a quadrilateral surface was fitted, and the trimmed surface was trimmed to generate a trimmed surface. Furthermore, we have researched and developed a method for fitting a tangent plane and a trimmed surface that satisfies the curvature continuity to the triangular mesh data, and succeeded in making the tangent plane of the trimmed surface and the curvature continuous.
• A construction of the thery of homogeneous surfaces in Riemannian symmetric spaces

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2016/04 -2021/03 

  Author : Naitoh Hiroo

   

  This research is positioned as the initial research of a research project that considers the classification of homogeneous submanifolds in the Riemannian symmetric spaces from the viewpoint of the Grassmann geometry of submanifolds, and the target submanifolds are limited to surfaces. The results obtained in this research led to the construction of a general theory regarding the framework of the Grassmann geometry of surfaces, and as a related research, gave the completion of the surface theory of Grassmann geometry in the three-dimensional Riemannian homogeneous space.
• Development and Extension of Discrete Integrable Geometry

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2016/04 -2020/03 

  Author : Kajiwara Kenji

   

  Discrete integrable differential geometry and its application have been studied, focusing on the integrable structure behind the discrete geometric objects. We have obtained the results on the discrete surfaces/curves and their deformation theory, discrete holomorphic functions, construction of discrete models of curves/surfaces, and stable and precise numerical method for the surfaces and interfaces. In particular, regarding the discrete surfaces/curves and their deformation theory, we formulated a good framework for the log-aesthetic curves developed in the area of the industrial design by using the Klein geometry and succeeded in generalization. Based on those results, we have proposed a project for JST CREST aiming at the development to the various areas of design, which has been successfully accepted.
• Construction of harmonic maps into non-compact symmetric spaces via loop groups and applications to surface theory

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2015/04 -2019/03 

  Author : Inoguchi Jun-ichi

   

  We gave a loop group method for constructing minimal surfaces with symmetry in the 3-dimensional Heisenberg group (the model space Nil of nilgeometry in the sense of Thurston). We also established loop group methods for constructing constant negative Gaussian curvature surfaces in the hyperbolic 3-space and maximal surfaces in the 3-dimensional anti de Sitter space-time. In addition, we generalized the Uhlenbeck-Segal theory for harmonic maps into compact semi-simple Lie groups (principal chiral models) to affine harmonic maps into general Lie groups equipped with natural bi-invariant torsion free connection.
• Applied analysis by discrete integrable systems and discrete differential geometry

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2011/04 -2015/03 

  Author : KAJIWARA Kenji, INOGUCHI Jun-ichi, NAKAYASHIKI Atsushi, MASUDA Tetsu, OHTA Yasuhiro, MATSUURA Nozomu

   

  By applying the theory of discrete integrable systems, studies on good discretization of geometric objects such as curves and surfaces have been carried out. The main results are as follows: (1) Discrete curve theory. Development of deformation theory of plane and space discrete curves and construction of explicit formula in terms of the tau functions. (2) Theory of discrete analytic functions. Construction of explicit formula for the discrete power function in terms of hypergetomtric tau function of the Painleve VI equation and generalization. (3) As an application, systematic construction of stable and highly accurate numerical scheme for nonlinear wave phenomena in terms of self-adaptive moving mesh scheme based on discretization of the Euler-Lagrange transformation.
• Research on classical differential geometry from modern view points and its applications

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2010/04 -2015/03 

  Author : KUROSE Takashi, SUYAMA Yoshihiko, HAMADA Tatsuyoshi, KAWAKUBO Satoshi, MATSUURA Nozomu, INOGUCHI Junichi, FURUHATA Hitoshi, FUJIOKA Atsushi

   

  In this research program, classical differential geometry, geometry of curves, surfaces and hypersurfaces in various spaces, have been studied, mainly with the method of the theory of integrable systems. Many results on classical differential geoemtry and its application have been achieved; for instance, through the observation that certain sorts of changes with time of curves yield equations dealt with in the theory of integrable systems, geometric descriptions and/or interpretations of several accomplishments of the theory have been given. Moreover, by applying geometry of hypersurfraces in affine spaces, new properties of statistical manifolds, which appear in informtion geometry, the study of mathematical statistics and information theory with differential geometric tools and methods, have been obtained and the statistical manifolds satisfying some curvature condition have been explicitely constructed and classified.
• Construction of surfaces in homogeneous spaces via spin geometry and loop groups

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2012/04 -2015/03 

  Author : INOGUCHI JUN-ICHI

   

  We showed that constancy of Gauss curvature of surfaces (of Gauss curvature less than 1) in the 3-sphere is characterized by the harmonicity of normal Gauss map. Based on this characterization, we established a loop group method for constructing negative constant Gauss curvature surfaces and surfaces of constant positive Gauss curvature (less than 1) in the 3-sphere simultaneously. We also obtain a loop group method for constructing surfaces of constant negative Gauss curvature (greather than -1) in hyperbolic 3-space. By combining spin geometry and loop group theory , we established a loop group method for constructing minimal surfaces in the 3-dimensional Heisenberg group. As an application, we give some new examples of minimal surfaces in the Heisenberg group.
• Global construction of constant mean curvature surfaces in terms of contact geometry and loop groups

  日本学術振興会：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2009/04 -2012/03 

  Author : INOGUCHI Jun-ichi

   

  We showed that minimal surfaces in hyperbolic 3-space are obtained as projections of f-holomorphic curves in the semi-Riemannian homogeneous contact space SL(2,C)/U(1). By using the appropriate loop group splitting, for any prescribed potential, we can construct f-holomorphic curves in SL(2,C)/U(1). It is shown that non-minimal constant mean curvature surfaces with mean curvature less than 1 can be obtained from f-holomorphic curves. By using this loop group method (new DPW-method), we constructed radially symmetric constant mean curvature surfaces in hyperbolic 3-space. We classified minimal translation surfaces in the 3-dimensional Heisenberg group.
• New development of harmonic maps

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2009 -2012 

  Author : URAKAWA Hajime, ICHIYAMA Toshiyuki, ITOH Jinichi, OBATA Nobuaki, INOGUCHI Junichi, HIAI Fumio

   

  In 1986, the concept of the bi-harmonic map which is an extension of harmonic maps was introduced. We raised the new notion of the bi-Yang-Mills field, which is an analog of the bi-harmonic map, and showed its isolation phenomena. That is, bi-Yang-Mills fields with some square-integral norm over compact manifolds with positive Ricci curvature must be Yang-Mills fields. We showed bi-harmonic maps which have a bounded square-integral norm must be harmonic if the target space has non-positive curvature. We classified all the bi-harmonic maps for cases where the target space is a compact Lie group or compact symmetric spaces.
• Research on constructions of constant mean curvature surfaces in terms of conformal geometry and loop groups

  日本学術振興会：Grant-in-Aid for Scientific Research(C)

  Date (from‐to) : 2006/04 -2009/03 

  Author : INOGUCHI Jun-ichi

   

  2 次複素特殊線型群 SL(2,C)のループ群を用いて5次元等質空間 SL(2,C)/U(1)に値をもつルジャンドル調和写像に対するループ群論的構成法(DPW 法)を確 立した。ルジャンドル調和写像と3次元双曲空間内の平均曲率一定曲面との対応により、ルー プ群論的構成法を用いて、3次元双曲空間内の、指定された臍点をもち、平均曲率が一定値で、 その絶対値が1未満の曲面を局所的に構成することが可能になった。また極小曲面も同時に構 成することが可能になった。
• Classical differential geometry from the modern viewpoint and its application

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2006 -2009 

  Author : KUROSE Takashi, SUAYMA Yoshihiko, HAMADA Tatsuyoshi, KAWAKUBO Satoshi, MATSUURA Nozomu, YAMADA Kotaro, INOGUCHI Junichi, FURUHATA Hitoshi

   

  In this research, we studied classical differential geometry from modern viewpoints, such as of the theory of integral systems and of the theory of singularities ; we obtained results on various fields of classical differential geometry and their applications, in particular, the motions of curves associated with integrable systems, explicit construction and the classification of conformally flat hypersurfaces of four-dimensional space forms, real hypersurfaces of complex space forms, surfaces of three-dimensional spaces, affine differential geometry and its applications to Hessian geometry and information geometry, and so on.
• 双曲空間内の曲面の無限次元リー群による構成の研究 研究課題

  日本学術振興会：科学研究費 若手研究(B)

  Date (from‐to) : 2004/04 -2006/03 

  Author : 井ノ口順一

   

  1)2003年に発表した論文Minimal surfaces in 3-dimensional solvable Lie groups, Chinise Annals of Mathematics B24(2003),73-84において3次元ユークリッド空間・3次元双曲空間・双曲平面と直線の直積,これらをすべて含む3次元等質空間の2径数族を構成した。族内の空間はすべて可解リー群である。 この2経数族に属する各空間内の極小曲面に対するガウス写像の満たす積分可能条件を求めた.この積分可能条件を用いて,ガウス写像とある複素数値函数の組が極小曲面を定めるための必要十分条件である偏微分方程式系を導出した.その偏微分方程式の解から極小曲面を与える積分表示公式を与えた。この公式はユークリッド空間内の極小曲面に対するWeierstrass-Enneper公式を一般化したものである。論文:Minimal surfaces in 3-dimensional solvable Lie groups IIとしてBullentin of the Australian Mathematical society誌に掲載が決定した。 2)極小はめこみ・調和写像の拡張概念である重調和写像・重調和はめ込みの具体例の構成を研究した。3次元双曲空間・3次元ユークリッド空間には極小でない重調和曲面が存在せず,3次元球面には極小でない重調和曲面は特定の半径をもつ小球のみであることが知られている。これらの事実に立脚し,極小でない重調和曲線・重調和曲面を許容する3次元等質空間を考察した。 とくに3次元既約標準簡約等質空間内の重調和曲線を分類した。この成果はJong Taek Cho氏,Jin-Eum Lee氏との共著論文Biharmonic curves in 3-dimensional Sasakain space formsとしてAnnali di Matematica et pura Applicata誌に掲載が決定した。
• The global behavior of curves and surfaces in space forms

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2003 -2006 

  Author : UMEHARA Masaaki, KOISO Norihito, YAMADA Kotaro, ROSSMAN Wayne F, KOKUBU Masatoshi, INOGUCHI Junichi

   

  We get the following results : 1.A maximal surface which is given by the real part of holomorphic isotropic immersion into C^3 is called a maxface. As a joint work with K.Yamada, the head investigator Umehara gave a Weierstrass-type representation formula for maxfaces, and gave an Osserman-type ineqality for complete maxfaces. The equality holds if and only if all ends of the surfaces are properly embedded. Moreover, as a joint work with K.Saji, S.Fujimori, and K.Yamada, the head investigator Umehara gave a criterion for the cuspidal cross cap, and showed that generic singular points for maxfaces consists of cuspidal edge, swallowtail and cuspidal cross cap. 2.As a joint work with K.Saji and K.Yamada, the head investigator Umehara studied the behavior of Gaussian curvature near the cuspidal edge and the swallowtail. In particular, the new geometric invariant on cuspidal edges called the singular curvature is introduced, and show that the integration of the singular curvature on the singular set is closely related to the Euler number of the surface. 3.A curve γ in the real projective plane is called anti-convex if for each point p on the curve, there exists a line passing through the point which does not meet y other than p. As a joint work with G.Thorbergsson, the head investigator Umehara studied the inflection points on anti-convex curves, and showed that the number of inflection points I and the number of the independent double tangents D satisfies the relation I-2D=3.
• Application of integrable systems methods to surfaces with particular variational properties

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2003 -2006 

  Author : ROSSMAM W.F., OHNITA Yoshihiro, GUEST M., YAMADA Kotaro, KOKUBU Masatoshi, INOGUCHI Jun-ichi

   

  The following results were obtained: 1) In a joint research project with U. Hertrich-Jeromin, S. Santos and F. Burstall, a suitable definition for discrete constant mean curvature surfaces in 3 dimensional space forms was obtained. Those 3 dimensional space forms consist of Euclidean 3-space, spherical 3-space and hyperbolic 3-space. It was shown that this new definition matches the old definition that is known for the Euclidean case, and this definition is new in the hyperbolic case. Using this definition, discrete Delaunay surfaces were studied, along with their discrete Darboux and Backlund transformations. An important tool in this research was the notion of conserved quantities. The case of smooth surfaces was developed by S. Santos and F. Burstall, while the discrete case was developed by U. Hertrich-Jeromin and myself. 2) In a joint research project with my Ph.D. graduate student N. Sultana, the stability and Morse index of constant mean curvature surfaces of revolution in spherical 3-space was studied. Because the axis of such a surface is a closed loop, these surfaces can become close tori, and then they will have finite index. It was shown that all such surfaces are unstable, and that they all have index at least 5, and (depending on the choice of surface) the index can be arbitrarily large. The index is the number of negative eigenvalues of the associated Jacobi operator. 3) In a continuation of a project with M. Kokubu, M. Umehara and K. Yamada, surfaces with constant Gauss curvature 0 in hyperbolic 3-space (flat fronts, which can have singularities) were studied. In particular, this year, it was shown that the caustics of such surfaces can have ends with asymptotic behavior described by cycloids.
• Classical differential geometry from the modern viewpoint and its applications

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2003 -2005 

  Author : KUROSE Takashi, SUYAMA Yoshihiko, HAMADA Tatsuyoshi, YAMADA Kotaro, INOGUCHI Jun-ichi, FURUHATA Hitoshi

   

  In this research, we planned to give a now development of the theories of classical differential geometry by restructuring them from the modern viewpoint, particularly, of the theories of integrable systems and of singularities. Our main results are the following : 1.(1)In affine differential geometry, one of the core theories of classical differential geometry, we mainly studied the geometry of affine hyperspheres and their representation formulae, and showed a relationship with the geometry of holomorphic statistical manifolds and the several properties of the center maps. We also studied the discretization of affine or centroaffine plane curves and gave a description of their time-evolution following discrete soliton equations ; (2)we characterized the classical examples of conformally flat hypersurfaces in 4-dimensional Euclidean space and constructed new examples ; (3)for real hypersurfaces in complex space forms, we introduced a new geometric invariant and classified Hopf real hypersurfaces using the invariant. 2.We studied the geometric properties of surfaces with singularities and obtained the following results : (1)We constructed the theory of flat fronts, the flat surfaces with singularities of a certain kind in 3-dimensional hyperbolic space. In particular, we defined (weak) completeness of flat fronts and showed their global properties ; (2)investigating the properties of the singularities of maximal surfaces in 3-dimensional Minkowski space, we constructed the theory of maxfaces, the spacelike maximal surfaces allowing singularities of a certain kind. 3.We studied transformations of surfaces and showed that the transformations given by the sphere congruences in Moebius geometry are obtained by the complexified line congruences in Euclidean space. We also investigated biharmonic curves in 3-dimensional homogeneous spaces and determined such curves when the homogeneous spaces are irreducible and reductive.
• Geometry of the flat tori in the sphere and non- linear wave equations

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2003 -2005 

  Author : KITAGAWA Yoshihisa, SAKAI Kazuhiro, INOGUCHI Jun-ichi, AIHARA Yoshihiro

   

  In this research, we studied geometry of flat tori in the 3-sphere, meromorphic mappings, surfaces of constant mean curvature and dynamical systems. The main results of this reseach are summarized as follows. 1.Studies on flat tori in the 3-sphere. In this research, Y.Kitagawa studied the conjecture that any isometric deformation of compact surface in $S^3$ preserves the enclosed volume. As a result, he proved that the conjecture is ture for all flat tori in $S^3$. 2.Studies on meromorphic mappings. In this research, Y.Aihara proved that for every hypersurface $D$ of degree $d$ in a complex projective space, there exists a holomorphic curve from the complex plane into the projective space whose deficiency for $D$ is positive and less than one. 3.Studies on constant mean curvature surfaces and Backlund transformations. In this research, J.Inoguchi proved that Bianchi-Backlund transformation of a constant mean curvature surface is equivalent to the Darboux transformation and the simple type dressing. 4.Studies on dynamical systems. In this research, K. Sakai proved that the $C^1$ interior of the set of expansive vector fields on a manifold is characterized as the set of vector fields without singularities satisfying both Axiom A and the quasi-transversality condition.
• Generalizations of Weierstrass-type representation formulae and applications

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2002 -2005 

  Author : YAMADA Kotaro, MIYAOKA Reiko, SAEKI Osamu, UMEHARA Masaaki, KUROSE Takashi, TAKAHASHI Masaro

   

  1.W rewrote the Weierstrass-type representation formula for flat surfaces in hyperbolic 3-space in the form without integration (Darboux-type formula), and classified complete flat surfaces with small numbers of ends. 2.We pointed out the class of ambient spaces for which an analogue of Weierstrass-type (Bryant) representation formula for mean curvature one surfaces in hyperbolic 3-space holds. 3.We found criteria for singularities (cuspidal edges, swallowtails, cuspidal cross caps) which are generic singularities of fronts or frontals. 4.We established fundamental notions of flat fronts in hyperbolic 3-space, and investiagted properties of singularities of such surfaces. 5.We defined a certain class of maximal surfaces with singularities in Minkowski 3-space (called maxface), and investigated their singularities.
• 定曲率空間内の曲面の無限次元リー群による構成の研究

  日本学術振興会：科学研究費助成事業

  Date (from‐to) : 2002 -2003 

  Author : 井ノ口 順一

   

  3次元定曲率空間内の「可積分系構造を持つ曲面」を無限次元リー群論的に構成する研究を継続して行なった。本年度は平均曲率一定曲面の変換論を小林真平氏(神戸大学・ミュンヘン工科大学)と共同で研究した。3次元ユークリッド空間内の平均曲率曲面は線叢による変換(Backlund変換)を許容しない。19世紀にL.Bianchiは線叢の複素化を考察し平均曲率一定曲面から新たな平均曲率一定曲面を得る操作を得た。この操作をBianchi-Backlund変換(BB変換)とよぶ。自明解である円柱面にBB変換を施して得られる平均曲率一定曲面をバブルトン(bubbleton)と呼ぶ。一方、平均曲率一定曲面は双等温曲面(isothermic surface)の典型例である。双等温という性質は共形変換で不変であり「共形幾何における球叢」を用いた変換論が展開できる。球叢による双等温曲面の変換はDarboux変換とよばれる。Darboux変換は複素一径数に依存する。複素一径数は実または準虚数でなければならない。 1997年に出版された論文でUdo Hertich-JerominとFranz Peditは「平均曲率一定曲面に対するDarboux変換で実一径数に依存するものはBianchi-Backlund変換と一致すること」を示した。更に次の予想を提出した。"純虚数に依存するDarboux変換はBB変換に由来しないであろう" (1)複素線叢を詳細に再検討し変換にはもう一種,「平均曲率一定曲面の変換」を与えるものがあることを発見した。 (2)従来の研究において漏れていたBB変換は「純虚数型のDarboux変換」であることを示した。 以上2つの研究結果によりHertrich-Jeromin・Peditの予想に対し否定的解答を与えた。 (論文:Shimpei Kobayashi and Jun-ichi Inoguchi,"Another bubbletons"として発表予定.2003年7月の国際会議で口頭発表)今年度の成果は従来から期待されている複素ドレッシング変換論構築への道標に相当することから注目を浴びている。
• Modern Research of Affine and Projective Geometry and its Applications

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2000 -2002 

  Author : KUROSE Takashi, YAMADA Kotaro, HAMADA Tatsuyoshi, SUYAMA Yoshihiko, FURUHATA Hitoshi, INOGUCHI Jun-ichi

   

  In this research, we studied classical differential geometries, theory of integral systems and information geometry. 1. Classical Differential Geometries (1) We characterized minimal affine hypersurfaces and minimal centroaffine immersions of codimension two. Moreover, we gave an explicit method of constructing self-dual minimal centroaffine surfaces of codimension two. (2) We studied manifolds with projectively flat torsion-free affine connection whose Ricci curvature is symmetric and definite, and showed fundamental results on the injectivity of the projective developing maps of such manifolds and the convexity of their image. (3) For conformally flat hypersurfaces of a 4-dimensional sphere, we defined a new conformal invariant. Using the invariant, we characterized the classical examples and constructed new examples. (4) We developed a very concrete and comprehensive theory on curves and surfaces in 3-dimensional homogeneous spaces. 2. Integrable Systems We investigated various integrable systems appeared in classical differential geometries. We obtained representation formulae for minimal surfaces in 3-dimensional solvable Lie groups and flat surfaces in a 3-dimensional hyperbolic space. We also developed a comprehensive theory of (spacelike) surfaces with harmonic inverse mean curvature in 3-dimensional Riemannian space forms and Lorentzian space forms. 3. Information Geometry and Statistical Manifolds (1) We defined complex statistical manifolds and studied them from the view points of affine differential geometry and of information geometry, especially of quantum estimation theory. (2) As a generalization of special Kahler manifolds, we defined statistical manifolds with compatible complex structure and investigated their fundamental properties. (3) On (-1)-conformally flat statistical manifolds, we gave an explicit method of constructing the Volonoi diagrams.
• 定曲率空間内の曲面に対する無限次元群作用の研究

  日本学術振興会：科学研究費助成事業

  Date (from‐to) : 2000 -2001 

  Author : 井ノ口 順一

   

  前年度に引き続きBacklund変換の変換群論的把握に向けて無限次元リー群論の観点から研究を行った。また対称空間ではない等質空間内の曲面・調和写像の構成についても研究を行った。 (1)Chaohao Gu氏(谷超豪),Hesheng Hu氏(胡和生)(中国・Fudan University)と共同研究を行い以下の成果を得た。 Liouville方程式・cosh-Gordon方程式に対するBacklund変換を与えた。さらにこれらのBacklund変換を負定値計量をもつ3次元空間(ミンコフスキー空間)内の時間的曲面間の空間的線叢および時間的線叢として幾何学的に定義できることを示した。 上述のBacklund変換を「フレームに対する変換」として再定式化しループ群論的解釈を与えた。 2)J.Dorfmeister, F.Pedit, H.Wuによる「リーマン面からコンパクト・リーマン対称空間への対称空間」に対するループ群論的WeierstraB構成法(非線型ダランベール公式)を対称ではない標準簡約等質空間(naturally reductive homogeneous space)への拡張を研究した。その成果として実Stiefel多様体への水平的調和写像に対しWeierstraB構成法が適用できることがわかった。この成果は3次元定曲率空間内の平均曲率一定曲面の構成に応用できる。 (3)3次元ユークリッド空間内の極小曲面に対するWeierstraB-Enneper表現公式を3次元可解リー群に対し拡張した。この拡張版の公式は國分雅敏氏による「3次元双曲空間内の極小曲面に対する表現公式」をも含む。 (4)実特殊線型群SL(2,R)内の平均曲率一定曲面のガウス写像の調和性を研究した。とくに平均曲率一定曲面でガウス写像が鉛直調和(vertically harmonic)である曲面を分類・決定した。さらに平均曲率一定曲面でガウス写像が調和となるものを分類・決定した。 (5)Mohamed Belkhelfa氏,Franki Dillen氏(KU Leuven,ベルギー)と共同研究を行い3次元標準簡約等質空間(naturally reductive homogeneous space)内の第二基本形式が平行な曲面を分類・決定した。
• Applications of integrable systems in geometry and topology

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2000 -2001 

  Author : GUEST Martin, KAMISHIMA Yoshinobu, OKA Mutsuo, OHNITA Yoshihiro, INOGUCHI Junichi, UDAGAWA Seiichi

   

  Results were obtained on the geometry and topology of harmonic maps and spaces of harmonic maps, especially in the case where the domain is a Riemann surface and the target space is a compact Lie group or symmetric space. Guest used a generalization of the Weierstrass representation for minimal surfaces to study harmonic maps from the two-dimensional sphere (or, more generally, harmonic maps of finite uniton number, from any Riemann surface) to the unitary group. Earlier results of Uhlenbeck, Segal, Dorfmeister-Pedit-Wu, Burstall-Guest were developed into an effective tool for describing such maps. In particular, an explicit canonical form was obtained, and this was used to study the space of all such maps. The main application was a description of the connected components of the space of harmonic maps from the two-dimensional sphere to the unitary group. Ohnita used a different approach, based on earlier work of Hitchin in gauge theory, to obtain a framework for studying the geometry (in particular, the pre-symplectic geometry) of spaces of harmonic maps. The harmonic map equation can be regarded as an integrable system, and the above work sheds light on other integrable systems. Two other examples of integrable systems were studied from this point of view, and preliminary results obtained. The first example, studied by Guest, was the theory of quantum differential equations. Parallels with harmonic maps were established, forming the basis for future work in this direction. Results on quantum cohomology of symmetric spaces were obtained also by Ohnita and Nishimori, and on quantum cohomology of flag manifolds by Guest and Otofuji. The second example, studied by Burstall and Calderbank, was the integrable systems aspect of conformal and Mobius geometry, and a new approach was initiated.
• Geometry of surfaces in space forms

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 1999 -2000 

  Author : UMEHARA Masaaki, HONDA Nobuhiro, KANNO Hiroaki, MATSUMOTO Takao, INOGUCHI Junichi, KOKUBU Masatoshi

   

  We get the following results : 1. The head investigator Umehara gave a classification for complete constant mean curvature 1 surfaces (i.e. CMC-1 surfaces) in the hyperbolic 3-space H^3 of total absolute curvature (resp. the dual total absolute) curvature less than or equal to 4π. Moreover, he gave non-existence and existence results when the surfaces has dual total curvature less than or equal to 8π. These results are shown in a joint work with Rossman and Yamada. 2. The head investigator Umehara, Kokubu, Takahashi and Yamada gave a theory of surfaces with holomorphic Gauss maps in the duals of compact semisimple Lie groups, which is a generalization of CMC-1 surfaces in H^3, and show an analogue of Chern-Osserman Inequality for minimal surfaces in the Euclidean π-space. Moreover, they gave several non-trivial examples of such surfaces and showed mean curvature of these surfaces are all proportional to the sectional curvature of the ambient space. 3. The head investigator Umehara and Bobenko investigated the monodromy of constant mean curvatures in H^3 and showed that the number of isometric immersions with a prescribed constant mean curvature into H^3 on a given Riemannian 2-manifold is finite.
• Research for manifolds with conformal structure

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 1997 -1999 

  Author : SUYAMA Yoshihiko, KUROSE Takashi, AKUTAGAWA Kazuo, SHIOHAMA Katsuhiro, INOGUCHI Jun-ichi, YAMADA Kataro

   

  1. Conformably flat hypersurfaces. We studied conformally flat, hypersurfaces in the space forms of dimension 4, and found a good structure on the 4-dimensional standard sphere for each hypersurface. According to the structure, the set of conformally flat hypersurfaces is divided into three classes : the parabolic class, the elliptic class, and the hyperbolic class. We showed that the classes are invariant under conformal transformations of the sphere and the respective class consists of conformally flat hypersurfaces constructed by surfaces of constant curvature in one of the 3-dimensional space forms : the Euclidean space, the hyperbolic space, or the sphere. 2. Conformal-projective transformations of statistical manifolds. In this study, we obtained the following result : A conformal-projective transformation of a statistical manifold leaves all umbilical points and the skew-symmetric component of the Ricci curvature of any hypersurfaces ; moreover, this property characterizes the conformal-projective transformations when the dimension of the statistical manifold is greater than 2. We also found a tensor field that is invariant under any conformal-projective transformations and that reduces to the conformal curvature tensor if the underlying statistical manifold is a usual Riemannian manifold. 3. A representation formula of surfaces with constant mean curvature (CMC surfaces) in a 3-dimensional space form and their Gauss map. The existence problem of harmonic maps was studied in the case where the destination is a non-complete Riemannian space with non-positive curvature unbounded from below. In this situation, we showed tile existence and the uniqueness theorems of harmonic maps for a Dirichlet problem at infinity. As an application, we constructed CMC surfaces in the 3-dimensional hyperbolic space form. 4. An extension of the class of CMC surfaces from the viewpoint of the theory of integrable systems. We defined surfaces with harmonic inverse mean curvature (HIMC surfaces) in the 3-dimentional space forms, and showed that there exists a correspondence among the HIMC surfaces similar to the Lawson correspondence, one of the features of the class of CMC surfaces. We also studied the relation between the class of HIMC surfaces and the class of H-surfaces, which is an extension of the class of CMC surfaces from the variational viewpoint. As a result, we proved that HIMC surfaces are obtained from the gauge-theoretic equation for H-surfaces with a certain condition of reduction.

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