Event horizons of higher dimensional space-time and the theory of Lagrange/Legendrian singularities

• Project/Area Number: 24654008
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Geometry
• Research Institution: Hokkaido University
• Principal Investigator: IZUMIYA Shyuichi 北海道大学, 理学(系)研究科(研究院), 教授 (80127422)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Project Status: Completed (Fiscal Year 2014)
• Budget Amount: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000); Fiscal Year 2014: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2013: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2012: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: ルジャンドル特異点 / ラグランジュ特異点 / ブレーン宇宙論 / 微分幾何学 / 焦点集合 / 波面 / 世界面 / 時空 / 反ド・ジッター空間 / ルジャンドル多様体 / 波面の伝播 / 特異点 / ローレンツ空間形 / 光的超曲面 / コーシー地平面

Outline of Final Research Achievements

We constructed a geometric framework for describing the caustics of world sheets in the Minkowski space-time and gave a classification of the singularities of the caustics in the 3-dimensional Minkowski space-time. The caustics of world sheets are considered to be boundaries of one of the "multiverse" models in theoretical Physics.

Research Products

• [Journal Article] Anti de Sitter horospherical flat timelike surfaces (2014)
  • Author(s): Liang Chen,Shyuichi Izumiya、Donghe Pei and Kentaro Saji
  • Journal Title: Science China Mathematics Volume: 57 Issue: 9 Pages: 1841-1866
  • DOI: 10.1007/s11425-014-4870-7
  • Peer Reviewed
• [Journal Article] Total lightcone curvatures of spacelike submanifolds in Lorentz-Minkowski space (2014)
  • Author(s): Shyuichi Izumiya
  • Journal Title: Differential Geometry and its Applications Volume: 34 Pages: 103-127
  • DOI: 10.1016/j.difgeo.2014.03.012
  • Peer Reviewed
• [Journal Article] Lightlike hypersurfaces along spacelike submanifolds in de Sitter space (2014)
  • Author(s): Shyuichi Izumiya and Takami Sato
  • Journal Title: Journal of Singularities Volume: 10 Pages: 157-173
  • DOI: 10.5427/jsing.2014.10j
  • Peer Reviewed / Open Access
• [Journal Article] Lightlike hypersurfaces along spacelike submanifolds in Minowski space-time (2013)
  • Author(s): Shyuichi Izumiya, Takami Sato
  • Journal Title: Journal of Geometry and Physics Volume: 71 Pages: 30-52
  • DOI: 10.1016/j.geomphys.2013.03.005
  • Peer Reviewed
• [Journal Article] Lightcone dualities for curves in the lightcone unit 3-sphere (2013)
  • Author(s): Shyuichi Izumiya, Yang Jiang, Takami Sato
  • Journal Title: Journal of Mathematical Physics Volume: 54 Issue: 6 Pages: 63511-63511
  • DOI: 10.1063/1.4811451
  • Peer Reviewed
• [Journal Article] Apparent contours in the Minkowski 3-space and first order ordinary differential equations (2013)
  • Author(s): Shyuichi Izumiya and Farid Tari
  • Journal Title: Nonlinearity Volume: 26 Issue: 4 Pages: 911-932
  • DOI: 10.1088/0951-7715/26/4/911
  • Peer Reviewed
• [Journal Article] Pedal foliations and Gauss maps of hypersurfaces in Euclidean space (2012)
  • Author(s): Shyuichi Izumiya and Masatomo Takahashi
  • Journal Title: Journal of Singularities Volume: ６ Pages: 84-97
  • Peer Reviewed
• [Journal Article] Slant geometry of spacelike hypersurfaces in Hyperbolic space and de Sitter space (2012)
  • Author(s): M. Asayama, S. Izumiya, A. Tamaoki and H. Yildirim
  • Journal Title: Revista Matemarica Iberoamericana Volume: 28 Issue: 2 Pages: 371-400
  • DOI: 10.4171/rmi/681
  • Peer Reviewed
• [Journal Article] Extensions of the mandala of Legendrian, dualities for pseudo-spheres in Lorentz Minkowski space (2012)
  • Author(s): Shyuichi Izumiya and Handan Yildirim
  • Journal Title: Topology and its Applications Volume: 159(2) Issue: 2 Pages: 509-518
  • DOI: 10.1016/j.topol.2011.09.026
  • Peer Reviewed
• [Presentation] Caustics of world hyper sheets in Loretnz-Minkowski space (2014)
  • Author(s): Shyuichi Izumiya
  • Organizer: International Conference on Geometry and its Applications
  • Place of Presentation: Jadavpur University (INDIA)
  • Year and Date: 2014-10-16
  • Invited
• [Presentation] The theory of wave front propagations and its applications (overview) (2014)
  • Author(s): Shyuichi Izumiya
  • Organizer: 13th International Workshop on Real and Complex Singulariteis
  • Place of Presentation: Sao Paulo University (Brazil)
  • Year and Date: 2014-07-31
  • Invited
• [Presentation] Flat approximations of surfaces along curves
  • Author(s): 泉屋周一
  • Organizer: Geometric Singularity Theory, Polish-Japanese Singuarity Theory Working Days
  • Place of Presentation: Banach Center (Warsaw, Poland)
  • Invited
• [Presentation] Caustics of world sheets in anti-de Sitter space
  • Author(s): 泉屋周一
  • Organizer: Singularities in geometry and appliccations III
  • Place of Presentation: International Center for Mathematics (Edinburgh, Scotland)
  • Invited
• [Presentation] Geometry of world sheets in Minkowski space-time
  • Author(s): 泉屋周一
  • Organizer: 数理科学研究所研究集会
  • Place of Presentation: 京都大学数理科学研究所（京都市）
  • Invited
• [Presentation] Total lightcone curvatures of spacelike submanifolds in Lorentz-Minkowski space
  • Author(s): 泉屋周一
  • Organizer: The International Conference of Differential Geometry and Dynamical Systems
  • Place of Presentation: "Callatis" High School, Mangalia (Rumania)
  • Invited
• [Presentation] The lightlike geometry of spacelike submanifolds in Lorentz-Minkowslki space
  • Author(s): 泉屋周一
  • Organizer: Bruce 60 & Wall 75, Workshop in Singularity Theory, its Applications and Future Prospects
  • Place of Presentation: Liverpool University (UK)
  • Invited