Development of Theory of Harmonic Maps

• Project/Area Number: 18K03352
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12020:Mathematical analysis-related
• Research Institution: Tohoku University
• Principal Investigator: Urakawa Hajime 東北大学, 情報科学研究科, 名誉教授 (50022679)
• Project Period (FY): 2018-04-01 – 2022-03-31
• Project Status: Completed (Fiscal Year 2021)
• Budget Amount: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2021: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2020: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2019: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
• Keywords: 調和写像 / 極小部分多様体 / 二重調和写像 / リーマン多様体 / リーマン・ベクトル束 / 沈め込み / 二重調和ベクトル場 / 捻れ積 / 正曲率空間 / 等質空間 / 単連結 / 調和射 / 部分群 / 超曲面 / ファイバー束 / 二重調和 / コンパクト / 第一固有値 / 射影 / リッチ曲率 / 等長はめ込み / G-主束 / 群作用 / 軌道 / コンパクト対称三対 / ベクトル束 / 葉層構造 / 調和ベクトル場 / 二調和ベクトル場 / リーマン主束 / リーマン沈め込み / ２－調和写像 / ｋ－調和写像

Outline of Final Research Achievements

During the period of our research on the biharmonic maps, we have proceeded the following studies: (1) the biharmonic sub manifolds, (2) the biharmonic warped products, (3) the biharmonic principal G-bundles, (4) the biharmonic vector fields on pseudo Riemannian manifolds, (5) the poly-harmonic maps (including biharmonic) mappings, (6) the transversally biharmonic mappings, (7) the biharmonic projections of the principal bundles over the Riemannian manifolds of nonpositive Ricci curvature, (8) the biharmonic Riemannian submersions, (9) the biharmonic double fibrations of compact Lie groups, (10) the harmonic morphisms of compact homogeneous spaces of positive curvature, (11) the biharmonic mappings for double vibrations of compact Lie groups.

Academic Significance and Societal Importance of the Research Achievements

二重調和写像の研究は、極小部分多様体や調和写像の研究の自然な一般化であり、多くの微分幾何学者らの関心を引き、1980年代から、日本、アメリカ、イギリス、フランス、イタリア、ルーマニア、ロシア、中国、ブラジルなどの研究者らによって推進されてきた。二重調和写像の研究の成果は、微分幾何学への応用があると期待され極めて重要である。この研究は我が国においても、当該研究者を中心に多くの研究者らによって活発に推進されている。

Research Products

• [Journal Article] Harmonic maps and biharmonic maps for double liberations of compact Lie groups (2022)
  • Author(s): Hajime Urakawa
  • Journal Title: Hokkaido Mathematical Journal Volume: 51
  • Peer Reviewed
• [Journal Article] Harmonic morphisms of compact homogeneous spaces of positive curvature (2021)
  • Author(s): Hajime Urakawa
  • Journal Title: Note di Matematica Volume: 41
  • Peer Reviewed
• [Journal Article] Biharmonic Hermitian vector bundles over compact Kaehler manifolds and compact Einstein Riemannian manifolds (2020)
  • Author(s): Urakawa, Hajime
  • Journal Title: Note di Matematica Volume: 40 Pages: 1-16
  • Peer Reviewed
• [Journal Article] Biharmonic maps on principal G-bundles over complete Riemannian manifolds of nonpositive Ricci curvature (2019)
  • Author(s): Urakawa, Hajime
  • Journal Title: Michigan Mathematical Journal Volume: 68 Pages: 47-115
  • Peer Reviewed
• [Journal Article] Biharmonic homogeneous submanifolds in comopact symmetric spaces and compact Lie groups (2019)
  • Author(s): Ohno, Shinji, Sakai, Takashi and Urakawa, Hajime
  • Journal Title: Hiroshima Mathematical Journal Volume: 49 Pages: 47-115
  • Peer Reviewed
• [Journal Article] Harmonic maps and biharmonic Riemannian submersions (2019)
  • Author(s): Urakawa, Hajime
  • Journal Title: Note di Mathematica Volume: 39 Pages: 1-23
  • Peer Reviewed
• [Journal Article] Biharmonic homogeneous submanifolds in compact symmetric spaces and compact Lie groups (2019)
  • Author(s): Shinji Ohno, Takashi Sakai and Hajime Urakawa
  • Journal Title: Hiroshima Mathematical Journal Volume: 49
  • Peer Reviewed
• [Journal Article] Rigidity of transversally biharmonic maps between foliated Riemannian manifolds (2018)
  • Author(s): Shinji Ohno, Takashi Sakai and Hajime Urakawa
  • Journal Title: Hokkaido Mathematical Journal Volume: 47
  • Peer Reviewed
• [Journal Article] Polyharmonic maps into the Euclidean space (2018)
  • Author(s): Nobumitsu Nakauchi and Hajime Urakawa
  • Journal Title: Note di Matematica Volume: 38
  • Peer Reviewed
• [Journal Article] Biharmonic vector fields on pseudo-Riemannian manifolds (2018)
  • Author(s): Michael Markellos and Hajime Urakawa
  • Journal Title: Journal of Geometry and Physics Volume: 130
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Harmonic maps and biharmonic maps on principal bundles and warped products (2018)
  • Author(s): Hajime Urakawa
  • Journal Title: Journal of Korean Mathematical Society Volume: 55
  • Peer Reviewed
• [Journal Article] Biharmonic submanifolds in a Riemannian manifold (2018)
  • Author(s): Norihito Koiso and Hajime Urakawa
  • Journal Title: Osaka Journal of Mathematics Volume: 55
  • NAID: 120006478932
  • Peer Reviewed
• [Presentation] Harmonic maps and biharmonic maps between Riemannian manifolds (2022)
  • Author(s): Hajime Urakawa
  • Organizer: The 2022 the First International Conference on Applied Mathematics and Digital Simulation (AMDS 2022)
  • Int'l Joint Research / Invited
• [Presentation] Geometry of harmonic maps and biharmonic maps (2019)
  • Author(s): Hajime Urakawa
  • Organizer: International conference of differential geometry
  • Int'l Joint Research / Invited
• [Presentation] Biharmonic principal G-bundles and vector bundles (2019)
  • Author(s): Hajime Urakawa
  • Organizer: Differential Equations over Manifolds
  • Int'l Joint Research / Invited
• [Presentation] 調和写像と２調和写像の幾何学 (2019)
  • Author(s): 浦川肇
  • Organizer: 第１８回秋葉原微分幾何学セミナー
  • Invited
• [Presentation] Harmonic and biharmonic Riemannian submersions (2018)
  • Author(s): Hajime Urakawa
  • Organizer: Differential Geometry Workshop 2018--Harmonic maps, biharmonic maps, harmonic morphisms and related topics--
  • Int'l Joint Research / Invited
• [Presentation] Biharmonic Riemannian submersions の幾何 (2018)
  • Author(s): 浦川肇
  • Organizer: 研究会「多様体上の微分方程式」
  • Invited
• [Presentation] On biharmonic Riemannian submersions (2018)
  • Author(s): 浦川肇
  • Organizer: 福岡大学微分幾何研究会
  • Invited
• [Presentation] Biharmonic maps and harmonic morphisms from Riemannian manifolds of positive sectional curvature (2018)
  • Author(s): 浦川肇
  • Organizer: 北海道大学研究集会「幾何学のスペクトル Spectrum of Geometry」
  • Invited
• [Book] 代数学 第2巻 (2020)
  • Author(s): 藤原松三郎、浦川肇、高木泉、藤原毅夫
  • Total Pages: 693
  • Publisher: 内田老鶴圃
  • ISBN: 9784753601622
• [Book] Geometry of biharmonic mappings, Differential geometry of variational methods (2019)
  • Author(s): Urakawa, Hajime
  • Total Pages: 348
  • Publisher: World Scientific Publishing Co. Pte. Ltd.
  • ISBN: 9789813236394
• [Book] Geometry of Biharmonic Mappings--Differential Geometry of Variational Methods-- (2019)
  • Author(s): Hajime Urakawa
  • Total Pages: 348
  • Publisher: World Scientific Publishing, Co. Pte. Ltd.
  • ISBN: 9813236418