Klein-Cartan program for geometry and differential equations

2021 Fiscal Year Final Research Report

• Project/Area Number: 17K05232
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Yokkaichi University
• Principal Investigator: Morimoto Tohru 四日市大学, 関孝和数学研究所, 研究員 (80025460)
• Project Period (FY): 2017-04-01 – 2022-03-31
• Keywords: 幾何構造の同値問題 / 外在的幾何 / 内在的幾何 / 包合的な微分方程式系 / 不変量

Outline of Final Research Achievements

We, joint with Boris Doubrov and Yoshinori Machida, have established a unified general theory of extrinsic geometry and linear differential equations on a basis of nilpotent geometry and nilpotent analysis, and published a paper entitled:
Extrinsic geometry and linear differential equations, in Sigma. As an application of the general theory, we then have carried out detailed studies on extrinsic geometry and linear differential equations of sl(3)type and have classified all the transitive geometric structures.
We joint with Jaehyun Hong, have established a unified general theory of intrinsic geometry by integrating and improving the previous works, and put a preprint : Prolongations, invariants, and fundamental identities, in ArXiv math.In particular we have given a general algorithm to find a fundamental system of invariant of an arbitrary G-structure( in an extended sense)on a filtered manifold. We are then applying the above theory to complex geometry and subRiemannian geometry.

Free Research Field

幾何学

Academic Significance and Societal Importance of the Research Achievements

幾何には外在的幾何と内在的幾何の区別がある。
先に述べた、外在的幾何における統一理論と内在的幾何における統一理論は、その双方が互いに連携し、古典微分幾何の基本原理を与えるものであり、幾何における深い理解と広い応用をもたらすものであると思われる。