Study on identites for generalized gradients associated to geometric structures and their applications

2017 Fiscal Year Final Research Report

• Project/Area Number: 15K04858
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Waseda University
• Principal Investigator: HOMMA Yasushi 早稲田大学, 理工学術院, 教授 (50329108)
• Project Period (FY): 2015-04-01 – 2018-03-31
• Keywords: スピン幾何学 / ラリタ-シュインガー作用素 / ディラック作用素 / ワイゼンベック公式 / 幾何構造 / ラプラス作用素

Outline of Final Research Achievements

The purpose of this project is to find new identities for generalized gradients related to geometric structures and apply them to geometry, harmonic analysis and theoretical physics.Here,a generalized gradient is a conformally covariant 1st differential operator on a spin manifold such as the Dirac operator and the Rarita-Schwinger operator. Doing an international joint research, we have the following results:
(1) We give the twisted Weitzenb\"ock formula explicitly which is a unique relation for generalized gradients on two different vector bundles. We also give some applications such as a commutative relation for Lichnerowicz Laplacian and a generalized gradient.
(2) We clarify a relation between Rarita-Schwinger fields and some geometric structures. We also give a classification of spin manifolds admitting parallel 3/2-spin fields.

Free Research Field

数物系科学（幾何学）