Moving frame and spin field representations of submanifolds in flat space
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866913397271953408 |
|---|---|
| author | Zou, Shou-Jyun |
| author_facet | Zou, Shou-Jyun |
| contents | We introduce a spin field approach, that is compatible with the Cartan moving frame method, to describe the submanifold in a flat space. In fact, we consider a kind of spin field $ψ$, that satisfies a Killing spin field equation (analogous to a Killing spinor equation) written in terms of the Clifford algebra, and we use the spin field to locally rotate the orthonormal basis $\{\hat{e}_\mathtt{I}\}$. Then, the deformed orthonormal frame $\{\tildeψ\hat{e}_\mathtt{I}ψ\}$ can be seen as the moving frame of a submanifold. We find some solutions to the Killing spin field equation and demonstrate an explicit example. Using the product of the spin fields, one can easily generate a new immersion submanifold, and this technique should be useful for studies in geometry and physics. Through the spin field, we find a linear relation between the connection and the extrinsic curvature of the submanifold. We propose a conjecture that any solution of the Killing spin field equation can be locally written as the product of the solutions we find. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_12855 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Moving frame and spin field representations of submanifolds in flat space Zou, Shou-Jyun General Mathematics General Relativity and Quantum Cosmology High Energy Physics - Theory Mathematical Physics We introduce a spin field approach, that is compatible with the Cartan moving frame method, to describe the submanifold in a flat space. In fact, we consider a kind of spin field $ψ$, that satisfies a Killing spin field equation (analogous to a Killing spinor equation) written in terms of the Clifford algebra, and we use the spin field to locally rotate the orthonormal basis $\{\hat{e}_\mathtt{I}\}$. Then, the deformed orthonormal frame $\{\tildeψ\hat{e}_\mathtt{I}ψ\}$ can be seen as the moving frame of a submanifold. We find some solutions to the Killing spin field equation and demonstrate an explicit example. Using the product of the spin fields, one can easily generate a new immersion submanifold, and this technique should be useful for studies in geometry and physics. Through the spin field, we find a linear relation between the connection and the extrinsic curvature of the submanifold. We propose a conjecture that any solution of the Killing spin field equation can be locally written as the product of the solutions we find. |
| title | Moving frame and spin field representations of submanifolds in flat space |
| topic | General Mathematics General Relativity and Quantum Cosmology High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2406.12855 |