The Gauss formulas for Laplacians on submanifolds
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866912402674548736 |
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| author | Chan, Chi Hin Czubak, Magdalena |
| author_facet | Chan, Chi Hin Czubak, Magdalena |
| contents | There are several types of Laplacians of a vector field on a Riemannian manifold. These include the Bochner and the Hodge Laplacian. The Gauss formula for the Levi-Civita connection relates the extrinsic connection to the intrinsic connection. We extend the Gauss formula for the connection to formulas for the different types of Laplacians of a vector field on a submanifold of any codimension $k\geq 1$. In the process, we derive a Gauss formula for the Ricci operator, formulas for the divergence of the second fundamental form, and a formula for the Laplacian of a $1$-form on a surface of revolution in terms of the Lie derivatives. The formulas have applications to the study of the formulation of the incompressible Navier-Stokes equations on a Riemannian manifold. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_11928 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The Gauss formulas for Laplacians on submanifolds Chan, Chi Hin Czubak, Magdalena Differential Geometry Analysis of PDEs There are several types of Laplacians of a vector field on a Riemannian manifold. These include the Bochner and the Hodge Laplacian. The Gauss formula for the Levi-Civita connection relates the extrinsic connection to the intrinsic connection. We extend the Gauss formula for the connection to formulas for the different types of Laplacians of a vector field on a submanifold of any codimension $k\geq 1$. In the process, we derive a Gauss formula for the Ricci operator, formulas for the divergence of the second fundamental form, and a formula for the Laplacian of a $1$-form on a surface of revolution in terms of the Lie derivatives. The formulas have applications to the study of the formulation of the incompressible Navier-Stokes equations on a Riemannian manifold. |
| title | The Gauss formulas for Laplacians on submanifolds |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2212.11928 |