Attached Submanifolds Beyond Symmetric Spaces
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2026
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866908775470858240 |
|---|---|
| author | Kerr, Megan M. Payne, Tracy L. |
| author_facet | Kerr, Megan M. Payne, Tracy L. |
| contents | We study submanifold geometry in the presence of symmetry, focusing on submanifolds of solvmanifolds with an unusual property relative to Ricci curvature. We generalize work of H. Tamaru \cite{tamaru-11} in which he explores the geometry of submanifolds of symmetric spaces of noncompact type constructed from parabolic subgroups of the isometry group. He calls these attached submanifolds. The Ricci curvatures of attached submanifolds coincide with the restrictions of the Ricci curvatures of ambient symmetric spaces.
We broaden Tamaru's construction by weakening the hypotheses on the ambient space, allowing a pseudo-Riemannian scalar product, and defining attached submanifolds in terms of root spaces. We demonstrate that in this setting, the Ricci curvature restriction property for attached submanifolds holds if and only if the submanifold satisfies an algebraic criterion that we call the Jacobi Star Condition. Like attached submanifolds of symmetric spaces, our attached submanifolds are minimal, and are only totally geodesic under hypotheses analogous to hypotheses in the symmetric space case.
Finally, we give an example of a solvmanifold that has an attached submanifold and is not a symmetric space, demonstrating that attached submanifolds are not unique to symmetric spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_13461 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Attached Submanifolds Beyond Symmetric Spaces Kerr, Megan M. Payne, Tracy L. Differential Geometry 53C30, 53C40 53C25 52C35 22E25 We study submanifold geometry in the presence of symmetry, focusing on submanifolds of solvmanifolds with an unusual property relative to Ricci curvature. We generalize work of H. Tamaru \cite{tamaru-11} in which he explores the geometry of submanifolds of symmetric spaces of noncompact type constructed from parabolic subgroups of the isometry group. He calls these attached submanifolds. The Ricci curvatures of attached submanifolds coincide with the restrictions of the Ricci curvatures of ambient symmetric spaces. We broaden Tamaru's construction by weakening the hypotheses on the ambient space, allowing a pseudo-Riemannian scalar product, and defining attached submanifolds in terms of root spaces. We demonstrate that in this setting, the Ricci curvature restriction property for attached submanifolds holds if and only if the submanifold satisfies an algebraic criterion that we call the Jacobi Star Condition. Like attached submanifolds of symmetric spaces, our attached submanifolds are minimal, and are only totally geodesic under hypotheses analogous to hypotheses in the symmetric space case. Finally, we give an example of a solvmanifold that has an attached submanifold and is not a symmetric space, demonstrating that attached submanifolds are not unique to symmetric spaces. |
| title | Attached Submanifolds Beyond Symmetric Spaces |
| topic | Differential Geometry 53C30, 53C40 53C25 52C35 22E25 |
| url | https://arxiv.org/abs/2601.13461 |