Einstein Lie groups, geodesic orbit manifolds and regular Lie subgroups
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866911755398021120 |
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| author | Souris, Nikolaos Panagiotis |
| author_facet | Souris, Nikolaos Panagiotis |
| contents | We study the relation between two special classes of Riemannian Lie groups $G$ with a left-invariant metric $g$: The Einstein Lie groups, defined by the condition $\operatorname{Ric}_g=cg$, and the geodesic orbit Lie groups, defined by the property that any geodesic is the integral curve of a Killing vector field. The main results imply that extensive classes of compact simple Einstein Lie groups $(G,g)$ are not geodesic orbit manifolds, thus providing large-scale answers to a relevant question of Y. Nikonorov. Our approach involves studying and characterizing the $G\times K$-invariant geodesic orbit metrics on Lie groups $G$ for a wide class of subgroups $K$ that we call (weakly) regular. By-products of our work are structural and characterization results that are of independent interest for the classification problem of geodesic orbit manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_08946 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Einstein Lie groups, geodesic orbit manifolds and regular Lie subgroups Souris, Nikolaos Panagiotis Differential Geometry 53C25, 53C30 We study the relation between two special classes of Riemannian Lie groups $G$ with a left-invariant metric $g$: The Einstein Lie groups, defined by the condition $\operatorname{Ric}_g=cg$, and the geodesic orbit Lie groups, defined by the property that any geodesic is the integral curve of a Killing vector field. The main results imply that extensive classes of compact simple Einstein Lie groups $(G,g)$ are not geodesic orbit manifolds, thus providing large-scale answers to a relevant question of Y. Nikonorov. Our approach involves studying and characterizing the $G\times K$-invariant geodesic orbit metrics on Lie groups $G$ for a wide class of subgroups $K$ that we call (weakly) regular. By-products of our work are structural and characterization results that are of independent interest for the classification problem of geodesic orbit manifolds. |
| title | Einstein Lie groups, geodesic orbit manifolds and regular Lie subgroups |
| topic | Differential Geometry 53C25, 53C30 |
| url | https://arxiv.org/abs/2109.08946 |