Einstein metrics on homogeneous spaces $H\times H/ΔK$
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866912072847065088 |
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| author | Lauret, Jorge Will, Cynthia |
| author_facet | Lauret, Jorge Will, Cynthia |
| contents | Given any compact homogeneous space $H/K$ with $H$ simple, we consider the new space $M=H\times H/ΔK$, where $ΔK$ denotes diagonal embedding, and study the existence, classification and stability of $H\times H$-invariant Einstein metrics on $M$, as a first step into the largely unexplored case of homogeneous spaces of compact non-simple Lie groups. We find unstable Einstein metrics on $M$ for most spaces $H/K$ such that their standard metric is Einstein (e.g., isotropy irreducible) and the Killing form of $\mathfrak{k}$ is a multiple of the Killing form of $\mathfrak{h}$ (e.g., $K$ simple), a class which contains $17$ families and $50$ individual examples. A complete classification is obtained in the case when $H/K$ is an irreducible symmetric space with $K$ simple. We also study the behavior of the scalar curvature function on the space of all normal metrics on $M=H\times H/ΔK$ (none of which is Einstein), obtaining that the standard metric is a global minimum. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_13407 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Einstein metrics on homogeneous spaces $H\times H/ΔK$ Lauret, Jorge Will, Cynthia Differential Geometry Given any compact homogeneous space $H/K$ with $H$ simple, we consider the new space $M=H\times H/ΔK$, where $ΔK$ denotes diagonal embedding, and study the existence, classification and stability of $H\times H$-invariant Einstein metrics on $M$, as a first step into the largely unexplored case of homogeneous spaces of compact non-simple Lie groups. We find unstable Einstein metrics on $M$ for most spaces $H/K$ such that their standard metric is Einstein (e.g., isotropy irreducible) and the Killing form of $\mathfrak{k}$ is a multiple of the Killing form of $\mathfrak{h}$ (e.g., $K$ simple), a class which contains $17$ families and $50$ individual examples. A complete classification is obtained in the case when $H/K$ is an irreducible symmetric space with $K$ simple. We also study the behavior of the scalar curvature function on the space of all normal metrics on $M=H\times H/ΔK$ (none of which is Einstein), obtaining that the standard metric is a global minimum. |
| title | Einstein metrics on homogeneous spaces $H\times H/ΔK$ |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2402.13407 |