Generalized Ricci flow on aligned homogeneous spaces
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866910843471396864 |
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| author | Gutiérrez, Valeria |
| author_facet | Gutiérrez, Valeria |
| contents | The fixed points of the generalized Ricci flow are the Bismut Ricci flat metrics, i.e., a generalized metric $(g,H)$ on a manifold $M$, where $g$ is a Riemannian metric and $H$ a closed $3$-form, such that $H$ is $g$-harmonic and $\operatorname{Rc}(g)=\tfrac{1}{4} H_g^2$. Given two standard Einstein homogeneous spaces $G_i/K$, where each $G_i$ is a compact simple Lie group and $K$ is a closed subgroup of them holding some extra assumption, we consider $M = G_1 \times G_2 / ΔK$. Recently, Lauret and Will proved the existence of a Bismut Ricci flat metric on any of these spaces. We proved that this metric is always asymptotically stable for the generalized Ricci flow on $M$ among a subset of $G$-invariant metrics and, if $G_1 = G_2$, then it is globally stable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_03332 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalized Ricci flow on aligned homogeneous spaces Gutiérrez, Valeria Differential Geometry The fixed points of the generalized Ricci flow are the Bismut Ricci flat metrics, i.e., a generalized metric $(g,H)$ on a manifold $M$, where $g$ is a Riemannian metric and $H$ a closed $3$-form, such that $H$ is $g$-harmonic and $\operatorname{Rc}(g)=\tfrac{1}{4} H_g^2$. Given two standard Einstein homogeneous spaces $G_i/K$, where each $G_i$ is a compact simple Lie group and $K$ is a closed subgroup of them holding some extra assumption, we consider $M = G_1 \times G_2 / ΔK$. Recently, Lauret and Will proved the existence of a Bismut Ricci flat metric on any of these spaces. We proved that this metric is always asymptotically stable for the generalized Ricci flow on $M$ among a subset of $G$-invariant metrics and, if $G_1 = G_2$, then it is globally stable. |
| title | Generalized Ricci flow on aligned homogeneous spaces |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2401.03332 |