Metric operator and geodesic orbit property for a standard homogeneous Finsler metric
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910415690137600 |
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| author | Zhang, Lei Xu, Ming |
| author_facet | Zhang, Lei Xu, Ming |
| contents | In this paper, we introduce the metric operator for a compact homogeneous Finsler space, and use it to investigate the geodesic orbit property. We define the notion of standard homogeneous $(α_1,\cdots,α_s)$-metric which generalizes the notion of standard homogeneous $(α_1,α_2)$-metric. We classify all connected simply connected homogeneous manifold $G/H$ with a compact connected simple Lie group $G$ and two irreducible summands in its isotropy representation, such that there exists a standard homogeneous $(α_1,α_2)$-metric which is g.o. but not naturally reductive on $G/H$. We also prove that on a generalized Wallach space which is not a product of three symmetric spaces, any standard homogeneous $(α_1,α_2,α_3)$-metric $F$ with respect to the canonical decomposition is g.o. on $G/H$ if and only if $F$ is a normal homogeneous Riemannian metric. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_13367 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Metric operator and geodesic orbit property for a standard homogeneous Finsler metric Zhang, Lei Xu, Ming Differential Geometry In this paper, we introduce the metric operator for a compact homogeneous Finsler space, and use it to investigate the geodesic orbit property. We define the notion of standard homogeneous $(α_1,\cdots,α_s)$-metric which generalizes the notion of standard homogeneous $(α_1,α_2)$-metric. We classify all connected simply connected homogeneous manifold $G/H$ with a compact connected simple Lie group $G$ and two irreducible summands in its isotropy representation, such that there exists a standard homogeneous $(α_1,α_2)$-metric which is g.o. but not naturally reductive on $G/H$. We also prove that on a generalized Wallach space which is not a product of three symmetric spaces, any standard homogeneous $(α_1,α_2,α_3)$-metric $F$ with respect to the canonical decomposition is g.o. on $G/H$ if and only if $F$ is a normal homogeneous Riemannian metric. |
| title | Metric operator and geodesic orbit property for a standard homogeneous Finsler metric |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2404.13367 |