Lie groups with a bi-invariant distance
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917115269742592 |
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| author | Larotonda, Gabriel Rey, Iván |
| author_facet | Larotonda, Gabriel Rey, Iván |
| contents | We show that a Lie group $G$ admitting a bi-invariant distance must be the product $G=H\times K$ of an abelian group $H$ and a compact group $K$ with discrete center. Moreover, the distance in $G$ must come from the infima of lengths of paths for a unique infinitesimal metric (a Finsler norm) defined in the Lie algebra of $G$. From this we derive the distance minimizing paths which are left or right translations of one-parameter groups (though these are not the unique minizing paths if the norm is not smooth or strictly convex). Then we introduce a notion of sectional curvature $sec(π)$ for a bi-invariant distance, following Milnor's ideas, and we show that this curvature is bounded and non-negative, and it is null when the $2$-plane $π$ is an abelian Lie subalgebra of $Lie(G)$. We show that when the distance is strictly convex, our sectional curvature vanishes if and only if the $2$-plane is abelian. We give finer characterizations for the case of vanishing curvature, for the case of non-strictly convex norms |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_09699 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lie groups with a bi-invariant distance Larotonda, Gabriel Rey, Iván Differential Geometry 22E65 (Primary) 58B20, 53C22 (Secondary) We show that a Lie group $G$ admitting a bi-invariant distance must be the product $G=H\times K$ of an abelian group $H$ and a compact group $K$ with discrete center. Moreover, the distance in $G$ must come from the infima of lengths of paths for a unique infinitesimal metric (a Finsler norm) defined in the Lie algebra of $G$. From this we derive the distance minimizing paths which are left or right translations of one-parameter groups (though these are not the unique minizing paths if the norm is not smooth or strictly convex). Then we introduce a notion of sectional curvature $sec(π)$ for a bi-invariant distance, following Milnor's ideas, and we show that this curvature is bounded and non-negative, and it is null when the $2$-plane $π$ is an abelian Lie subalgebra of $Lie(G)$. We show that when the distance is strictly convex, our sectional curvature vanishes if and only if the $2$-plane is abelian. We give finer characterizations for the case of vanishing curvature, for the case of non-strictly convex norms |
| title | Lie groups with a bi-invariant distance |
| topic | Differential Geometry 22E65 (Primary) 58B20, 53C22 (Secondary) |
| url | https://arxiv.org/abs/2511.09699 |