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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.06760 |
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| _version_ | 1866910235369668608 |
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| author | Liu, Chong Wang, Shi |
| author_facet | Liu, Chong Wang, Shi |
| contents | For any compact connected Lie group $G$, we introduce a novel notion of average signature $\mathbb A(G)$ valued in its tensor Lie algebra, by taking the average value of the signature of the unique length-minimizing geodesics between all pairs of generic points in $G$. we prove that using the average signature together with the trace operation with respect to the given bi-invariant Riemannian metric on $G$, one can recover certain geometric quantities of $G$, including the dimension, the diameter, the volume and the scalar curvature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_06760 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Average signature of geodesic paths in compact Lie groups Liu, Chong Wang, Shi Differential Geometry Probability 60L10, 22E15 For any compact connected Lie group $G$, we introduce a novel notion of average signature $\mathbb A(G)$ valued in its tensor Lie algebra, by taking the average value of the signature of the unique length-minimizing geodesics between all pairs of generic points in $G$. we prove that using the average signature together with the trace operation with respect to the given bi-invariant Riemannian metric on $G$, one can recover certain geometric quantities of $G$, including the dimension, the diameter, the volume and the scalar curvature. |
| title | Average signature of geodesic paths in compact Lie groups |
| topic | Differential Geometry Probability 60L10, 22E15 |
| url | https://arxiv.org/abs/2411.06760 |