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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2412.02872 |
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| _version_ | 1866914072805507072 |
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| author | Nguyen, Duc Toan |
| author_facet | Nguyen, Duc Toan |
| contents | Geodesic nets are types of graphs in Riemannian manifolds where each edge is a geodesic segment. One important object used in the construction of geodesic nets is a balanced vertex, where the sum of unit tangent vectors along adjacent edges is zero. We prove the existence of a balanced vertex of a triangle (with three unbalanced vertices) on a general two-dimensional Riemannian surface when all angles measure less than $2π/3$, if the length of the sides of the triangle is not too large. This property is a generalization for the existence of the Fermat point of a planar triangle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_02872 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the existence of a balanced vertex in geodesic nets with three boundary vertices Nguyen, Duc Toan Differential Geometry Geodesic nets are types of graphs in Riemannian manifolds where each edge is a geodesic segment. One important object used in the construction of geodesic nets is a balanced vertex, where the sum of unit tangent vectors along adjacent edges is zero. We prove the existence of a balanced vertex of a triangle (with three unbalanced vertices) on a general two-dimensional Riemannian surface when all angles measure less than $2π/3$, if the length of the sides of the triangle is not too large. This property is a generalization for the existence of the Fermat point of a planar triangle. |
| title | On the existence of a balanced vertex in geodesic nets with three boundary vertices |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2412.02872 |