Riemannian distance and symplectic embeddings in cotangent bundle
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866913345490124800 |
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| author | Broćić, Filip |
| author_facet | Broćić, Filip |
| contents | Given an open neighborhood $W$ of the zero section in the cotangent bundle of $N$ we define a distance-like function $ρ_W$ on $N$ using certain symplectic embeddings from the standard ball $B^{2n}(r)$ to $W$. We show that when $W$ is the unit disc-cotangent bundle of a Riemannian metric on $N$, $ρ_W$ recovers the metric. As an intermediate step, we give a new construction of the ball of capacity 4 to the product of Lagrangian discs $P_L := B^n(1)\times B^n(1)$, and we give a new proof of the strong Viterbo conjecture about normalized capacities for $P_L$. We also give bounds of the symplectic packing number of two balls in a unit disc-cotangent bundle relative to the zero section $N$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_12752 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Riemannian distance and symplectic embeddings in cotangent bundle Broćić, Filip Symplectic Geometry Differential Geometry Given an open neighborhood $W$ of the zero section in the cotangent bundle of $N$ we define a distance-like function $ρ_W$ on $N$ using certain symplectic embeddings from the standard ball $B^{2n}(r)$ to $W$. We show that when $W$ is the unit disc-cotangent bundle of a Riemannian metric on $N$, $ρ_W$ recovers the metric. As an intermediate step, we give a new construction of the ball of capacity 4 to the product of Lagrangian discs $P_L := B^n(1)\times B^n(1)$, and we give a new proof of the strong Viterbo conjecture about normalized capacities for $P_L$. We also give bounds of the symplectic packing number of two balls in a unit disc-cotangent bundle relative to the zero section $N$. |
| title | Riemannian distance and symplectic embeddings in cotangent bundle |
| topic | Symplectic Geometry Differential Geometry |
| url | https://arxiv.org/abs/2303.12752 |