A Conformal Co-Symplectic Structure on the Space of Pseudo-Riemannian Geodesics
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914078782390272 |
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| author | Iglesias-Zemmour, Patrick |
| author_facet | Iglesias-Zemmour, Patrick |
| contents | The classical construction of the symplectic structure on the space of geodesic trajectories via Hamiltonian reduction fails in the pseudo-Riemannian setting due to a dimensional mismatch created by the null geodesics. This paper proposes a new, unified approach. We first construct the space of all geodesic trajectories $\mathcal{G}_\text{traj}$ directly as the quotient of the space of geodesics curves $\mathcal{G}_\text{curv}$ by the affine reparametrization group. The analysis of the orbits of this group action reveals a key geometric distribution. To describe this distribution globally, we introduce a canonical object, the "conformal co-symplectic structure" $σ$, defined by pushing forward the conformal class of the inverse $ω^{-1}$ of the original symplectic form $ω$. We prove that the image of this structure coincides with the geometric distribution identified previously. On the subspace of time-like and space-like geodesics, this structure is non-degenerate and defines a conformal class of symplectic forms. On the null subspace, its image is a codimension-$1$ distribution that we prove is the canonical contact structure on the space of light rays. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_05155 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Conformal Co-Symplectic Structure on the Space of Pseudo-Riemannian Geodesics Iglesias-Zemmour, Patrick Differential Geometry 53C50, 53D05, 53D10 The classical construction of the symplectic structure on the space of geodesic trajectories via Hamiltonian reduction fails in the pseudo-Riemannian setting due to a dimensional mismatch created by the null geodesics. This paper proposes a new, unified approach. We first construct the space of all geodesic trajectories $\mathcal{G}_\text{traj}$ directly as the quotient of the space of geodesics curves $\mathcal{G}_\text{curv}$ by the affine reparametrization group. The analysis of the orbits of this group action reveals a key geometric distribution. To describe this distribution globally, we introduce a canonical object, the "conformal co-symplectic structure" $σ$, defined by pushing forward the conformal class of the inverse $ω^{-1}$ of the original symplectic form $ω$. We prove that the image of this structure coincides with the geometric distribution identified previously. On the subspace of time-like and space-like geodesics, this structure is non-degenerate and defines a conformal class of symplectic forms. On the null subspace, its image is a codimension-$1$ distribution that we prove is the canonical contact structure on the space of light rays. |
| title | A Conformal Co-Symplectic Structure on the Space of Pseudo-Riemannian Geodesics |
| topic | Differential Geometry 53C50, 53D05, 53D10 |
| url | https://arxiv.org/abs/2510.05155 |