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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.08948 |
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| _version_ | 1866912374456320000 |
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| author | Ravanpak, Zohreh Vizman, Cornelia |
| author_facet | Ravanpak, Zohreh Vizman, Cornelia |
| contents | For a Poisson manifold endowed with a pseudo-Riemannian metric, we investigate degeneracies arising when the metric is restricted to symplectic leaves. Central to this work is the generalized double bracket (GDB) vector field-a geometric construct introduced in our earlier work-which generalizes gradient dynamics to indefinite metric settings. We identify admissible regions where the so-called double bracket metric remains non-degenerate on symplectic leaves, enabling the GDB vector field to function as a gradient flow on the admissible regions with respect to this metric. We illustrate these concepts with a variety of examples and carefully discuss the complications that arise when the pseudo-Riemannian metric fails to induce a non-degenerate metric on certain regions of the symplectic leaves. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_08948 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Metric degeneracies and gradient flows on symplectic leaves Ravanpak, Zohreh Vizman, Cornelia Differential Geometry Mathematical Physics For a Poisson manifold endowed with a pseudo-Riemannian metric, we investigate degeneracies arising when the metric is restricted to symplectic leaves. Central to this work is the generalized double bracket (GDB) vector field-a geometric construct introduced in our earlier work-which generalizes gradient dynamics to indefinite metric settings. We identify admissible regions where the so-called double bracket metric remains non-degenerate on symplectic leaves, enabling the GDB vector field to function as a gradient flow on the admissible regions with respect to this metric. We illustrate these concepts with a variety of examples and carefully discuss the complications that arise when the pseudo-Riemannian metric fails to induce a non-degenerate metric on certain regions of the symplectic leaves. |
| title | Metric degeneracies and gradient flows on symplectic leaves |
| topic | Differential Geometry Mathematical Physics |
| url | https://arxiv.org/abs/2505.08948 |