Length spectrum rigidity and flexibility of spheres of revolution with one equator
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866909999042658304 |
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| author | Abbondandolo, Alberto Mazzucchelli, Marco |
| author_facet | Abbondandolo, Alberto Mazzucchelli, Marco |
| contents | We define a notion of marked length spectrum for $S^1$-symmetric Riemannian metrics on the two-sphere having only one equator. We prove that isospectral metrics in this class have conjugate geodesic flows. Under a further $\mathbb{Z}_2$-symmetry assumption, we show that the marked length spectrum determines the metric. Finally, we show that every isospectral class of metrics contains a unique $\mathbb{Z}_2$-symmetric metric and give an explicit description of this isospectral class as an infinite dimensional convex set, generalizing the known description of $S^1$-symmetric Zoll metrics. This paper contains also two appendices, in which we provide an elementary proof of the fact that a $C^2$ real valued function on an interval is determined by the set of tangent lines to its graph, and we classify a class of $S^1$-invariant contact forms on three-manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_16804 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Length spectrum rigidity and flexibility of spheres of revolution with one equator Abbondandolo, Alberto Mazzucchelli, Marco Differential Geometry Dynamical Systems Symplectic Geometry 53D25, 37J35, 37C27 We define a notion of marked length spectrum for $S^1$-symmetric Riemannian metrics on the two-sphere having only one equator. We prove that isospectral metrics in this class have conjugate geodesic flows. Under a further $\mathbb{Z}_2$-symmetry assumption, we show that the marked length spectrum determines the metric. Finally, we show that every isospectral class of metrics contains a unique $\mathbb{Z}_2$-symmetric metric and give an explicit description of this isospectral class as an infinite dimensional convex set, generalizing the known description of $S^1$-symmetric Zoll metrics. This paper contains also two appendices, in which we provide an elementary proof of the fact that a $C^2$ real valued function on an interval is determined by the set of tangent lines to its graph, and we classify a class of $S^1$-invariant contact forms on three-manifolds. |
| title | Length spectrum rigidity and flexibility of spheres of revolution with one equator |
| topic | Differential Geometry Dynamical Systems Symplectic Geometry 53D25, 37J35, 37C27 |
| url | https://arxiv.org/abs/2601.16804 |