Conjugate points on Lie groups with left-invariant metrics
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| author | Brigant, Alice Le Lichtenfelz, Leandro Preston, Stephen C. |
| author_facet | Brigant, Alice Le Lichtenfelz, Leandro Preston, Stephen C. |
| contents | We prove sufficient conditions for the existence of conjugate points along geodesics of a left-invariant metric on a Lie group, using a reformulation of the index form in terms of the adjoint action. In the compact semisimple case, with an arbitrary left-invariant metric, we show that all geodesics must have a conjugate point, and we give upper and lower bounds on conjugate times. In particular this applies to the left-invariant metrics on $SU(n)$ and $SO(n)$ which are of importance in fluid dynamics and rigid body motion, and yields estimates for the diameter and injectivity radius. We also establish criteria in the noncompact case: we show that every closed nonhomogeneous geodesic has a conjugate point, and determine explicit conditions for them in the three-dimensional unimodular case. For homogeneous geodesics, we relate conjugate points to Lagrangian stability, and Eulerian stability of the corresponding steady velocity. Finally, we obtain as by-products criteria for conjugate points in general homogeneous spaces, by lifting the problem to the total Lie group of the quotient and using a result of O'Neill. Through several examples, we show that our theorems apply when well-known criteria relying on positive Ricci curvature or other curvature bounds fail, and in some cases even when Ricci curvature is negative in all directions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_03854 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Conjugate points on Lie groups with left-invariant metrics Brigant, Alice Le Lichtenfelz, Leandro Preston, Stephen C. Differential Geometry We prove sufficient conditions for the existence of conjugate points along geodesics of a left-invariant metric on a Lie group, using a reformulation of the index form in terms of the adjoint action. In the compact semisimple case, with an arbitrary left-invariant metric, we show that all geodesics must have a conjugate point, and we give upper and lower bounds on conjugate times. In particular this applies to the left-invariant metrics on $SU(n)$ and $SO(n)$ which are of importance in fluid dynamics and rigid body motion, and yields estimates for the diameter and injectivity radius. We also establish criteria in the noncompact case: we show that every closed nonhomogeneous geodesic has a conjugate point, and determine explicit conditions for them in the three-dimensional unimodular case. For homogeneous geodesics, we relate conjugate points to Lagrangian stability, and Eulerian stability of the corresponding steady velocity. Finally, we obtain as by-products criteria for conjugate points in general homogeneous spaces, by lifting the problem to the total Lie group of the quotient and using a result of O'Neill. Through several examples, we show that our theorems apply when well-known criteria relying on positive Ricci curvature or other curvature bounds fail, and in some cases even when Ricci curvature is negative in all directions. |
| title | Conjugate points on Lie groups with left-invariant metrics |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2408.03854 |