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Using Besson's characterization, we prove the Strong Arnold Hypothesis is satisfied for all metrics except for a set of infinite codimension, and use this to obtain the precise codimension of the set of metrics admitting an eigenvalue of any given multiplicity. Furthermore, we show that the Strong Arnold Hypothesis is satisfied for all metrics admitting eigenvalues of multiplicity at most six, and discuss several examples of metrics violating it.
| Comments: | 41 pages, changes in v4: It came to our attention that the simplified transversality condition (Theorem 1.1) was previously obtained by G. Besson. We updated the preprint accordingly. Secondly, a discussion of the (non)-manifold structure of the set of metrics with an eigenvalue of a given multiplicity near metrics where transversality fails was added |
| Subjects: | Differential Geometry (math.DG); Analysis of PDEs (math.AP); Spectral Theory (math.SP) |
| MSC classes: | 58J50 (Primary) 58C40, 58J37, 47A55, 53B20 (Secondary) |
| Cite as: | arXiv:2312.16939 [math.DG] |
| (or arXiv:2312.16939v4 [math.DG] for this version) | |
| https://doi.org/10.48550/arXiv.2312.16939 arXiv-issued DOI via DataCite |
From: Josef Eberhard Greilhuber [view email]
[v1]
Thu, 28 Dec 2023 10:22:27 UTC (50 KB)
[v2]
Mon, 4 Mar 2024 08:14:16 UTC (44 KB)
[v3]
Tue, 5 Mar 2024 20:49:45 UTC (44 KB)
[v4]
Mon, 25 May 2026 06:12:20 UTC (54 KB)
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