Higher order obstructions to Riccati-type equations
| Autor: | Kim, Jihun, Nagy, Paul-Andi, Park, JeongHyeong |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | We develop new techniques in order to deal with Riccati-type equations, subject to a further algebraic constraint, on Riemannian manifolds $(M^3,g)$. We find that the obstruction to solve the aforementioned equation has order $4$ in the metric coefficients and is fully described by an homogeneous polynomial in $\mathrm{Sym}^{16}TM$. Techniques from real algebraic geometry, reminiscent of those used for the "PositiveStellen-Satz " problem, allow determining the geometry in terms of explicit exterior differential systems. Analysis of the latter shows flatness for the metric $g$; in particular we complete the classification of asymptotically harmonic manifolds of dimension $3$, establishing those are either flat or real hyperbolic spaces. Comment: minor changes, a few typos caught |
| Databáze: | arXiv |
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