Higher order obstructions to Riccati-type equations
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866915501511278592 |
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| author | Kim, Jihun Nagy, Paul-Andi Park, JeongHyeong |
| author_facet | Kim, Jihun Nagy, Paul-Andi Park, JeongHyeong |
| contents | 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 the connection coefficients and a class of Hessian-type equations. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_16915 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Higher order obstructions to Riccati-type equations Kim, Jihun Nagy, Paul-Andi Park, JeongHyeong Differential Geometry Algebraic Geometry 53C25, 53C21 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 the connection coefficients and a class of Hessian-type equations. 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. |
| title | Higher order obstructions to Riccati-type equations |
| topic | Differential Geometry Algebraic Geometry 53C25, 53C21 |
| url | https://arxiv.org/abs/2407.16915 |