Harmonic-curvature warped products over surfaces
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866909432460345344 |
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| author | Derdzinski, Andrzej Piccione, Paolo |
| author_facet | Derdzinski, Andrzej Piccione, Paolo |
| contents | For warped products with harmonic curvature, nonconstant warping functions $ϕ$, and compact two-dimensional bases $(M,h)$, we establish a dichotomy: either the Gaussian curvature $K$ of the metric $g=ϕ^{-2}h$ is constant and negative, or $ϕ$ equals a specific elementary function of $K$, also depending on the dimension $p$ and Einstein constant $\varepsilon$ of the fibre. In both cases the fibre must be an Einstein manifold with $p>1$ and $\varepsilon>0$, while the function $f=ϕ^{p/2}$ satisfies a Yamabe-type second-order differential equation on $(M,g)$. We prove that both possibilities are realized on every closed orientable surface of genus greater than $1$, and in the latter case -- which also occurs on the $2$-sphere and real projective plane -- the metrics in question constitute uncountably many distinct homothety types. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_01695 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Harmonic-curvature warped products over surfaces Derdzinski, Andrzej Piccione, Paolo Differential Geometry 53C25, 53B20 For warped products with harmonic curvature, nonconstant warping functions $ϕ$, and compact two-dimensional bases $(M,h)$, we establish a dichotomy: either the Gaussian curvature $K$ of the metric $g=ϕ^{-2}h$ is constant and negative, or $ϕ$ equals a specific elementary function of $K$, also depending on the dimension $p$ and Einstein constant $\varepsilon$ of the fibre. In both cases the fibre must be an Einstein manifold with $p>1$ and $\varepsilon>0$, while the function $f=ϕ^{p/2}$ satisfies a Yamabe-type second-order differential equation on $(M,g)$. We prove that both possibilities are realized on every closed orientable surface of genus greater than $1$, and in the latter case -- which also occurs on the $2$-sphere and real projective plane -- the metrics in question constitute uncountably many distinct homothety types. |
| title | Harmonic-curvature warped products over surfaces |
| topic | Differential Geometry 53C25, 53B20 |
| url | https://arxiv.org/abs/2201.01695 |