Harmonic morphisms on $(\mathbb{S}^4
In this paper we study examples of harmonic morphisms due to Burel from $(\mathbb{S}^4,g_{k,l})$ into $\mathbb{S}^2$ where $(g_{k,l})$ is a family of conformal metrics on $\mathbb{S}^4$. To do this construction we define two maps, $F$ from $(\mathbb{S}^4,g_{k,l})$ to $(\mathbb{S}^3,\bar{g_{k,l}})$ a...
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ftunitours:oai:HAL:hal-01912983v1 2024-05-19T07:45:41+00:00 Harmonic morphisms on $(\mathbb{S}^4 Makki, Ali Soret, Marc Ville, Marina Laboratoire de Mathématiques et Physique Théorique (LMPT) Université de Tours (UT)-Centre National de la Recherche Scientifique (CNRS) 2018-11-05 https://hal.science/hal-01912983 https://hal.science/hal-01912983/document https://hal.science/hal-01912983/file/article%202.pdf en eng HAL CCSD hal-01912983 https://hal.science/hal-01912983 https://hal.science/hal-01912983/document https://hal.science/hal-01912983/file/article%202.pdf info:eu-repo/semantics/OpenAccess https://hal.science/hal-01912983 2018 [MATH]Mathematics [math] [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] info:eu-repo/semantics/preprint Preprints, Working Papers, . 2018 ftunitours 2024-04-25T00:00:22Z In this paper we study examples of harmonic morphisms due to Burel from $(\mathbb{S}^4,g_{k,l})$ into $\mathbb{S}^2$ where $(g_{k,l})$ is a family of conformal metrics on $\mathbb{S}^4$. To do this construction we define two maps, $F$ from $(\mathbb{S}^4,g_{k,l})$ to $(\mathbb{S}^3,\bar{g_{k,l}})$ and $\phi_{k,l}$ from $(\mathbb{S}^3,\bar{g_{k,l}})$ to $(\mathbb{S}^2,can)$; the two maps are both horizontally conformal and harmonic. Let $\Phi_{k,l}=\phi_{k,l} \circ F$. It follows from Baird-Eells that the regular fibres of $\Phi_{k,l}$ for every $k,l$ are minimal. If $|k|=|l|=1$, the set of critical points is given by the preimage of the north pole : it consists in two 2-spheres meeting transversally at 2 points. If $k,l\neq1$ the set of critical points are the preimages of the north pole (the same two spheres as for $k=l=1$ but with multiplicity $l$) together with the preimage of the south pole (a torus with multiplicity $k$). Report North Pole South pole Université François-Rabelais de Tours: HAL |
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Open Polar |
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Université François-Rabelais de Tours: HAL |
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English |
topic |
[MATH]Mathematics [math] [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] |
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[MATH]Mathematics [math] [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] Makki, Ali Soret, Marc Ville, Marina Harmonic morphisms on $(\mathbb{S}^4 |
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[MATH]Mathematics [math] [MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] |
description |
In this paper we study examples of harmonic morphisms due to Burel from $(\mathbb{S}^4,g_{k,l})$ into $\mathbb{S}^2$ where $(g_{k,l})$ is a family of conformal metrics on $\mathbb{S}^4$. To do this construction we define two maps, $F$ from $(\mathbb{S}^4,g_{k,l})$ to $(\mathbb{S}^3,\bar{g_{k,l}})$ and $\phi_{k,l}$ from $(\mathbb{S}^3,\bar{g_{k,l}})$ to $(\mathbb{S}^2,can)$; the two maps are both horizontally conformal and harmonic. Let $\Phi_{k,l}=\phi_{k,l} \circ F$. It follows from Baird-Eells that the regular fibres of $\Phi_{k,l}$ for every $k,l$ are minimal. If $|k|=|l|=1$, the set of critical points is given by the preimage of the north pole : it consists in two 2-spheres meeting transversally at 2 points. If $k,l\neq1$ the set of critical points are the preimages of the north pole (the same two spheres as for $k=l=1$ but with multiplicity $l$) together with the preimage of the south pole (a torus with multiplicity $k$). |
author2 |
Laboratoire de Mathématiques et Physique Théorique (LMPT) Université de Tours (UT)-Centre National de la Recherche Scientifique (CNRS) |
format |
Report |
author |
Makki, Ali Soret, Marc Ville, Marina |
author_facet |
Makki, Ali Soret, Marc Ville, Marina |
author_sort |
Makki, Ali |
title |
Harmonic morphisms on $(\mathbb{S}^4 |
title_short |
Harmonic morphisms on $(\mathbb{S}^4 |
title_full |
Harmonic morphisms on $(\mathbb{S}^4 |
title_fullStr |
Harmonic morphisms on $(\mathbb{S}^4 |
title_full_unstemmed |
Harmonic morphisms on $(\mathbb{S}^4 |
title_sort |
harmonic morphisms on $(\mathbb{s}^4 |
publisher |
HAL CCSD |
publishDate |
2018 |
url |
https://hal.science/hal-01912983 https://hal.science/hal-01912983/document https://hal.science/hal-01912983/file/article%202.pdf |
genre |
North Pole South pole |
genre_facet |
North Pole South pole |
op_source |
https://hal.science/hal-01912983 2018 |
op_relation |
hal-01912983 https://hal.science/hal-01912983 https://hal.science/hal-01912983/document https://hal.science/hal-01912983/file/article%202.pdf |
op_rights |
info:eu-repo/semantics/OpenAccess |
_version_ |
1799485781056684032 |