Harmonic morphisms, Hermitian structures and symmetric spaces

Martin Svensson

Harmonic morphisms, Hermitian structures and symmetric spaces

Martin Svensson

Forskningsoutput: Avhandling › Doktorsavhandling (sammanläggning)

Sammanfattning

Popular Abstract in Swedish
En harmonisk morfism är en funktion mellan två Riemannmångfalder, med den egenskapen att dess sammansättning med en lokal harmonisk funktion på bildmängden ånyo är en lokal harmonisk funktion på definitionsmängden. En sådan funktion är automatiskt en harmonisk avbildning, som dessutom är (svagt) horisontellt konform. I lokala koordinater betyder detta att funktionen måste uppfylla ett icke-lineärt, överbestämt system av partiella differentialekvationer. Därför är i allmänhet existensen av sådana funktioner, till och med i det lokala fallet, mycket svår att påvisa. Dock, när mångfalderna har Hermitska strukturer, är det naturligt att söka efter kandidater för harmoniska morfismer bland de holomorfa avbildningarna.

I denna avhandling studerar vi i en serie artiklar strukturen hos harmoniska morfismer mellan Hermitska mångfalder. I det sammanhanget undersöker vi också konforma folieringar, vilket är ett besläktat område. Vi bevisar flera resultat för holomorfa harmoniska morfismer mellan Kähler mångfalder, speciellt när definitionsmängden har icke-negativ krökning. Många av resultaten visas också vara sanna för holomorfa konforma folieringar. Ett viktigt verktyg är en formel av Walzcak, som, i detta sammanhang, väsentligen förenklas.

Genom en konstruktion som härrör från twistorteorin producerar vi dessutom det första kända exempel på en harmonisk morfism vars fibrer inte är fullständigt geodetiska från det hyperboliska rummet av dimension större än 4. Dessa idéer generaliseras sedan till en rad andra Riemannska symmetriska rum av typ I och III, varvid vi kan påvisa existensen av sådana funktioner från dessa rum. Genom att utnyttja dualitetsbegreppet konstruerar vi också harmoniska morfismer från flera Riemannska symmetriska rum av typ II och IV.

Originalspråk	engelska
Kvalifikation	Doktor
Tilldelande institution	Matematik (naturvetenskapliga fakulteten)
Handledare	Gudmundsson, Sigmundur, handledare
Tilldelningsdatum	2004 mars 16
Status	Published - 2004

Bibliografisk information

Defence details

Date: 2004-03-16
Time: 13:15
Place: Matematikhuset Sal C

External reviewer(s)

Name: Burstall, Fran
Title: Professor
Affiliation: University of Bath, UK.

---

Article: M. Svensson, "On holomorphic harmonic morphisms", Manuscripta Math. 107 (2002), 1-13.

Article: M. Svensson, "Harmonic morphisms from even-dimensional hyperbolic spaces", Math. Scand. 92 (2003), 246-260.

Article: M. Svensson, "Holomorphic foliations, harmonic morphisms and the Walczak formula", J. London Math. Soc. 68 (2003), 781-794.

Article: M. Svensson, "Harmonic morphisms in Hermitian geometry", J. Reine Angew. Math. (to appear).

Article: S. Gudmundsson and M. Svensson, "Harmonic morphisms from the Grassmannians and their non-compact duals", preprint, Lund University (2003).

Article: S. Gudmundsson and M. Svensson, "Harmonic morphisms from the compact simple Lie groups and their non-compact duals", preprint, Lund University (2003).

Ämnesklassifikation (UKÄ)

Matematik

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@phdthesis{91248d3e2b4c4f74b700d6cdcfd7bfa5,

title = "Harmonic morphisms, Hermitian structures and symmetric spaces",

abstract = "A harmonic morphism is a map between two Riemannian manifolds with the property that its composition with a local harmonic function on the codomain is a local harmonic function on the domain. Such a map is automatically a harmonic map, satisfying an additional partial conformality condition called horizontal (weak) conformality. In local coordinates, this amounts to a non-linear, over-determined system of partial differential equations. Therefore, the question of the existence of such maps, even in the local case, is very hard to answer in general. However, when the manifolds involved carry Hermitian structures, it is natural to search among the holomorphic maps for candidates for harmonic morphisms. In this thesis, we study in a series of articles the geometry of harmonic morphisms and conformal foliations, a closely related topic, in the context of Hermitian geometry. Several results are proved for holomorphic harmonic morphisms between K{\"a}hler manifolds, in particular when the domain is of non-negative sectional curvature or constant holomorphic sectional curvature. Many of these results also extend to holomorphic conformal foliations. A principal tool we use is a formula by Walczak which, in this context, takes a particularly simple form. We make a brief digression to apply this formula to harmonic morphisms of warped product type, which results in a decomposition theorem for such maps. We also study the four-dimensional, conformally flat case in detail. By a construction which originates in the ideas of twistor theory, we produce the first known examples of harmonic morphisms without totally geodesic fibres from real hyperbolic spaces of even dimension greater than four. These ideas are then generalized to obtain harmonic morphisms from several other Riemannian symmetric spaces of type I and III, such as the Grassmannians, where the question of local existence was an open problem. By further exploiting the concept of duality for Riemannian symmetric spaces, we also obtain harmonic morphisms from several Riemannian symmetric spaces of type II and IV.",

keywords = "isotropic subbundles, conformal foliations, holomorphic maps, harmonic morphism",

author = "Martin Svensson",

note = "Defence details Date: 2004-03-16 Time: 13:15 Place: Matematikhuset Sal C External reviewer(s) Name: Burstall, Fran Title: Professor Affiliation: University of Bath, UK. --- Article: M. Svensson, {"}On holomorphic harmonic morphisms{"}, Manuscripta Math. 107 (2002), 1-13. Article: M. Svensson, {"}Harmonic morphisms from even-dimensional hyperbolic spaces{"}, Math. Scand. 92 (2003), 246-260. Article: M. Svensson, {"}Holomorphic foliations, harmonic morphisms and the Walczak formula{"}, J. London Math. Soc. 68 (2003), 781-794. Article: M. Svensson, {"}Harmonic morphisms in Hermitian geometry{"}, J. Reine Angew. Math. (to appear). Article: S. Gudmundsson and M. Svensson, {"}Harmonic morphisms from the Grassmannians and their non-compact duals{"}, preprint, Lund University (2003). Article: S. Gudmundsson and M. Svensson, {"}Harmonic morphisms from the compact simple Lie groups and their non-compact duals{"}, preprint, Lund University (2003).",

year = "2004",

language = "English",

type = "Doctoral Thesis (compilation)",

school = "Mathematics (Faculty of Sciences)",

}

TY - THES

T1 - Harmonic morphisms, Hermitian structures and symmetric spaces

AU - Svensson, Martin

N1 - Defence details Date: 2004-03-16 Time: 13:15 Place: Matematikhuset Sal C External reviewer(s) Name: Burstall, Fran Title: Professor Affiliation: University of Bath, UK. --- Article: M. Svensson, "On holomorphic harmonic morphisms", Manuscripta Math. 107 (2002), 1-13. Article: M. Svensson, "Harmonic morphisms from even-dimensional hyperbolic spaces", Math. Scand. 92 (2003), 246-260. Article: M. Svensson, "Holomorphic foliations, harmonic morphisms and the Walczak formula", J. London Math. Soc. 68 (2003), 781-794. Article: M. Svensson, "Harmonic morphisms in Hermitian geometry", J. Reine Angew. Math. (to appear). Article: S. Gudmundsson and M. Svensson, "Harmonic morphisms from the Grassmannians and their non-compact duals", preprint, Lund University (2003). Article: S. Gudmundsson and M. Svensson, "Harmonic morphisms from the compact simple Lie groups and their non-compact duals", preprint, Lund University (2003).

PY - 2004

Y1 - 2004

N2 - A harmonic morphism is a map between two Riemannian manifolds with the property that its composition with a local harmonic function on the codomain is a local harmonic function on the domain. Such a map is automatically a harmonic map, satisfying an additional partial conformality condition called horizontal (weak) conformality. In local coordinates, this amounts to a non-linear, over-determined system of partial differential equations. Therefore, the question of the existence of such maps, even in the local case, is very hard to answer in general. However, when the manifolds involved carry Hermitian structures, it is natural to search among the holomorphic maps for candidates for harmonic morphisms. In this thesis, we study in a series of articles the geometry of harmonic morphisms and conformal foliations, a closely related topic, in the context of Hermitian geometry. Several results are proved for holomorphic harmonic morphisms between Kähler manifolds, in particular when the domain is of non-negative sectional curvature or constant holomorphic sectional curvature. Many of these results also extend to holomorphic conformal foliations. A principal tool we use is a formula by Walczak which, in this context, takes a particularly simple form. We make a brief digression to apply this formula to harmonic morphisms of warped product type, which results in a decomposition theorem for such maps. We also study the four-dimensional, conformally flat case in detail. By a construction which originates in the ideas of twistor theory, we produce the first known examples of harmonic morphisms without totally geodesic fibres from real hyperbolic spaces of even dimension greater than four. These ideas are then generalized to obtain harmonic morphisms from several other Riemannian symmetric spaces of type I and III, such as the Grassmannians, where the question of local existence was an open problem. By further exploiting the concept of duality for Riemannian symmetric spaces, we also obtain harmonic morphisms from several Riemannian symmetric spaces of type II and IV.

AB - A harmonic morphism is a map between two Riemannian manifolds with the property that its composition with a local harmonic function on the codomain is a local harmonic function on the domain. Such a map is automatically a harmonic map, satisfying an additional partial conformality condition called horizontal (weak) conformality. In local coordinates, this amounts to a non-linear, over-determined system of partial differential equations. Therefore, the question of the existence of such maps, even in the local case, is very hard to answer in general. However, when the manifolds involved carry Hermitian structures, it is natural to search among the holomorphic maps for candidates for harmonic morphisms. In this thesis, we study in a series of articles the geometry of harmonic morphisms and conformal foliations, a closely related topic, in the context of Hermitian geometry. Several results are proved for holomorphic harmonic morphisms between Kähler manifolds, in particular when the domain is of non-negative sectional curvature or constant holomorphic sectional curvature. Many of these results also extend to holomorphic conformal foliations. A principal tool we use is a formula by Walczak which, in this context, takes a particularly simple form. We make a brief digression to apply this formula to harmonic morphisms of warped product type, which results in a decomposition theorem for such maps. We also study the four-dimensional, conformally flat case in detail. By a construction which originates in the ideas of twistor theory, we produce the first known examples of harmonic morphisms without totally geodesic fibres from real hyperbolic spaces of even dimension greater than four. These ideas are then generalized to obtain harmonic morphisms from several other Riemannian symmetric spaces of type I and III, such as the Grassmannians, where the question of local existence was an open problem. By further exploiting the concept of duality for Riemannian symmetric spaces, we also obtain harmonic morphisms from several Riemannian symmetric spaces of type II and IV.

KW - isotropic subbundles

KW - conformal foliations

KW - holomorphic maps

KW - harmonic morphism

M3 - Doctoral Thesis (compilation)

ER -

Harmonic morphisms, Hermitian structures and symmetric spaces

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