*Paul Baird and John C. Wood*

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198503620
- eISBN:
- 9780191708435
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198503620.003.0012
- Subject:
- Mathematics, Pure Mathematics

This chapter shows that a harmonic morphism from a manifold of dimension n+1 to a manifold of dimension n is, locally or globally, a principal bundle with a certain metric. When n = 3, in a ...
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This chapter shows that a harmonic morphism from a manifold of dimension n+1 to a manifold of dimension n is, locally or globally, a principal bundle with a certain metric. When n = 3, in a neighbourhood of a critical point, a harmonic morphism behaves like the Hopf polynomial map; when n > 3, there can be no critical points. A factorization theorem and a circle action are obtained in all cases, leading to topological restrictions. Given a nowhere-zero Killing field V, it is shown how to find harmonic morphisms with fibres tangent to V. Harmonic morphisms of warped product type are discussed; these are related to isoparametric functions. These two types are the only types that can occur on a space form or on an Einstein manifold when n > 3. When n = 3, a third type of harmonic morphism is found related to the Beltrami fields equation of hydrodynamics.Less

This chapter shows that a harmonic morphism from a manifold of dimension *n+1* to a manifold of dimension *n* is, locally or globally, a principal bundle with a certain metric. When *n* = 3, in a neighbourhood of a critical point, a harmonic morphism behaves like the Hopf polynomial map; when *n* > 3, there can be no critical points. A factorization theorem and a circle action are obtained in all cases, leading to topological restrictions. Given a nowhere-zero Killing field *V*, it is shown how to find harmonic morphisms with fibres tangent to *V*. Harmonic morphisms of warped product type are discussed; these are related to isoparametric functions. These two types are the only types that can occur on a space form or on an Einstein manifold when *n* > 3. When *n* = 3, a third type of harmonic morphism is found related to the Beltrami fields equation of hydrodynamics.

*Paul Baird and John C. Wood*

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198503620
- eISBN:
- 9780191708435
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198503620.003.0010
- Subject:
- Mathematics, Pure Mathematics

This chapter shows that a non-constant harmonic morphism with compact fibres from a 3-manifold endows it with the structure of a Seifert fibre space — a 3-manifold with a certain type of ...
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This chapter shows that a non-constant harmonic morphism with compact fibres from a 3-manifold endows it with the structure of a Seifert fibre space — a 3-manifold with a certain type of one-dimensional foliation with leaf space an orbifold. Conversely, any Seifert fibre space can be obtained from a harmonic morphism by smoothing the orbifold leaf space. A global version of the factorization theorem is provided, then the metrics on a 3-manifold which support a harmonic morphism to a surface are described locally and globally. It is shown how fundamental invariants of a one-dimensional foliation propagate along its leaves. The necessary curvature conditions where a Riemannian 3-manifold supports a non-constant harmonic morphism to a surface are given. These show that there are never more than two such harmonic morphisms up to equivalence, even locally, unless it is a space form.Less

This chapter shows that a non-constant harmonic morphism with compact fibres from a 3-manifold endows it with the structure of a Seifert fibre space — a 3-manifold with a certain type of one-dimensional foliation with leaf space an orbifold. Conversely, any Seifert fibre space can be obtained from a harmonic morphism by smoothing the orbifold leaf space. A global version of the factorization theorem is provided, then the metrics on a 3-manifold which support a harmonic morphism to a surface are described locally and globally. It is shown how fundamental invariants of a one-dimensional foliation propagate along its leaves. The necessary curvature conditions where a Riemannian 3-manifold supports a non-constant harmonic morphism to a surface are given. These show that there are never more than two such harmonic morphisms up to equivalence, even locally, unless it is a space form.

*Alfred Galichon*

- Published in print:
- 2016
- Published Online:
- January 2018
- ISBN:
- 9780691172767
- eISBN:
- 9781400883592
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691172767.003.0006
- Subject:
- Economics and Finance, Microeconomics

This chapter considers the case when the attributes are d-dimensional vectors and the surplus is the scalar product; it assumes that the distribution of the workers' attributes is continuous, but it ...
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This chapter considers the case when the attributes are d-dimensional vectors and the surplus is the scalar product; it assumes that the distribution of the workers' attributes is continuous, but it relaxes the assumption that the distribution of the firms' attributes is discrete. This setting allows us to entirely rediscover convex analysis, which is introduced from the point of view of optimal transport. As a consequence, Brenier's polar factorization theorem is given, which provides a vector extension for the scalar notions of quantile and rearrangement.Less

This chapter considers the case when the attributes are *d*-dimensional vectors and the surplus is the scalar product; it assumes that the distribution of the workers' attributes is continuous, but it relaxes the assumption that the distribution of the firms' attributes is discrete. This setting allows us to entirely rediscover convex analysis, which is introduced from the point of view of optimal transport. As a consequence, Brenier's polar factorization theorem is given, which provides a vector extension for the scalar notions of quantile and rearrangement.

*John Campbell, Joey Huston, and Frank Krauss*

- Published in print:
- 2017
- Published Online:
- March 2018
- ISBN:
- 9780199652747
- eISBN:
- 9780191749148
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780199652747.003.0002
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

The hard scattering formalism is introduced, starting from a physical picture based on the idea of equivalent quanta borrowed from QED, and the notion of characteristic times. Contact to the standard ...
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The hard scattering formalism is introduced, starting from a physical picture based on the idea of equivalent quanta borrowed from QED, and the notion of characteristic times. Contact to the standard QCD treatment is made after discussing the running coupling and the Altarelli–Parisi equations for the evolution of parton distribution functions, both for QED and QCD. This allows a development of a space-time picture for hard interactions in hadron collisions, integrating hard production cross sections, initial and final state radiation, hadronization, and multiple parton scattering. The production of a W boson at leading and next-to leading order in QCD is used to exemplify characteristic features of fixed-order perturbation theory, and the results are used for some first phenomenological considerations. After that, the analytic resummation of the W boson transverse momentum is introduced, giving rise to the notion of a Sudakov form factor. The probabilistic interpretation of the Sudakov form factor is used to discuss patterns in jet production in electron-positron annihilation.Less

The hard scattering formalism is introduced, starting from a physical picture based on the idea of equivalent quanta borrowed from QED, and the notion of characteristic times. Contact to the standard QCD treatment is made after discussing the running coupling and the Altarelli–Parisi equations for the evolution of parton distribution functions, both for QED and QCD. This allows a development of a space-time picture for hard interactions in hadron collisions, integrating hard production cross sections, initial and final state radiation, hadronization, and multiple parton scattering. The production of a W boson at leading and next-to leading order in QCD is used to exemplify characteristic features of fixed-order perturbation theory, and the results are used for some first phenomenological considerations. After that, the analytic resummation of the W boson transverse momentum is introduced, giving rise to the notion of a Sudakov form factor. The probabilistic interpretation of the Sudakov form factor is used to discuss patterns in jet production in electron-positron annihilation.