*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.

*Clifford Henry Taubes*

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780199605880
- eISBN:
- 9780191774911
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199605880.003.0011
- Subject:
- Mathematics, Geometry / Topology, Mathematical Physics

This chapter examines the related notions of covariant derivative and connection. It covers the space of covariant derivatives. It also gives a relatively straightforward construction of a covariant ...
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This chapter examines the related notions of covariant derivative and connection. It covers the space of covariant derivatives. It also gives a relatively straightforward construction of a covariant derivative on a given vector bundle E → M with fiber 𝕍n = ℝnRn or ℂn. It looks at principal bundles and connections; connections and covariant derivatives; and horizontal lifts. The chapter gives an application to the classification of principal G-bundles up to isomorphism and explains connections, covariant derivatives, and pull-back bundles.Less

This chapter examines the related notions of covariant derivative and connection. It covers the space of covariant derivatives. It also gives a relatively straightforward construction of a covariant derivative on a given vector bundle E → M with fiber 𝕍^{n} = ℝ^{n}Rn or ℂ^{n}. It looks at principal bundles and connections; connections and covariant derivatives; and horizontal lifts. The chapter gives an application to the classification of principal G-bundles up to isomorphism and explains connections, covariant derivatives, and pull-back bundles.

*Clifford Henry Taubes*

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780199605880
- eISBN:
- 9780191774911
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199605880.003.0013
- Subject:
- Mathematics, Geometry / Topology, Mathematical Physics

This chapter examines flat connections. A connection on a principal bundle is said to be flat when its curvature 2-form is identically zero. The discussions cover flat connections on bundles over the ...
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This chapter examines flat connections. A connection on a principal bundle is said to be flat when its curvature 2-form is identically zero. The discussions cover flat connections on bundles over the circle; foliations; automorphisms of a principal bundle; the fundamental group of a manifold; the flat connections on bundles over M; the universal covering space; holonomy and curvature; and proof of the classification theorem for flat connections.Less

This chapter examines flat connections. A connection on a principal bundle is said to be flat when its curvature 2-form is identically zero. The discussions cover flat connections on bundles over the circle; foliations; automorphisms of a principal bundle; the fundamental group of a manifold; the flat connections on bundles over M; the universal covering space; holonomy and curvature; and proof of the classification theorem for flat connections.

*Ercüment H. Ortaçgil*

- Published in print:
- 2018
- Published Online:
- September 2018
- ISBN:
- 9780198821656
- eISBN:
- 9780191860959
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198821656.003.0017
- Subject:
- Mathematics, Pure Mathematics

Up to now, the discussion has been mainly concerned with Lie groups and their curved analogs, namely, parallelizable manifolds and their curvatures. The problem is to generalize this construction to ...
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Up to now, the discussion has been mainly concerned with Lie groups and their curved analogs, namely, parallelizable manifolds and their curvatures. The problem is to generalize this construction to arbitrary geometric structures. The first step is to study the flat case, and this is the subject of this chapter.Less

Up to now, the discussion has been mainly concerned with Lie groups and their curved analogs, namely, parallelizable manifolds and their curvatures. The problem is to generalize this construction to arbitrary geometric structures. The first step is to study the flat case, and this is the subject of this chapter.