Harmonic morphisms and circle actions on 3- and 4-manifolds

Baird, Paul

doi:10.5802/aif.1210

Cited by 20 publications

(56 citation statements)

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“…Also, because M Ricci(X,X) = M Ricci(Y, Y) from (1.1) and (1.5), it follows that X(a) 2 -Y(a) 2 is a basic function. Hence X(cr),y(cr) are basic functions and, moreover, outside the set…”

Section: X(a)y(a) Y(a)z(a) Z(a)x(a)mentioning

confidence: 99%

“…This is also illustrated by the following example. For a E R let g a be the Riemannian metric on M 4 \{0} = (0, oo) x S 3 defined by g a = p 2 h + p-2 {pdp + aA) 2 .…”

Section: Proposition 21 Let Tp : (M 4 ^) -» (N 3 H) Be a Harmonicmentioning

confidence: 99%

“…Then ^a is a harmonic morphism with totally geodesic fibres. Any fibre of it is isometric with (R 2 \{0},7 a ) where 7 0 , in polar coordinates {p,6), is given by la = p 2 d0 2 + p-2 (pdp + ad0) 2 .…”

Section: Proposition 21 Let Tp : (M 4 ^) -» (N 3 H) Be a Harmonicmentioning

confidence: 99%

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Harmonic morphisms with 1-dimensional fibres on 4-dimensional Einstein manifolds

Pantilie¹

2002

Communications in Analysis and Geometry

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“…Also, because M Ricci(X,X) = M Ricci(Y, Y) from (1.1) and (1.5), it follows that X(a) 2 -Y(a) 2 is a basic function. Hence X(cr),y(cr) are basic functions and, moreover, outside the set…”

Section: X(a)y(a) Y(a)z(a) Z(a)x(a)mentioning

confidence: 99%

“…This is also illustrated by the following example. For a E R let g a be the Riemannian metric on M 4 \{0} = (0, oo) x S 3 defined by g a = p 2 h + p-2 {pdp + aA) 2 .…”

Section: Proposition 21 Let Tp : (M 4 ^) -» (N 3 H) Be a Harmonicmentioning

confidence: 99%

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Harmonic morphisms with 1-dimensional fibres on 4-dimensional Einstein manifolds

Pantilie¹

2002

Communications in Analysis and Geometry

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“…For this we need the following, which improves one of the statements of [28 Proof. By a result of P. Baird [1], a harmonic morphism (p : (M, g) -> (JV, h) with one-dimensional fibres is submersive if dimM > 4, whilst, if dimM = 4, the set of critical points of cp is discrete. Now Theorem 1.5 is equivalent to a classification result for harmonic morphisms with one-dimensional fibres on Einstein four-manifolds [27, Corollary 1.9], [26, Corollary 3.4.5].…”

mentioning

confidence: 99%

“…Because the critical points of (p are isolated, from [10] it follows that, in the neighbourhood of a critical point, ip is topologically equivalent to the cone over the Hopf fibration S' 3 -> 5 2 . Therefore there exists a connected component of a fibre of ip which is diffeomorphic to 5 1 , and hence at some point we must have V(X~2) = 0. Thus, if (iii) of Theorem 1.5 holds, then ip is submersive.…”

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confidence: 99%

A new construction of Einstein self-dual metrics

Pantilie¹,

Wood²

2002

Asian Journal of Mathematics

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Abstract. We give a new construction of Ricci-flat self-dual metrics which is a natural extension of the Gibbons-Hawking ansatz. We also give characterisations of both these constructions, and explain how they come from harmonic morphisms.Introduction. In [15,18], G.W. Gibbons and S.W. Hawking introduced a construction of Einstein self-dual metrics with zero scalar curvature (see [24] for a thorough discussion of this ansatz). The construction is in the spirit of Kaluza-Klein theory with the projection of the (local) bundle being a Riemannian submersion followed by a conformal transformation.A harmonic morphism is a map between Riemannian manifolds which preserves Laplace's equation (see Section 2 below). In [7], R.L. Bryant gave a local normal form for the metric on the domain of a submersive harmonic morphism with one-dimensional fibres. When the domain is four-dimensional, this local normal form includes that of the Gibbons-Hawking ansatz.In [26,27], it is shown that, from an Einstein four-manifold, there are precisely three types of harmonic morphism with one-dimensional fibres. The first two types are due to R.L. Bryant and to P. Baird and J. Eells, respectively, and lead to the Gibbons-Hawking construction and the well-known warped product construction of Einstein metrics (see Theorem 1.1 and Theorem 1.2 below). The third type can be seen as a construction of Ricci-flat self-dual metrics. We present this new construction in Section 1 (Theorem 1.3) together with the above mentioned result from [26,27] reformulated as a classification result for Einstein four-manifolds whose metric can be written in Bryant's local normal form (Theorem 1.5). In Section 2 we review some facts on harmonic morphisms. In Section 3 we give proofs of the results of Section 1 and show that our construction is a natural extension of the Gibbons-Hawking ansatz; indeed all three constructions are characterised by an equation (3.1) which generalizes the monopole equation. We also classify the harmonic morphisms with one-dimensional fibres on compact Einstein four-manifolds. Our new construction involves solving equation (1.5) below, which is a particular case of the Beltrami fields equation of hydrodynamics (see [20]). In Section 4 we describe all solutions of (1.5), both locally and globally, on 5 3 by a method similar to the one used in [20] to describe solutions of the Beltrami fields equation on R 3 (see Remark 4.2), giving all Einstein metrics of the form (1.4) below; in fact we show that for such metrics the Einstein condition is equivalent to the Beltrami fields equation.

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