Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-27T02:01:19.974Z Has data issue: false hasContentIssue false
  • English
  • Français

Harmonic morphisms with one-dimensional fibres on conformally-flat Riemannian manifolds

Published online by Cambridge University Press:  01 July 2008

RADU PANTILIE
RADU PANTILIE*
Affiliation:
Institute of Mathematics “Simion Stoilow” of the Romanian Academy, P.O. Box 1-764, RO-014700 Bucharest, Romania. e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We classify the harmonic morphisms with one-dimensional fibres (1) from real-analytic conformally-flat Riemannian manifolds of dimension at least four (Theorem 3.1), and (2) between conformally-flat Riemannian manifolds of dimensions at least three (Corollaries 3.4 and 3.6).

Also, we prove (Proposition 2.5) an integrability result for any real-analytic submersion, from a constant curvature Riemannian manifold of dimension n+2 to a Riemannian manifold of dimension 2, which can be factorised as an n-harmonic morphism with two-dimensional fibres, to a conformally-flat Riemannian manifold, followed by a horizontally conformal submersion, (n≥4).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 145 , Issue 1 , July 2008 , pp. 141 - 151
Copyright
Copyright © Cambridge Philosophical Society 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Baird, P. and Wood, J. C.. Harmonic morphisms between Riemannian manifolds. London Math. Soc. Monogr. (N.S.), no. 29 (Oxford University Press, 2003).CrossRefGoogle Scholar
[2]Bryant, R. L.. Harmonic morphisms with fibres of dimension one. Comm. Anal. Geom. 8 (2000), 219–265.CrossRefGoogle Scholar
[3]Calderbank, D. M. J.. The Faraday 2-form in Einstein–Weyl geometry. Math. Scand. 89 (2001), 97–116.CrossRefGoogle Scholar
[4]Lafontaine, J.. Conformal geometry from the Riemannian viewpoint. Conformal geometry (Bonn, 1985/1986), Aspects Math. E12, (Vieweg, Braunschweig, 1988), 65–92.CrossRefGoogle Scholar
[5]Loubeau, E.. The Fuglede–Ishihara and Baird–Eells theorems for p > 1. Contemp. Math. 288 (2001), 376–380.CrossRefGoogle Scholar
[6]Loubeau, E. and Pantilie, R.. Harmonic morphisms between Weyl spaces and twistorial maps, Comm. Anal. Geom. 14 (2006), 847881.CrossRefGoogle Scholar
[7]Loubeau, E. and Pantilie, R.. Harmonic morphisms between Weyl spaces and twistorial maps II. Preprint, IMAR, Bucharest (2006), (math.DG/0610676).CrossRefGoogle Scholar
[8]Pantilie, R.. Harmonic morphisms with one-dimensional fibres. Internat. J. Math. 10 (1999), 457–501.CrossRefGoogle Scholar
[9]Pantilie, R.. Submersive harmonic maps and morphisms. PhD Thesis (University of Leeds, 2000).Google Scholar
[10]Pantilie, R.. Harmonic morphisms with 1-dimensional fibres on 4-dimensional Einstein manifolds. Comm. Anal. Geom. 10 (2002), 779–814.CrossRefGoogle Scholar
[11]Pantilie, R.. Harmonic morphisms between Weyl spaces. Modern Trends in Geometry and Topology. Proceedings of the Seventh International Workshop on Differential Geometry and Its Applications. (Deva, Romania, 5-11 September, 2005), 321–332.Google Scholar
[12]Pantilie, R. and Wood, J. C.. Harmonic morphisms with one-dimensional fibres on Einstein manifolds. Trans. Amer. Math. Soc. 354 (2002), 4229–4243.CrossRefGoogle Scholar
[13]Pantilie, R. and Wood, J. C.. A new construction of Einstein self-dual manifolds. Asian J. Math. 6 (2002) 337–348.CrossRefGoogle Scholar
[14]Pantilie, R. and Wood, J. C.. Twistorial harmonic morphisms with one-dimensional fibres on self-dual four-manifolds. Quart. J. Math. 57 (2006), 105–132.CrossRefGoogle Scholar
[15]Wood, J. C.. Harmonic morphisms and Hermitian structures on Einstein 4-manifolds. Internat. J. Math. 3 (1992), 415–439.CrossRefGoogle Scholar