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Keywords:
Einstein manifold; conformal foliation; semi-conformal map; biconformal deformation
Einstein manifold; conformal foliation; semi-conformal map; biconformal deformation
Summary:
Biconformal deformations take place in the presence of a conformal foliation, deforming by different factors tangent to and orthogonal to the foliation. Four-manifolds endowed with a conformal foliation by surfaces present a natural context to put into effect this process. We develop the tools to calculate the transformation of the Ricci curvature under such deformations and apply our method to construct Einstein $4$-manifolds. Examples of one particular family have ends which collapse asymptotically to $\mathbb{R}^2$.
References:
[1] Baird, P., Wood, J.C.: Harmonic morphisms between Riemannian manifolds. London Math. Soc. Monographs, New Series, vol. 29, Oxford Univ. Press, 2003. MR 2044031
[2] Besse, A.: Einstein Manifolds. Springer-Verlag, 1987. Zbl 0613.53001
[3] Danielo, L.: Structures Conformes, Harmonicité et Métriques d’Einstein. Ph.D. thesis, Université de Bretagne Occidentale, 2004.
[4] Danielo, L.: Construction de métriques d’Einstein à partir de transformations biconformes. Ann. Fac. Sci. Toulouse (6) 15 (3) (2006), 553–588. DOI 10.5802/afst.1129 | MR 2246414
[5] Dieudonné, J.: Foundations of Modern Analysis. Academic Press, 1969.
[6] Hebey, E.: Scalar curvature type problems in Riemannian geometry. Notes of a course given at the University of Rome 3. http://www.u-cergy.fr/rech/pages/hebey/
[7] Hebey, E.: Introduction à l’analyse non linéaire sur les variétés. Diderot, Paris, 1997.
[8] Hilbert, D.: Die Grundlagen der Physik. Nachr. Ges. Wiss. Göttingen (1915), 395–407.
[9] Hitchin, N.J.: On compact four-dimensional Einstein manifolds. J. Differential Geom. 9 (1974), 435–442. DOI 10.4310/jdg/1214432419
[10] Schoen, R.: Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differential Geom. 20 (1984), 479–495. DOI 10.4310/jdg/1214439291 | Zbl 0576.53028
[11] Spivak, M.: A Comprehensive Introduction to Riemannian Geometry. 2nd ed., Publish or Perish, Wilmongton DE, 1979.
[12] Thorpe, J.A.: Some remarks on the Gauss-Bonnet formula. J. Math. Mech. 18 (1969), 779–786.
[13] Vaisman, I.: Conformal foliations. Kodai Math. J. 2 (1979), 26–37. DOI 10.2996/kmj/1138035963
[14] Yamabe, H.: On a deformation of Riemannian structures on compact manifolds. Osaka Math. J. 12 (1960), 21–37. Zbl 0096.37201
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