Non-compact cohomogeneity one Einstein manifolds

Böhm, Christoph

Bulletin de la Société Mathématique de France, Tome 127 (1999) no. 1, pp. 135-177.

[1] Atiyah (M.), Hitchin (N.). - The geometry and dynamics of magnetic monopoles. - Princeton University Press, 1988. | MR | Zbl

[2] Alekseevsky (D.V.). - Classification of quaternionic spaces with a transitive solvable group of motions, Izv. Akad. Nauk SSSR Ser. Mat., t. 39, 1975; English transl. Math. USSR-Izv., t. 9, 1975, p. 297-339, p. 315-362. | Zbl

[3] Alekseevsky (D.V.), Kimel'Fel'D (B.N.). - Structure of homogeneous riemannian spaces with zero Ricci curvature, Funct. Analysis Appl., t. 9, 1975, p. 97-102. | MR | Zbl

[4] Bérard-Bergery (L.). - Sur de nouvelles variétés riemannienes d'Einstein, Inst. E. Cartan, t. 4, 1982, p. 1-60. | MR | Zbl

[5] Besse (A.L.). - Einstein Manifolds. - Springer-Verlag, Berlin Heidelberg, 1987. | MR | Zbl

[6] Böhm (C.). - Inhomogeneous Einstein metrics on low-dimensional spheres and other low-dimensional spaces, Invent. Math., t. 134, 1998, p. 145-176. | MR | Zbl

[7] Böhm (C.). - Non-existence of cohomogeneity one Einstein metrics, to appear in Math. Ann. | Zbl

[8] Bourguignon (J.-P.), Karcher (H.). - Curvature operators: Pinching estimates and geometric examples, Ann. Sci. École Norm. Sup., t. 11, 1978, p. 71-92. | Numdam | MR | Zbl

[9] Bredon (G.E.). - Introduction to compact transformation groups. - Acad. Press, New York, London, 1972. | MR | Zbl

[10] Bryant (R.), Salamon (S.). - On the construction of some complete metrics with exceptional holonomy, Duke Math. J., t. 58, 1989, p. 829-850. | MR | Zbl

[11] Calabi (E.). - A construction of nonhomogeneous Einstein metrics, Differential Geometry, Proc. Symp. Pure Math., t. 27, 1975, p. 17-24. | MR | Zbl

[12] Calabi (E.). - Métriques kählériennes et fibrés holomorphes, Ann. Sci. École Norm. Sup., 4e série, t. 12, 1979, p. 269-294. | Numdam | MR | Zbl

[13] Cartan (E.). - Sur une classe remarquable d'espaces de Riemann, Bull. Soc. Math. France, t. 54, 1926; t. 55, 1927, p. 114-134, p. 214-264. | JFM | Numdam

[14] Cartan (E.). - Sur la structure des groupes de transformations finis et continus. - Thèse, Paris, 1894. | JFM

[15] Cheeger (J.), Colding (T.H.). - Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. of Math., t. 144, 1996, p. 189-237. | MR | Zbl

[16] Cheeger (J.), Tian (G.). - On the cone structure at infinity of Ricci flat manifolds with Euclidean volume growth and quadratic curvature decay, Invent. Math., t. 118, 1994, p. 493-571. | MR | Zbl

[17] Dancer (A.), Wang (M.Y.). - Kähler-Einstein manifolds of cohomogeneity one, to appear in Math. Ann. | Zbl

[18] Eguchi (T.), Hanson (A.). - Asymptotically flat self-dual solutions to Euclidean gravity, Phys. Lett. B, t. 74, 1978, p. 249-251.

[19] Eschenburg (J.-H.). - Lecture Notes on Symmetric spaces. - Preprint, 1997.

[20] Eschenburg (J.-H.), Wang (M.Y.). - The Initial Value Problem for Cohomogeneity One Einstein Metrics, to appear in J. Geom. Analysis. | Zbl

[21] Ferus (D.), Karcher (H.). - Non-rotational minimal spheres and minimizing cones, Comm. Math. Helv., t. 60, 1985, p. 247-269. | MR | Zbl

[22] Gallot (S.), Hulin (D.), Lafontaine (J.). - Riemannian Geometry. - Springer Verlag, New York, Berlin, Heidelberg, 1990. | MR | Zbl

[23] Gibbons (G.), Page (D.), Pope (C.N.). - Einstein metrics on S3, R3 and R4 bundles, Comm. Math. Phys., t. 127, 1990, p. 529-553. | MR | Zbl

[24] Hawking (S.W.). - Gravitational Instantons, Phys. Lett. A, t. 60, 1977, p. 81-83. | MR

[25] Heber (J.). - Non-compact homogeneous Einstein spaces, Invent. Math., t. 133, 1998, p. 279-352. | MR | Zbl

[26] Iwasawa (K.). - On some types of topological spaces, Ann. of Math., t. 50, 1949, p. 507-558. | MR | Zbl

[27] Jensen (G.R.). - Einstein metrics on principal fibre bundles, J. Differential Geometry, t. 8, 1973, p. 599-614. | MR | Zbl

[28] Malgrange (B.). - Sur les points singuliers des équations différentielles, L'Enseignement Math., t. 20, 1974, p. 147-176. | MR | Zbl

[29] Milnor (J.). - Morse Theory. - Annals of Mathematics Studies, Princeton University Press, 1963. | MR | Zbl

[30] Milnor (J.). - Curvatures of Left Invariant Metrics on Lie Groups, Advances in Math., t. 21, 1976, p. 293-329. | MR | Zbl

[31] Montgomery (D.), Samelson (H.). - Groups transitive on the n-dimensional torus, Bull. Amer. Math. Soc., t. 49, 1943, p. 455-456. | MR | Zbl

[32] Page (D.), Pope (C.N.). - Einstein metrics on quaternionic line bundles, Class. Quantum Grav., t. 3, 1986, p. 249-259. | MR | Zbl

[33] Page (D.), Pope (C.N.). - Inhomogeneous Einstein metrics on complex line bundles, Class. Quantum Grav., t. 4, 1987, p. 213-225. | MR | Zbl

[34] Perko (L.). - Differential equations and Dynamical Systems, TAM 7. - Springer-Verlag, New York, Berlin, Heidelberg.

[35] Stenzel (M.). - Ricci-flat metrics on the complexification of a compact rank one symmetric space, Manu. Math., t. 80, 1993, p. 151-163. | MR | Zbl

[36] Wang (J.). - Einstein metrics on bundles. - Ph. D. thesis, McMaster Univ., 1996.

[37] Wang (J.), Wang (M.Y.). - Einstein metrics on S2-bundles, Math. Ann., t. 310, 1998, p. 497-526. | MR | Zbl

[38] Wang (M.Y.), Ziller (W.). - Existence and non-existence of homogeneous Einstein metrics, Invent. Math., t. 84, 1986, p. 177-194. | MR | Zbl

[39] Wang (M.Y.), Ziller (W.). - On isotropy irreducible Riemannian manifolds, Acta Math., t. 166, 1991, p. 223-261. | MR | Zbl

[40] Wolf (J.A.). - Spaces of constant curvature. - McGraw-Hill Series in Higher Math., 1967. | MR | Zbl

[41] Wolf (J.A.). - The structure of isotropy irreducible homogeneous spaces, Acta Math., t. 120, 1968 p. 59-148; correction, Acta Math., t. 152, 1984, p. 141-142. | MR | Zbl

[42] Ziller (W.). - Homogeneous Einstein Metrics on Spheres and Projective Spaces, Math. Ann., t. 259, 1982, p. 351-358. | MR | Zbl