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Ricci-flat metrics on the complexification of a compact rank one symmetric space. (English) Zbl 0811.53049
The author shows that the tangent bundle and cotangent bundle of a compact rank one symmetric space admits a complete Ricci-flat Kähler metric. These bundles have a natural complex structure, and the term Kähler refers to this structure.
Reviewer: W.Ballmann (Bonn)

MSC:
53C35 Differential geometry of symmetric spaces
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References:
[1] Abraham, R. and Marsden, J.: Foundations of Mechanics, 2nd ed. Benjamin/Cummings (1978) · Zbl 0393.70001
[2] Bando, S. and Kobayashi, R.: Ricci-flat Kähler metrics on affine algebraic manifolds.II. Math. Ann.287, 175–180 (1990) · Zbl 0701.53083 · doi:10.1007/BF01446884
[3] Eguchi, T. and Hanson, A. J.: Asymptotically flat self-dual solutions to Euclidean gravity. Phys. Lett.74B (3), 249–251 (1978)
[4] Gibbons, G. W. and Pope, C. N.: The positive action conjecture and asymptotically Euclidean metrics in quantum gravity. Commun. Math. Phys.66, 267–290 (1979) · doi:10.1007/BF01197188
[5] Guillemin, V. and Stenzel, M.: Grauert tubes and the homogeneous Monge-Ampère equation. J. Differential Geometry34, 561–570 (1992) · Zbl 0746.32005
[6] Guillemin, V and Stenzel, M.: Grauert tubes and the homogeneous Monge-Ampère equation. II. J. Differential Geometry35, 627–641 (1992) · Zbl 0789.32010
[7] Guillemin, V. and Sternberg, S.: Geometric quantization and multiplicities of group representations. Invent. Math.67, 515–538 (1982) · Zbl 0503.58018 · doi:10.1007/BF01398934
[8] Guillemin, V. and Sternberg, S.: Convexity and the moment map.II. Invent. Math.77, 533–546 (1984) · Zbl 0561.58015 · doi:10.1007/BF01388837
[9] Helgason, S.: Differential geometry, Lie groups, and symmetric spaces, 2nd ed. Academic Press (1978) · Zbl 0451.53038
[10] Kobayashi, S. and Nomizu, K.: Foundations of Differential Geometry, Volume II. John Wiley and Sons (1969) · Zbl 0175.48504
[11] Kobayashi, R.: Ricci-Flat Kähler metrics on affine algebraic manifolds and degenerations of Kähler-Einstein K3 surfaces. Advanced Studies in Pure Mathematics18–II, 137–228. San Diego: Academic Press (1990) · Zbl 0754.53051
[12] Lempert, L. and Szöke, R.: Global solutions of the homogeneous complex Monge-Ampère equation and complex structures on the tangent bundle of Riemannian manifolds. Math. Ann.290, 689–712 (1991) · Zbl 0752.32008 · doi:10.1007/BF01459268
[13] LeBrun, C.: Complete Ricci-flat metrics on \(\mathbb{C}\) n need not be flat. Proceedings of Symposia in Pure Mathematics52 (2), 297–304 (1991) · Zbl 0739.53053
[14] Matsushima, Y.: Espaces homogènes de Stein des groupes de Lie complexes. Nagoya Math. J.16, 205–218 (1960) · Zbl 0094.28201
[15] Morimoto, A. and Nagano, T.: On pseudo-conformal transformations of hypersurfaces. J. Math. Soc. Japan15 (3), 289–300 (1963) · Zbl 0119.06701 · doi:10.2969/jmsj/01530289
[16] Patrizio, G. and Wong, P. M.: Stein manifolds with compact symmetric centers. Math. Ann.289 (3), 355–382 (1991) · Zbl 0729.32003 · doi:10.1007/BF01446577
[17] Stenzel, M.: An adapted complex structure on the cotangent bundle of a compact Riemannian homogeneous space. To appear in the Proceedings of the AMS.
[18] Stenzel, M.: Kähler structures on cotangent bundles of real analytic Riemannian manifolds. Ph.D. thesis, MIT (1990)
[19] Szöke, R.: Complex structures on tangent bundles of Riemannian manifolds Math. Ann.291, 409-428 (1991) · Zbl 0749.53021
[20] Tian, G. and Yau, S. T.: Complete Kähler manifolds with zero Ricci curvature. I. J. Am. Math. Soc.3 (3), 579–609 (1990) · Zbl 0719.53041
[21] Tian, G. and Yau, S. T.: Complete Kähler manifolds with zero Ricci curvature.II. Invent. Math.100 (1991) · Zbl 0766.53053
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