# zbMATH — the first resource for mathematics

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
Full Text:
##### 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.