Hostname: page-component-848d4c4894-xm8r8 Total loading time: 0 Render date: 2024-07-06T06:56:02.282Z Has data issue: false hasContentIssue false
  • English
  • Français

Adapted complex structures and geometric quantization

Published online by Cambridge University Press:  22 January 2016

Róbert Szőke
Róbert Szőke*
Affiliation:
1088 Budapest, Rákoáczi u. 5, rszoke@cs.elte.hu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A compact Riemannian symmetric space admits a canonical complexification. This so called adapted complex manifold structure JA is defined on the tangent bundle. For compact rank-one symmetric spaces another complex structure JS is defined on the punctured tangent bundle. This latter is used to quantize the geodesic flow for such manifolds. We show that the limit of the push forward of JA under an appropriate family of diffeomorphisms exists and agrees with JS.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 154 , 1999 , pp. 171 - 183
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1999

References

[Ag] Aguilar, R. M., Dual canonical structures on tangent bundles of Riemannian manifolds, Preprint (1997).Google Scholar
[F-T] Furutani, K. and Tanaka, R., A Kähler structure on the punctured cotangent bundle of complex and quaternion projective spaces and its application to geometric quantization I, J. Math. Kyoto Univ., 34-4 (1994), 719–737.Google Scholar
[F-Y] Furutani, K. and Yoshizawa, S., A Kähler structure on the punctured cotangent bundle of complex and quaternion projective spaces and its application to geometric quantization II, Jap. J. Math., 21 (1995), 355–392.Google Scholar
[G-S] Guillemin, V. and Stenzel, M., Grauert tubes and the homogeneous Monge-Ampère equation I, J. Diff. Geom., 34 (1991), 561–570.Google Scholar
[Ii] Ii, K., Kähler structures on tangent bundles of Riemannian manifolds of constant positive curvature, Preprint (1996).Google Scholar
[Ka] Kan, S. J., The asymptotic expansion of a CR invariant and Grauert tubes, Math. Ann., 304 (1996), 63–92.Google Scholar
[L-Sz] Lempert, L. and Szőke, R., Global solutions of the homogeneous complex Monge-Ampère equation and complex structures on the tangent bundle of Riemannian manifolds, Math. Ann., 290 (1991), 689–712.CrossRefGoogle Scholar
[Ra1] Rawnsley, J. H., Coherent states and Kähler manifolds, Quart. J. Math. Oxford, 28 (1977), 403–415.CrossRefGoogle Scholar
[Ra2], A nonunitary pairing of polarizations for the Kepler problem, Trans. Amer. Math. Soc., 250 (1979), 167–180.CrossRefGoogle Scholar
[Sh] Shabat, B. V., An introduction to complex analysis, Nauka, Moscow, 1976, (Russian).Google Scholar
[So] Souriau, J. M., Sur la variété de Kepler, Symposia Math., 14, Academic Press, London (1974), pp. 343–360.Google Scholar
[Sz] Szőoke, R., Complex structures on the tangent bundle of Riemannian manifolds, Math. Ann., 291 (1991), 409–428.Google Scholar
[Wo] Woodhouse, N. M. J., Geometric quantization, Oxford University Press, Oxford, 1992.Google Scholar