Skip to main content
SpringerLink
Log in

Adapted complex structures and the geodesic flow

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

In this paper, we give a new construction of the adapted complex structure on a neighborhood of the zero section in the tangent bundle of a compact, real-analytic Riemannian manifold. Motivated by the “complexifier” approach of T. Thiemann as well as certain formulas of V. Guillemin and M. Stenzel, we obtain the polarization associated to the adapted complex structure by applying the “imaginary-time geodesic flow” to the vertical polarization. Meanwhile, at the level of functions, we show that every holomorphic function is obtained from a function that is constant along the fibers by “composition with the imaginary-time geodesic flow.” We give several equivalent interpretations of this composition, including a convergent power series in the vector field generating the geodesic flow.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Axelrod S., Della Pietra S., Witten E.: Geometric quantization of Chern–Simons gauge theory. J. Diff. Geom. 33, 787–902 (1991)

    MathSciNet  MATH  Google Scholar 

  2. Aguilar R.M.: Symplectic reduction and the homogeneous complex Monge-Ampère equation. Ann. Global Anal. Geom. 19(4), 327–353 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Burns D., Hind R.: Symplectic geometry and the uniqueness of Grauert tubes. Geom. Funct. Anal. 11(1), 1–10 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Florentino C., Matias P., Mourão J., Nunes J.P.: Geometric quantization, complex structures and the coherent state transform. J. Func. Anal. 221, 303–322 (2005)

    Article  MATH  Google Scholar 

  5. Florentino C., Matias P., Mourão J., Nunes J.P.: On the BKS pairing for Kähler quantizations for the cotangent bundle of a Lie group. J. Func. Anal. 234, 180–198 (2006)

    Article  MATH  Google Scholar 

  6. Grauert H.: On Levi’s problem and the imbedding of real-analytic manifolds. Ann. Math. (2) 68, 460–472 (1958)

    Article  MathSciNet  Google Scholar 

  7. Guillemin V., Stenzel M.: Grauert tubes and the homogeneous Monge-Ampère equation. J. Differential Geom. 34(2), 561–570 (1991)

    MathSciNet  MATH  Google Scholar 

  8. Guillemin V., Stenzel M.: Grauert tubes and the homogeneous Monge-Ampère equation. II. J. Differential Geom. 35(3), 627–641 (1992)

    MathSciNet  MATH  Google Scholar 

  9. Hall B.C.: The Segal–Bargmann “coherent state” transform for compact Lie groups. J. Func. Anal. 122, 103–151 (1994)

    Article  MATH  Google Scholar 

  10. Hall B.C.: The inverse Segal-Bargmann transform for compact Lie groups. J. Funct. Anal. 143(1), 98–116 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hall B.C.: Geometric quantization and the generalized Segal–Bargmann transform for Lie groups of compact type. Commun. Math. Phys. 226, 233–268 (2002)

    Article  MATH  Google Scholar 

  12. Halverscheid S.: Complexifications of geodesic flows and adapted complex structures. Rep. Math. Phys. 50(3), 329–338 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hall B.C., Mitchell J.J.: Coherent states on spheres. J. Math. Phys. 43(3), 1211–1236 (2002)

    Article  MathSciNet  Google Scholar 

  14. Huebschmann J.: Kähler quantization and reduction. J. Reine Angew. Math. 591, 75–109 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jost, J.: Riemannian Geometry and Geometric Analysis, Universitext, Springer-Verlag, Berlin (1995)

  16. Kohno, M.: Global Analysis in Linear Differential Equations, Mathematics and its Applications, vol. 471, Kluwer Academic Publishers, Dordrecht (1999)

  17. Lempert L., Szőke R.: Global solutions of the homogeneous complex Monge-Ampère equation and complex structures on the tangent bundle of Riemannian manifolds. Math. Ann. 290(4), 689–712 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  18. Szőke R.: Complex structures on tangent bundles of Riemannian manifolds. Math. Ann. 291(3), 409–428 (1991)

    Article  MathSciNet  Google Scholar 

  19. Szőke R.: Automorphisms of certain Stein manifolds. Math. Z. 219(3), 357–385 (1995)

    Article  MathSciNet  Google Scholar 

  20. Szőke R.: Involutive structures on the tangent bundle of symmetric spaces. Math. Ann. 319(2), 319–348 (2001)

    Article  MathSciNet  Google Scholar 

  21. Thiemann T.: Reality conditions inducing transforms for quantum gauge field theory and quantum gravity. Classical Quant Gravity 13, 1383–1403 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  22. Thiemann T.: Gauge field theory coherent states (GCS). I. General properties. Classical Quant Gravity 18(11), 2025–2064 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  23. Thiemann T.: Complexifier coherent states for quantum general relativity. Classical Quant Gravity 23(6), 2063–2117 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Totaro B.: Complexifications of nonnegatively curved manifolds. J. Eur. Math. Soc. (JEMS) 5(1), 69–94 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  25. Whitney H., Bruhat F.: Quelques propriétés fondamentales des ensembles analytiques-réels. Comment. Math. Helv. 33, 132–160 (1959)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Notre Dame, 255 Hurley Building, Notre Dame, IN, 46556, USA

    Brian C. Hall

  2. Center for Mathematical Analysis, Geometry, and Dynamical Systems, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001, Lisbon, Portugal

    William D. Kirwin

Authors
  1. Brian C. Hall
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. William D. Kirwin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to William D. Kirwin.

Additional information

Brian C. Hall was supported in part by NSF Grant DMS-0555862. William D. Kirwin would like to thank the University of Hong Kong and the Max Planck Institute for Mathematics in the Sciences for their hospitality during the preparation of this paper.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hall, B.C., Kirwin, W.D. Adapted complex structures and the geodesic flow. Math. Ann. 350, 455–474 (2011). https://doi.org/10.1007/s00208-010-0564-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-010-0564-9

Mathematics Subject Classification (2000)

Search

Navigation