Skip to main content
SpringerLink
Log in

Chapter 7 Symmetric Spaces and Kähler Manifolds

  • Chapter
  • First Online:
Riemannian Geometry and Geometric Analysis

Part of the book series: Universitext ((UTX))

  • 7055 Accesses

Abstract

This chapter treats Kähler manifolds and symmetric spaces as important examples of Riemannian manifolds in detail.

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

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 99.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    This sequence was derived in the previous editions of this textbook, but for the present edition, we are not including an introduction to cohomology theory anymore as that can be readily found in standard textbooks on algebraic topology.

  2. 2.

    One may easily modify the proof at this place so as to avoid using the completeness of M. 

  3. 3.

    In the bibliography, a superscript will indicate the edition of a monograph. For instance,72017 means 7th edition, 2017.

Bibliography

  1. S. I. Amari and H. Nagaoka. Methods of information geometry. AMS and Oxford Univ. Press, 2000.

    MATH  Google Scholar 

  2. T. Aubin. Nonlinear analysis on manifolds. Monge–Ampère equations. Springer-Verlag, Berlin, 1982.

    Google Scholar 

  3. N. Ay, J. Jost, H.V.Lê, and L. Schwachhöfer. Information geometry. Springer, Berlin, 2017.

    MATH  Google Scholar 

  4. W. Ballmann, M. Gromov, and V. Schroeder. Manifolds of nonpositive curvature. Birkhäuser, 1985.

    Book  MATH  Google Scholar 

  5. J.P. Bourguignon. Eugenio Calabi and Kähler metrics, pages 61–85. In: P. de Bartolomeis (ed.), Manifolds and geometry, Symp. Math. 36, Cambridge Univ. Press, 1996.

    Google Scholar 

  6. E. Calabi and E. Vesentini. On compact, locally symmetric Kähler manifolds. Ann. Math, 71:472–507, 1960.

    Article  MathSciNet  MATH  Google Scholar 

  7. S.Y. Cheng and S.T. Yau. The real Monge-Ampère equation and affine flat structures. In W.T. Wu S.S. Chern, editor, Differential geometry and differential equations, Proc.Beijing Symp.1980, pages 339–370. Springer, 1982.

    Google Scholar 

  8. P. Eberlein. Rigidity problems for manifolds of nonpositive curvature. Springer LNM 1156, 1984.

    Google Scholar 

  9. P. Eberlein. Geometry of nonpositively curved manifolds. Univ. Chicago Press, 1996.

    MATH  Google Scholar 

  10. D. Freed. Special Kähler manifolds. Comm.Math.Phys., 203:31–52, 1999.

    Article  MathSciNet  MATH  Google Scholar 

  11. P. Griffiths and J. Harris. Principles of algebraic geometry. Wiley-Interscience, 1978.

    Google Scholar 

  12. S. Helgason. Differential Geometry, Lie Groups and Symmetric Spaces. Academic Press, 1978.

    Google Scholar 

  13. W. Hodge. The Theory and Applications of Harmonic Integrals. Cambr.Univ.Press, 1941,21952.

    Google Scholar 

  14. J. Hofrichter, J. Jost, and T.D. Tran. Information geometry and population genetics. Springer, 2017.

    Google Scholar 

  15. J. Jost. Nonlinear methods in Riemannian and Kählerian geometry. Birkhäuser,21991.

    Google Scholar 

  16. E. Kähler. Über eine bemerkenswerte Hermitesche Metrik. Abh. Math. Sem. Univ. Hamburg, 9:173–186, 1933.

    Article  MathSciNet  MATH  Google Scholar 

  17. E. Kähler. Mathematische Werke – Mathematical Works. Edited by R.Berndt and O.Riemenschneider; de Gruyter, 2003.

    Google Scholar 

  18. G.A. Margulis. Discrete groups of motion of manifolds of nonpositive curvature. AMS Transl., 190:33–45, 1977.

    MATH  Google Scholar 

  19. G.A. Margulis. Discrete subgroups of semisimple Lie groups. Springer, 1991.

    Google Scholar 

  20. G. Mostow. Strong rigidity of locally symmetric spaces. Ann. Math. Studies 78, Princeton Univ.Press, 1973.

    Google Scholar 

  21. A. Nadel. Multiplier ideal sheaves and Kähler-Einstein metrics of positive scalar curvature. Ann.Math., 132:549–596, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  22. H. Shima. The geometry of Hessian structures. World Scientific, 2007.

    Google Scholar 

  23. Y.T. Siu. The existence of Kähler-Einstein metrics on manifolds with positive anticanonical line bundle and suitable finite symmetry group. Ann.Math., 127:585–627, 1988.

    Article  MathSciNet  MATH  Google Scholar 

  24. G. Tian. On Kähler-Einstein metrics on certain Kähler manifolds with c 1(M) > 0. Inv. math., 89:225–246, 1987.

    Article  MATH  Google Scholar 

  25. G. Tian. Kähler-Einstein metrics with positive scalar curvature. Invent.Math., 130:1–39, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  26. G. Tian and S.T. Yau. Kähler-Einstein metrics on complex surfaces with c 1(M) > 0. Comm. Math. Phys., 112:175–203, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  27. R. Wells. Differential analysis on complex manifolds. Springer,21980.

    Google Scholar 

  28. S.T. Yau. Calabi’s conjecture and some new results in algebraic geometry. Proc.Nat.Acad.Sc., 74:1798–1801, 1977.

    Article  MathSciNet  MATH  Google Scholar 

  29. S.T. Yau. On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, I. Comm. Pure Appl. Math., 31:339–411, 1978.

    Article  MathSciNet  MATH  Google Scholar 

  30. S.T. Yau. Open problems in geometry. Proc. Symp. Pure Math., 54(I):1–28, 1993.

    MathSciNet  MATH  Google Scholar 

  31. R. Zimmer. Ergodic theory and semisimple groups. Birkhäuser, 1984.

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Max Planck Institute for Mathematics in the Sciences, Max Planck Society, Leipzig, Germany

    Jürgen Jost

Authors
  1. Jürgen Jost
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Jost, J. (2017). Chapter 7 Symmetric Spaces and Kähler Manifolds. In: Riemannian Geometry and Geometric Analysis. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-61860-9_8

Download citation

Publish with us

Policies and ethics

Search

Navigation