Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-12-02T21:31:39.121Z Has data issue: false hasContentIssue false
  • English
  • Français

Homogeneous structures on Kähler submanifolds of complex projective spaces*

Published online by Cambridge University Press:  20 January 2009

Sergio Console  and
Anna Fino
Sergio Console
Affiliation:
Dipartimento di Matematica Universita di Torino via Carlo Alberto, 10, I-10123 Torino e-mail [email protected] / [email protected]
Anna Fino
Affiliation:
Dipartimento di Matematica Universita di Torino via Carlo Alberto, 10, I-10123 Torino e-mail [email protected] / [email protected]
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.

In this paper we give a differential characterization of homogeneous Kähler submanifolds of complex projective spaces in terms of the existence of a tensor field, the homogeneous structure S. We show that for any m∈M, Sm determines a unitary representation whose orbit at m is a compact, complete Kähler submanifold which extends M. We consider the U(n) × U(N ~ n) (n = dim ℂM) module of the space of these tensors and we find its irreducible factors.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 39 , Issue 2 , June 1996 , pp. 381 - 395
Copyright
Copyright © Edinburgh Mathematical Society 1996

Footnotes

*

Work partially supported by the GNSAGA of CNR and by the MURST of Italy.

References

REFERENCES

1. Abbena, E. and Garbiero, S., Almost Hermitian homogeneous structures, Proc. Edinburgh Math. Soc. 31 (1988), 375–395.Google Scholar
2. Borel, A. et Weil, A., Représentations linéares et espaces homogènes kählériens des groupes de Lie compacts (Séminaire Bourbaki, exposé by J. P. Serre, 1954).Google Scholar
3. Calabi, E., Isometric imbedding of complex manifolds, Ann. of Math. 58 (1953), 1–23.Google Scholar
4. D'andrea, A. Carfagna, Mazzocco, R. and Romani, G., Some characterizations of 2-symmetric submanifolds in spaces of constant curvature, Czechoslovak Math. J. 44 (119) (1994), 691–711.CrossRefGoogle Scholar
5. Console, S., Infinitesimally homogeneous submanifolds of Euclidean spaces, Ann. Global Anal. Geom. 12 (1994), 313–334.CrossRefGoogle Scholar
6. Falcitelli, M., Farinola, A. and Salamon, S., Almost hermitian geometry, Diff. Ann. Global Anal. Geom. Appl. 4 (3) (1994), 259–282.Google Scholar
7. Fino, A., Almost contact homogeneous structures, Boll. Un. Mat. Ital., to appear.Google Scholar
8. Fulton, W. and Harris, J., Representation Theory (Springer, Berlin, New York, 1991).Google Scholar
9. Gray, A., Homogeneous almost Hermitian manifolds, Rend. Sem. Mat. Univ. Politec. Torino, Fasc. Spec. (1983), 17–58.Google Scholar
10. Ikawa, O., Harmonic mappings, minimal and totally geodesic immersions of compact Riemannian homogeneous spaces into Grassmann manifolds, Kodai Math. J. 16 (1993), 295–305.CrossRefGoogle Scholar
11. Kelly, E., Tight equivariant imbeddings of symmetric spaces, J. Diff. Geom. 7 (1972), 535–548.Google Scholar
12. Kowalski, O., Generalized Symmetric Spaces (Lecture Notes in Math. 805, Springer, Berlin, Heidelberg, New York, 1980).CrossRefGoogle Scholar
13. Kowalski, O. and Kulich, I., Generalized symmetric submanifolds of Euclidean Spaces, Math. Ann. 277 (1987), 67–78.CrossRefGoogle Scholar
14. Nakagawa, H. and Tagaki, R., On locally symmetric Kähler submanifolds in a complex projective space, J. Math. Soc. Japan 28 (1976), 638–667.Google Scholar
15. Nomizu, K., Invariant affine connections on homogeneous spaces, Amer. J. Math. 76 (1954), 33–65.CrossRefGoogle Scholar
16. Olmos, C., Isoparametric submanifolds and their homogeneous structures, J. Differential Geom. 38 (1993), 225–234.CrossRefGoogle Scholar
17. Olmos, C. and Sanchez, C., A geometric characterization of the orbits of s-representations, Differential J. Reine Angew. Math. 420 (1991), 195–202.Google Scholar
18. O'neill, B., The fundamental equations of a submersion, Michigan Math. J. 13 (1966), 459–469.Google Scholar
19. Salamon, S. M. Riemannian Geometry and holonomy groups (Pirman research notes, Longman, Harlow, Essex, UK, 1989).Google Scholar
20. Sekigawa, K., Notes on homogeneous almost Hermitian manifolds, Hokkaido Math. J. 1 (1978), 206–213.Google Scholar
21. Spivak, M., A Comprehensive Introduction to Differential Geometry, IV (Publish or Perish, Wilmington, Delaware, 1979).Google Scholar
22. Strübing, W., Symmetric submanifolds of Riemannian manifolds, Math. Ann. 245 (1979), 37–44.CrossRefGoogle Scholar
23. Takeuchi, M., Homogeneous Kähler submanifolds in complex projective spaces, Japan J. Math.s 4(1) (1978), 171–219.CrossRefGoogle Scholar
24. Tricerri, F. and Vanhecke, L., Homogeneous Structures on Riemannian Manifolds (London Mathematical Society Lecture Notes Series 83, Cambridge University Press, Cambridge, 1983).CrossRefGoogle Scholar
25. Weyl, H., Classical groups, their invariants and representations (Princeton University Press, Princeton, 1946).Google Scholar