Abstract
Let (X, g) be a compact Riemannian symmetric space. We say that a symmetric 2-form h on X satisfies the zero-energy condition if the integrals of h over the closed geodesies of X all vanish. We recall that the Lie derivative of the metric g along a vector field on X always verifies this condition. We say that (X, g) is infinitesimally rigid if the only symmetric 2-forms on X are the Lie derivatives of the metric g.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
L. BÈRARD BERGERY, J.P. BOURGUIGNON, J. LAFONTAINE, Déformations localement triviales des variétés riemanniennes, Differential Geometry,Proc. Sympos. Pure Math. Vol. XXVII, Part 1, Amer. Math. Soc., Providence, R.I., 1975, 3–32.
A.L. BESSE, Manifolds all of whose geodesies are closed, Ergeb. Math. Grenzgeb., Band 93, Springer-Verlag, Berlin, Heidelberg, New York, 1978.
E. CALABI, On compact, Riemannian manifolds with constant curvature. I, Differential Geometry, Proc. Sympos. Pure Math., Vol. III, Amer. Math. Soc., Providence, R.I., 1961, 155–180.
Y. DIENG, Quelques résultats de rigidité infinitésimale pour les quadriques complexes (to appear).
J. GASQUI, H. GOLDSCHMIDT, Déformations infinitésimales des espaces riemanniens localement symétriques. I,Adv. in Math. 48 (1983), 205–285.
J. GASQUI, H. GOLDSCHMIDT, Déformations infinitésimales des espaces riemanniens localement symétriques. II. La conjecture infinitésimale de Blaschke pour les espaces projectifs complexes, Ann. Inst. Fourier (Grenoble) 34 (1984), (2), 191–226.
J. GASQUI, H. GOLDSCHMIDT,Déformations infinitésimales des structures conformes plates, Progress in Mathematics, Vol. 52, Birkhàuser, Boston, Basel, Stuttgart, 1984.
J. GASQUI, H. GOLDSCHMIDT, Une caractérisation des formes exactes de degré 1 sur les espaces projectifs, Comment. Math. Helv. 60 (1985), 46–53.
J. GASQUI, H. GOLDSCHMIDT, Infinitesimal rigidity of S 1 ×RPn, Duke Math. J. 51 (1984), 675–690.
J. GASQUI, H. GOLDSCHMIDT, Infinitesimal rigidity of products of symmetric spaces (to appear).
J. GASQUI, H. GOLDSCHMIDT, Complexes of differential operators and symmetric spaces, this volume.
J. GASQUI, H. GOLDSCHMIDT, Rigidité infinitésimale des espaces projectifs et des quadriques complexes (to appear).
K. KIYOHARA, On deformations of the standard C m -metrics on projective spaces (to appear).
R. MICHEL, Problèmes d’analyse géométrique liés à la conjecture de Blaschke, Bull. Soc. Math. France, 101 (1973), 17–69.
R. MICHEL Un problème d’exactitude concernant les tenseurs symétriques et les géodésiques, C.R. Acad. Se. Paris Sér. A 284 (1977), 183–186;
R. MICHEL Tenseurs symétriques et géodésiques, C.R. Acad. Se. Paris Sér. A 284 (1977), 1065–1068.
R. MICHEL, Sur quelques problèmes de géométrie globale des géodésiques, Bol. Soc. Bras. Mat. 9 (1978), 19–38.
B. SMYTH, Differential geometry of complex hypersurfaces, Ann. of Math. 85 (1967), 246–266.
C. TSUKAMOTO, Infinitesimal Blaschke conjectures on projective spaces, Ann. Sc. Ecole Norm. Sup., (4) 14 (1981), 339–356.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1988 Kluwer Academic Publishers
About this chapter
Cite this chapter
Gasqui, J., Goldschmidt, H. (1988). Some Rigidity Results in the Deformation Theory of Symmetric Spaces. In: Hazewinkel, M., Gerstenhaber, M. (eds) Deformation Theory of Algebras and Structures and Applications. NATO ASI Series, vol 247. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-3057-5_16
Download citation
DOI: https://doi.org/10.1007/978-94-009-3057-5_16
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-7875-7
Online ISBN: 978-94-009-3057-5
eBook Packages: Springer Book Archive