ArticleOriginal scientific text
Title
Generalized symmetric spaces and minimal models
Authors 1, 1, 2
Affiliations
- Institute of Mathematics, Szczecin Technical University, Al. Piastów 17, 70-310 Szczecin, Poland
- Institute of Mathematics, Szczecin University, Wielkopolska 15, 70-451 Szczecin, Poland
Abstract
We prove that any compact simply connected manifold carrying a structure of Riemannian 3- or 4-symmetric space is formal in the sense of Sullivan. This result generalizes Sullivan's classical theorem on the formality of symmetric spaces, but the proof is of a different nature, since for generalized symmetric spaces techniques based on the Hodge theory do not work. We use the Thomas theory of minimal models of fibrations and the classification of 3- and 4-symmetric spaces.
Keywords
minimal model, Koszul complex, generalized symmetric space
Bibliography
- C. Allday and V. Puppe, Cohomology Theory of Transformation Groups, Cambridge Univ. Press, 1993.
- P. Deligne, P. Griffiths, J. Morgan and D. Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), 245-274.
- L. Flatto, Invariants of reflection groups, Enseign. Math. 28 (1978), 237-293.
- A. Gray, Riemannian manifolds with geodesic symmetries of order 3, J. Differential Geom. 7 (1972), 343-369.
- V. Greub, S. Halperin and R. Vanstone, Curvature, Connections and Cohomology, Vol. 3, Acad. Press, 1976.
- S. Halperin, Lectures on Minimal Models, Hermann, 1982.
- S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Acad. Press, 1978.
- J. A. Jiménez, Riemannian 4-symmetric spaces, Trans. Amer. Math. Soc. 306 (1988), 715-734.
- O. Kowalski, Classification of generalized Riemannian symmetric spaces of dimension ≤ 5, Rozpravy Československé Akad. Věd Řada Mat. Přírod. Věd 85 (1975).
- O. Kowalski, Generalized Symmetric Spaces, Springer, 1980.
- E. Kunz, Introduction to Commutative Algebra and Algebraic Geometry, Birkhäuser, 1985.
- D. Lehmann, Théorie homotopique des formes différentielles (d'après D. Sullivan), Astérisque 45 (1977).
- G. Lupton and J. Oprea, Symplectic manifolds and formality, J. Pure Appl. Algebra 91 (1994), 193-207.
- T. Miller and J. Neisendorfer, Formal and coformal spaces, Illinois J. Math. 22 (1978), 565-580.
- D. Sullivan, Infinitesimal computations in topology, Publ. IHES 47 (1977), 269-331.
- M. Takeuchi, On Pontrjagin classes of compact symmetric spaces, J. Fac. Sci. Univ. Tokyo Sect. I 9 (1962), 313-328.
- D. Tanré, Homotopie Rationnelle: Modèles de Chen, Quillen, Sullivan, Springer, 1988.
- J.-C. Thomas, Homotopie rationnelle des fibrés de Serre, Ph.D. Thesis, Université des Sciences et Techniques de Lille 1, 1980.
- J.-C. Thomas, Rational homotopy of Serre fibrations, Ann. Inst. Fourier (Grenoble) 31 (3) (1981), 71-90.
- M. Vigué-Poirrier and D. Sullivan, Cohomology theory of the closed geodesic problem, J. Differential Geom. 11 (1976), 633-644.
- R. O. Wells, Differential Analysis on Complex Manifolds, Prentice-Hall, 1973.
Pages:
17-35
Main language of publication
English
Received
1994-04-11
Accepted
1995-04-30
Published
1996
Exact and natural sciences