Infinite dimensional Riemannian symmetric spaces with fixed-sign curvature operator

[Espaces riemanniens symétriques de dimension infinie avec un opérateur de courbure de signe fixe]

Duchesne, Bruno

Annales de l'Institut Fourier, Tome 65 (2015), p. 211-244 / Harvested from Numdam

Résumé

Nous associons à tout espace riemannien symétrique (de dimension finie ou non) une $L^*$-algèbre dès lors que l’opérateur de courbure est de signe fixe. Les $L^*$-algèbres sont des algèbres de Lie avec une structure d’espace de Hilbert compatible. La $L^*$-algèbre que nous construisons est un invariant d’isomorphisme local et nous permet de classifier les espaces symétriques riemanniens simplement connexe avec un opérateur de courbure de signe fixe. Le cas de la courbure négative est mis en avant.

Abstract

We associate to any Riemannian symmetric space (of finite or infinite dimension) a L$^*$-algebra, under the assumption that the curvature operator has a fixed sign. L$^*$-algebras are Lie algebras with a pleasant Hilbert space structure. The L$^*$-algebra that we construct is a complete local isomorphism invariant and allows us to classify simply-connected Riemannian symmetric spaces with fixed-sign curvature operator. The case of nonpositive curvature is emphasized.

Détail

Publié le : 2015-01-01

Zbl 06496538

DOI : https://doi.org/10.5802/aif.2929
Classification:  53C35
Mots clés: Espaces riemanniens symétriques, $L^*$-algèbres, dimension infinie

Bibliographie

[1] Atkin, C. J. The Hopf-Rinow theorem is false in infinite dimensions, Bull. London Math. Soc., Tome 7 (1975) no. 3, pp. 261-266 | Article | MR 400283 | Zbl 0374.58006

[2] Balachandran, V. K. Real $L^{\ast } $-algebras, Indian J. Pure Appl. Math., Tome 3 (1972) no. 6, pp. 1224-1246 | MR 347920 | Zbl 0312.46060

[3] Bertram, Wolfgang The geometry of Jordan and Lie structures, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Tome 1754 (2000), pp. xvi+269 | Article | MR 1809879 | Zbl 1014.17024

[4] Borel, Armand Essays in the history of Lie groups and algebraic groups, American Mathematical Society, Providence, RI, History of Mathematics, Tome 21 (2001), pp. xiv+184 http://links.jstor.org/sici?sici=0002-9890(200111)108:9<879:TEOTTO>2.0.CO;2-7 | MR 1847105 | Zbl 1087.01011

[5] Bourbaki, N. Topological vector spaces. Chapters 1–5, Springer-Verlag, Berlin, Elements of Mathematics (Berlin) (1987), pp. viii+364 (Translated from the French by H. G. Eggleston and S. Madan) | MR 910295 | Zbl 0683.54004

[6] Bridson, Martin R.; Haefliger, André Metric spaces of non-positive curvature, Springer-Verlag, Berlin, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Tome 319 (1999), pp. xxii+643 | MR 1744486 | Zbl 0988.53001

[7] Caprace, Pierre-Emmanuel; Lytchak, Alexander At infinity of finite-dimensional CAT(0) spaces, Math. Ann., Tome 346 (2010) no. 1, pp. 1-21 | Article | MR 2558883 | Zbl 1184.53038

[8] Caprace, Pierre-Emmanuel; Monod, Nicolas Isometry groups of non-positively curved spaces: structure theory, J. Topol., Tome 2 (2009) no. 4, pp. 661-700 | Article | MR 2574740 | Zbl 1209.53060

[9] Chu, Cho-Ho Jordan triples and Riemannian symmetric spaces, Adv. Math., Tome 219 (2008) no. 6, pp. 2029-2057 | Article | MR 2456274 | Zbl 1233.17028

[10] Duchesne, Bruno Des espaces de Hadamard symétriques de dimension infinie et de rang fini, Université de Genève (2011) (Ph. D. Thesis)

[11] Duchesne, Bruno Infinite-dimensional nonpositively curved symmetric spaces of finite rank, Int. Math. Res. Not. IMRN (2013) no. 7, pp. 1578-1627 | MR 3044451 | Zbl 1315.53054

[12] Eells, James Jr. A setting for global analysis, Bull. Amer. Math. Soc., Tome 72 (1966), pp. 751-807 | Article | MR 203742

[13] Eells, James Jr.; Sampson, J. H. Harmonic mappings of Riemannian manifolds, Amer. J. Math., Tome 86 (1964), pp. 109-160 | Article | MR 164306 | Zbl 0122.40102

[14] Gallot, S.; Meyer, D. Opérateur de courbure et laplacien des formes différentielles d’une variété riemannienne, J. Math. Pures Appl. (9), Tome 54 (1975) no. 3, pp. 259-284 | MR 454884 | Zbl 0316.53036

[15] De La Harpe, Pierre Classification des $L^{\ast } $-algèbres semi-simples réelles séparables, C. R. Acad. Sci. Paris Sér. A-B, Tome 272 (1971), p. A1559-A1561 | MR 282218 | Zbl 0215.48501

[16] De La Harpe, Pierre Classical Banach-Lie algebras and Banach-Lie groups of operators in Hilbert space, Springer-Verlag, Berlin, Lecture Notes in Mathematics, Vol. 285 (1972), pp. iii+160 | MR 476820 | Zbl 0256.22015

[17] Helgason, Sigurdur Differential geometry, Lie groups, and symmetric spaces, American Mathematical Society, Providence, RI, Graduate Studies in Mathematics, Tome 34 (2001), pp. xxvi+641 (Corrected reprint of the 1978 original) | MR 1834454 | Zbl 0993.53002

[18] Kaup, Wilhelm Über die Klassifikation der symmetrischen hermiteschen Mannigfaltigkeiten unendlicher Dimension. I, Math. Ann., Tome 257 (1981) no. 4, pp. 463-486 | Article | MR 639580 | Zbl 0482.32010

[19] Kaup, Wilhelm Über die Klassifikation der symmetrischen hermiteschen Mannigfaltigkeiten unendlicher Dimension. II, Math. Ann., Tome 262 (1983) no. 1, pp. 57-75 | Article | MR 690007 | Zbl 0482.32011

[20] Kleiner, Bruce; Leeb, Bernhard Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings, Inst. Hautes Études Sci. Publ. Math. (1997) no. 86, p. 115-197 (1998) | Article | Numdam | MR 1608566 | Zbl 0910.53035

[21] Klingenberg, Wilhelm P. A. Riemannian geometry, Walter de Gruyter & Co., Berlin, de Gruyter Studies in Mathematics, Tome 1 (1995), pp. x+409 | MR 1330918 | Zbl 0911.53022

[22] Klotz, Michael Banach Symmetric Spaces (2009) (http://arxiv.org/abs/0911.2089) | MR 2859032

[23] Lang, Serge Fundamentals of differential geometry, Springer-Verlag, New York, Graduate Texts in Mathematics, Tome 191 (1999), pp. xviii+535 | MR 1666820 | Zbl 0932.53001 | Zbl 0995.53001

[24] Larotonda, Gabriel Nonpositive curvature: a geometrical approach to Hilbert-Schmidt operators, Differential Geom. Appl., Tome 25 (2007) no. 6, pp. 679-700 | Article | MR 2373944 | Zbl 1141.22006

[25] Mcalpin, John Harris Infinite dimensional manifolds and morse theory, ProQuest LLC, Ann Arbor, MI (1965), pp. 119 http://search.proquest.com/docview/302168992 (Thesis (Ph.D.)–Columbia University) | MR 2614999

[26] Monod, Nicolas Superrigidity for irreducible lattices and geometric splitting, J. Amer. Math. Soc., Tome 19 (2006) no. 4, pp. 781-814 | Article | MR 2219304 | Zbl 1105.22006

[27] Neeb, Karl-Hermann A Cartan-Hadamard theorem for Banach-Finsler manifolds, Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part II (Haifa, 2000), Tome 95 (2002), pp. 115-156 | Article | MR 1950888 | Zbl 1027.58003

[28] Petersen, Peter Riemannian geometry, Springer, New York, Graduate Texts in Mathematics, Tome 171 (2006), pp. xvi+401 | MR 2243772 | Zbl 1220.53002

[29] De Rham, Georges Sur la reductibilité d’un espace de Riemann, Comment. Math. Helv., Tome 26 (1952), pp. 328-344 | Article | MR 52177 | Zbl 0048.15701

[30] Schue, John R. Hilbert space methods in the theory of Lie algebras, Trans. Amer. Math. Soc., Tome 95 (1960), pp. 69-80 | Article | MR 117575 | Zbl 0093.30601

[31] Schue, John R. Cartan decompositions for $L^{\ast } $ algebras, Trans. Amer. Math. Soc., Tome 98 (1961), pp. 334-349 | MR 133408 | Zbl 0099.10205

[32] Shergoziev, B. U. Infinite-dimensional spaces with bounded curvature, Sibirsk. Mat. Zh., Tome 36 (1995) no. 5, p. 1167-1178, iv | Article | MR 1373605 | Zbl 0853.53053

[33] Simons, James On the transitivity of holonomy systems, Ann. of Math. (2), Tome 76 (1962), pp. 213-234 | Article | MR 148010 | Zbl 0106.15201

[34] Tumpach, Alice Barbara On the classification of infinite-dimensional irreducible Hermitian-symmetric affine coadjoint orbits, Forum Math., Tome 21 (2009) no. 3, pp. 375-393 | Article | MR 2526791 | Zbl 1166.58301

[35] Unsain, Ignacio Classification of the simple separable real $L^{^{\ast } }$-algebras, J. Differential Geometry, Tome 7 (1972), pp. 423-451 | MR 325721 | Zbl 0279.46044

[36] Upmeier, Harald Symmetric Banach manifolds and Jordan $C^\ast $-algebras, North-Holland Publishing Co., Amsterdam, North-Holland Mathematics Studies, Tome 104 (1985), pp. xii+444 (Notas de Matemática [Mathematical Notes], 96) | MR 776786 | Zbl 0561.46032