Infinite dimensional Riemannian symmetric spaces with fixed-sign curvature operator
Annales de l'Institut Fourier, Volume 65 (2015) no. 1, pp. 211-244.

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.

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.

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DOI: 10.5802/aif.2929
Classification: 53C35
Keywords: Riemannian symmetric spaces, L * -algebras, infinite dimension
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Duchesne, Bruno. Infinite dimensional Riemannian symmetric spaces with fixed-sign curvature operator. Annales de l'Institut Fourier, Volume 65 (2015) no. 1, pp. 211-244. doi : 10.5802/aif.2929. https://aif.centre-mersenne.org/articles/10.5802/aif.2929/

[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 * -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, Lecture Notes in Mathematics, Tome 1754, Springer-Verlag, Berlin, 2000, xvi+269 pages | Article | MR: 1809879 | Zbl: 1014.17024

[4] Borel, Armand Essays in the history of Lie groups and algebraic groups, History of Mathematics, Tome 21, American Mathematical Society, Providence, RI, 2001, xiv+184 pages 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, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1987, viii+364 pages (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, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Tome 319, Springer-Verlag, Berlin, 1999, xxii+643 pages | 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 (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 * -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, Lecture Notes in Mathematics, Vol. 285, Springer-Verlag, Berlin, 1972, iii+160 pages | MR: 476820 | Zbl: 0256.22015

[17] Helgason, Sigurdur Differential geometry, Lie groups, and symmetric spaces, Graduate Studies in Mathematics, Tome 34, American Mathematical Society, Providence, RI, 2001, xxvi+641 pages (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 | EuDML: 163571 | 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 | EuDML: 163690 | 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 | EuDML: 104123 | Numdam | MR: 1608566 | Zbl: 0910.53035

[21] Klingenberg, Wilhelm P. A. Riemannian geometry, de Gruyter Studies in Mathematics, Tome 1, Walter de Gruyter & Co., Berlin, 1995, x+409 pages | 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, Graduate Texts in Mathematics, Tome 191, Springer-Verlag, New York, 1999, xviii+535 pages | MR: 1666820 | 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, 119 pages 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, Graduate Texts in Mathematics, Tome 171, Springer, New York, 2006, xvi+401 pages | 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 | EuDML: 139052 | 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 * 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 * -algebras, J. Differential Geometry, Tome 7 (1972), pp. 423-451 | MR: 325721 | Zbl: 0279.46044

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

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