Abstract
In this paper we study Banach–Finsler manifolds endowed with a spray which have seminegative curvature in the sense that the corresponding exponential function has a surjective expansive differential in every point. In this context we generalize the classical theorem of Cartan–Hadamard, saying that the exponential function is a covering map. We apply this to symmetric spaces and thus obtain criteria for Banach–Lie groups with an involution to have a polar decomposition. Typical examples of symmetric Finsler manifolds with seminegative curvature are bounded symmetric domains and symmetric cones endowed with their natural Finsler structure which in general is not Riemannian.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[AB90] Alexander, S. B. and Bishop, R. L.: The Hadamard-Cartan theorem in locally convex metric spaces, Enseign. Math. 36 (1990), 309–320.
[Au55] Auslander, L.: On curvature in Finsler geometry, Trans. Amer. Math. Soc. 79 (1955), 378–388.
[BCS00] Bao, D., Chen, S.-S. and Shen, Z.: An Introduction to Riemann-Finsler Geometry, Grad. Texts in Math. 200, Springer, New York, 2000.
[BP80] Berkson, E., and Porta, H.: The group of isometries on Hardy spaces of the n-ball and the polydisc, Glasgow Math. J. 21 (1980), 199–204.
[Bou90] Bourbaki, N.: Groupes et alge´bres de Lie, chapitres 2 et 3, Masson, Paris, 1990.
[BH99] Bridson, M. R. and Haefliger, A.: Metric Spaces of Non-positive Curvature, Grundlehren Math. Wiss., Springer, Berlin, 1999.
[Ca63] Cartan, E.: Lec¸ ons sur la géométrie des espaces de Riemann, 2nd edn, Gauthier-Villars, Paris, 1963.
[CPR92] Corach, G., Porta, H. and Recht, L.: A geometric interpretation of Segal's inequality ‖eX+Y‖⩽‖eX/2eYeX/2‖ Proc. Amer. Math. Soc. 115(1) (1992), 229–231.
[CPR93] Corach, G., Porta, H. and Recht, L.: Geodesics and operator means in the space of positive operators, Internat. J. Math. 4(2) (1993), 193–202.
[CPR94] Corach, G., Porta, H. and Recht, L.: Convexity of the geodesics distance on spaces of positive operators, Illinois J. Math. 38:(1) (1994), 87–94.
[CGM90] Cuenca Mira, J. A., Garcia Martin, A. and Martin Gonzalez, C.: Structure theory of L*-algebras, Math. Proc. Cambridge Philos. Soc. 107 (1990), 361–365.
[Ee66] Eells, J.: A setting for global analysis, Bull. Amer. Math. Soc. 72 (1966), 751–807.
[El67] Eliasson, H.: Geometry of manifolds of maps, J. Differential Geom. 1 (1967), 169–194.
[FK94] Faraut, J. and Koranyi, A.: Analysis on Symmetric Cones, Oxford Math. Monogr., Oxford University Press, 1994.
[FR86] Friedman, Y. and Russo, B.: The Gelfand-Neimark theorem for JB*-triples, Duke Math. J. 53 (1986), 139–148.
[Gl99] Glückner, H.: Infinite-dimensional complex groups and semigroups: representations of cones, tubes and conelike semigroups, Dissertation, University of Technology, Darmstadt, 1999.
[GN01] Glückner, H. and Neeb, K.-H. Banach-Lie quotients, enlargibility, and Universal complexifications, J. Reine Angew. Math., to appear.
[Gr65] Grossman, N.: Hilbert manifolds without epiconjugate points, Proc. Amer. Math. Soc. 16 (1965), 1365–1371.
[Ha96] Hadamard, J.: Les surfaces à courbures opposées et leur lignes géodesiques, J. Math. Pures Appl. (5) 4 (1896), 27–73.
[Ha67] Halmos, P. R.: A Hilbert Space Problem Book, Grad. Texts in Math. 19, Springer, New York, 1967.
[dlH72] de la Harpe, P.: Classical Banach-Lie Algebras and Banach-Lie Groups of Operators in Hilbert Space, Lecture Notes in Math. 285, Springer, Berlin, 1972.
[dlH82] de la Harpe, P.: Classical groups and classical Lie algebras of operators, Proc. Sympos. Pure Math. 38(1) (1982), 477–513.
[Hel78] Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, London, 1978.
[HHL89] Hilgert, J., Hofmann, K. H. and Lawson, J. D.: Lie Groups, Convex Cones, and Semigroups, Oxford Univ. Press, 1989.
[HO96] Hilgert, J. and Ólafsson, G.: Causal Symmetric Spaces, Geometry and Harmonic Analysis, Academic Press, New York, 1996.
[Ka83] Kaup, W.: A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces, Math. Z. 183 (1983), 503–529.
[KM97] Kriegl, A. and Michor, P. The Convenient Setting for Global Analysis, Math. Surveys Monogr. 53, Amer. Math. Soc., Providence, 1997.
[KN96] Krütz, B. and Neeb, K.-H.: On hyperbolic cones and mixed symmetric spaces, J. Lie Theory 6(1) (1996), 69–146.
[La99] Lang, S.: Fundamentals of Differential Geometry, Grad. Texts in Math. 191, Springer, New York, 1999.
[Lim99a] Lim, Y.: Applications of geometric means on symmetric cones, submitted.
[Lim99b] Lim, Y.: Geometric means on symmetric cones, Arch. Math., to appear.
[Lim99c] Lim, Y.: Finsler metrics on symmetric cones, Math. Ann., to appear.
[Lo69] Loos, O.: Symmetric Spaces I:General Theory, Benjamin, New York, 1969.
[Ma62] Maissen, B.: Lie-Gruppen mit Banachräumen als Parameterräume, Acta Math. 108 (1962), 229–269.
[McA65] McAlpin, J.: Infinite dimensional manifolds and morse theory, Thesis, Columbia University, 1965.
[Ne99a] Neeb, K.-H.: Holomorphy and Convexity in Lie Theory, Expos. Math. 28, de Gruyter-Verlag, Berlin, 1999.
[Ne99b] Neeb, K.-H.: On the complex geometry of invariant domains in complexified symmetric spaces, Ann. Inst. Fourier 49(1) (1999), 177–225.
[Ne00a] Neeb, K.-H.: Infinite-dimensional Lie groups and their representations, Lectures at the European School in Group Theory, SDU-Odense Univ., August 2000 (available as Preprint), to appear in Progr. in Math., Birkhäuser-Verlag, Basel.
[Ne00b] Neeb, K.-H.: Compressions of infinite-dimensional bounded symmetric domains, Semigroup Forum, to appear.
[Ne01a] Neeb, K.-H.: Geometry and structure of L*-groups, in preparation.
[Ne01b] Neeb, K.-H.: Classical Hilbert-Lie groups, their extensions and their homotopy groups, In: Geometry and Analysis on Finite-and Infinite-Dimensional Lie Groups, A. Strasburger et al. (eds), Banach Center Publications 55 (2002), 87–151.
[Neh93] Neher, E.: Generators and relations for 3-graded Lie algebras, J. Algebra 155 (1993), 1–35.
[Paz83] Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, Berlin, 1983.
[PS86] Pressley, A. and Segal, G. Loop Groups, Oxford Univ. Press, Oxford, 1986.
[PW52] Putnam, C. R. and Winter, A.: The orthogonal group in Hilbert space, Amer. J. Math. 74 (1952), 52–78.
[RS78] Reed, S. and Simon, B. Methods of Modern Mathematical Physics IV: Analysis of Operators, Academic Press, New York, 1978.
[Re95] Remmert, R.: Funktionentheorie II, Springer Lehrbuch, Springer, Berlin, 1995.
[Ru73] Rudin, W.: Functional Analysis, McGraw Hill, New York, 1973.
[Ru86] Rudin, W.: Real and Complex Analysis, McGraw Hill, New York, 1986.
[St99] Stumme, N.: The structure of locally finite split Lie algebras, PhD thesis, Darmstadt University of Technology, 1999.
[Th71] Thompson, C. J.: Inequalities and partial orders on matrix spaces, Indiana Univ. Math. J. 21 (1971), 469–480.
[Up85] Upmeier, H.: Symmetric Banach Manifolds and Jordan C*-algebras, North-Holland, Amsterdam, 1985.
[Ve79] Vesentini, E.: Variations on a theme of Carathéodory, Ann. Sculoa Norm. Sup. Pisa 7(4) (1979), 39–68.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Neeb, KH. A Cartan–Hadamard Theorem for Banach–Finsler Manifolds. Geometriae Dedicata 95, 115–156 (2002). https://doi.org/10.1023/A:1021221029301
Issue Date:
DOI: https://doi.org/10.1023/A:1021221029301