Artículos de revistas
Properly Discontinuous Actions On Hilbert Manifolds
Registro en:
Bulletin Of The Brazilian Mathematical Society. Springer New York Llc, v. 45, n. 3, p. 433 - 452, 2014.
16787544
10.1007/s00574-014-0057-7
2-s2.0-84919927957
Autor
Biliotti L.
Mercuri F.
Institución
Resumen
In this article we study properly discontinuous actions on Hilbert manifolds giving new examples of complete Hilbert manifolds with nonnegative, respectively nonpositive, sectional curvature with infinite fundamental group. We also get examples of complete infinite dimensional Kähler manifolds with positive holomorphic sectional curvature and infinite fundamental group in contrastwith the finite dimensional case and we classify abelian groups acting linearly, isometrically and properly discontinuously on Stiefel manifolds. Finally, we classify homogeneous Hilbert manifolds with constant sectional curvature. 45 3 433 452 Atkin, C.J., The Hopf-Rinow theorem is false in infinite dimensions (1975) Bull. London Math. Soc., 7, pp. 261-266 Battaglia, F., A hypercomplex Stiefel manifold (1996) Differential Geom. Appl., 6 (2), pp. 121-128 Biliotti, L., Mercuri, F., Tausk, D., A note on tensor fields in Hilbert spaces (2002) An. Acad. Brasil. Ciênc., 74 (2), pp. 207-210 Biliotti, L., Properly Discontinuous isometric actions on the unith sphere of infinite dimensional Hilbert spaces (2004) Ann. Global Anal. Geom., 26, pp. 385-395 Biliotti, L., Exponential map of a weak Riemannian Hilbert manifold (2004) Illinois J. Math., 48, pp. 1191-1206 Biliotti, L., Exel, R., Piccione, P., Tausk, D., On the singularities of the exponentialmap in infinite dimensionalRiemannianmanifolds (2006) Math. Ann., 336 (2), pp. 247-267 Burghelea, D., Kuiper, N., Hilbertmanifolds (1968) Ann. of Math., 90, pp. 379-417 Cheeger, J., Ebin, D., (1975) Comparison theorems in Riemannian geometry, , North-Holland Publishing Co., Amsterdam: Dotti De Miatello, I.G., Extension of actions on Stiefel manifolds (1979) Pacific J. Math., 84 (1), pp. 155-169 Eells, J., A setting for global analysis (1966) Bull.Amer.Math. Soc., 72, pp. 751-807 Ekeland, I., The Hopf-Rinow Theorem in infinite dimension (1978) J. Diff. Geometry, 13, pp. 287-301 Edelman, A., Arias, T.A., Smith, S., The geometry of algorithms with orthogonality constraints (1998) Siam. J. Matrix Anal. Appl., 20 (2), pp. 303-353 Grossman, N., Hilbert manifolds without epiconjugate points (1965) Proc. of Amer. Math. Soc., 16, pp. 1365-1371 Harms, P., Mennucci, A., Geodesics in infinite dimensional Stiefel and Grassmannian manifolds (2013) C.R. Acad. Sci. Paris, 350, pp. 773-776 Huckleberry, A., Introduction to group actions in symplectic and complex geometry (2001) Infinite dimensional Kähler manifolds (Oberwolfach, 1995), pp. 1-129. , Birkhäuser, Basel: Klingenberg, W., (1982) Riemannian geometry, , De Gruyter studies in Mathemathics, New York: Kobayashi, S., Nomizu, K., Foundations of DifferentialGeometry. vol I (1963) Wiley-Interscience Kobayashi, S., Nomizu, K., Foundations of Differential Geometry. vol II (1963) Wiley-Interscience I.M. James. The topology of Stiefel manifolds. London Mathematical Society Lecture Note Series 24, Cambridge University Press, Cambridge — New York — Melbourne, (1976), viii+168 ppLang, S., Fundamentals of DifferentialGeometry. Third Edition (1999) Graduate Texts in Mathematics, 191. , Springer-Verlang, New York: Misiolek, G., Stability of flows of ideal fluids and the geometry of the group of diffeomorphisms (1993) Indiana Univ. Math. J., 42, pp. 215-235 Misiolek, G., Conjugate points in D µ(T 2) (1996) Proc. Amer. Math. Soc., 124, pp. 977-982 Misiolek, G., The exponential map on the free loop space is Fredholm (1997) Geom. Funct. Anal., 7, pp. 1-17 Misiolek, G., Exponential maps of Sobolev metrics on loop groups (1999) Proc. Amer. Math. Soc., 127, pp. 2475-2482 Neretin, Y.A., On Jordan angles and the Triangle Inequality in Grassmann Manifolds (2001) Geom. Dedicata, 86 (1-3), pp. 81-92 Rudin, W., (1991) Functional Analysis, , McGraw-Hill, Inc., New York: Tsukamoto, Y., On Kählerian manifolds with positive holomorphic sectional curvature (1957) Proc. Japan Acad., 33, pp. 333-335 Wolf, A.J., Sur la classification des variétés riemannienes homogénes á courbure constante (1960) Comptes rendus a l’Academie des Sciences á Paris, 250, pp. 3443-3445 Wolf, A.J., (2001) Spaces of constant curvature, , AMS Chelsea Publishing, Providence, RI: Wong, Y.-C., Sectional curvatures of Grassmann manifolds (1968) Proc. Natl. Acad. Sci. USA, 60, pp. 75-79 Younes, L., Michor, P., Shah, J., Mumford, D., A metric on shape space with explicit geodesics (2008) Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl., 19 (1), pp. 25-57