메뉴 건너뛰기




Volumn 56, Issue 1, 2000, Pages 1-66

Collapsing three-manifolds under a lower curvature bound

Author keywords

Alexandrov spaces; Seifert fibred spaces; The Gromov Hausdorff convergence; Topology of three manifolds

Indexed keywords


EID: 0039015730     PISSN: 0022040X     EISSN: 1945743X     Source Type: Journal    
DOI: 10.4310/jdg/1090347524     Document Type: Article
Times cited : (52)

References (40)
  • 2
    • 0003331011 scopus 로고
    • Manifolds of nonpositive curvature
    • Birkhauser, Boston
    • W. Ballmann, M. Gromov & V. Schroeder, Manifolds of nonpositive curvature, Progr. Math., Birkhauser, Boston, Vol. 61, 1985.
    • (1985) Progr. Math , vol.61
    • Ballmann, W.1    Gromov, M.2    Schroeder, V.3
  • 3
    • 84879168733 scopus 로고
    • A proof of the generalized schoenflies theorem
    • M. Brown, A proof of the generalized schoenflies theorem, Bull. Amer. Math. Soc. 66 (1960) 74-76.
    • (1960) Bull. Amer. Math. Soc , vol.66 , pp. 74-76
    • Brown, M.1
  • 4
    • 0001787207 scopus 로고
    • A. D. Aleksandrov spaces with curvatures bounded below
    • 222
    • Yu. Burago, M. Gromov & G. Perel’man, A. D. Aleksandrov spaces with curvatures bounded below, Uspekhi Mat. Nauk 47 (1992) (284), 3-51, 222
    • (1992) Uspekhi Mat. Nauk , vol.47 , Issue.284 , pp. 3-51
    • Burago, Y.U.1    Gromov, M.2    Perel’Man, G.3
  • 7
    • 0004268210 scopus 로고
    • Interscience Tracts in Pure and Applied Math., No. 6, Interscience Publishers, New York, London
    • H. Busemann, Convex surfaces, Interscience Tracts in Pure and Applied Math., No. 6, Interscience Publishers, New York, London, 1958.
    • (1958) Convex Surfaces
    • Busemann, H.1
  • 8
    • 0003232148 scopus 로고
    • Comparison theorems in Riemannian geometry
    • Amsterdam, North-Holland Math. Library
    • J. Cheeger & D. G. Ebin, Comparison theorems in Riemannian geometry, North-Holland Publishing Co., Amsterdam, North-Holland Math. Library, Vol. 9, 1975.
    • (1975) North-Holland Publishing Co , vol.9
    • Cheeger, J.1    Ebin, D.G.2
  • 9
    • 84968509403 scopus 로고
    • Nilpotent structures and invariant metrics on collapsed manifolds
    • J. Cheeger, K. Fukaya & M. Gromov, Nilpotent structures and invariant metrics on collapsed manifolds, J. Amer. Math. Soc. 5 (1992) 327-372.
    • (1992) J. Amer. Math. Soc , vol.5 , pp. 327-372
    • Cheeger, J.1    Fukaya, K.2    Gromov, M.3
  • 10
    • 0001265427 scopus 로고
    • On the structure of complete manifolds of nonnegative curvature
    • J. Cheeger & D. Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann. of Math. 96 (1972) 413-443.
    • (1972) Ann. of Math , vol.96 , pp. 413-443
    • Cheeger, J.1    Gromoll, D.2
  • 11
    • 84972538920 scopus 로고
    • Collapsing Riemannian manifolds while keeping their curvature bounded. I
    • J. Cheeger & M. Gromov, Collapsing Riemannian manifolds while keeping their curvature bounded. I, J. Differential Geom. 23 (1986) 309-346.
    • (1986) J. Differential Geom , vol.23 , pp. 309-346
    • Cheeger, J.1    Gromov, M.2
  • 12
    • 84972508732 scopus 로고
    • Collapsing Riemannian manifolds while keeping their curvature bounded. II
    • J. Cheeger & M. Gromov, Collapsing Riemannian manifolds while keeping their curvature bounded. II, J. Differential Geom. 32 (1990) 269-298.
    • (1990) J. Differential Geom , vol.32 , pp. 269-298
    • Cheeger, J.1    Gromov, M.2
  • 13
    • 84972535799 scopus 로고
    • A boundary of the set of the Riemannian manifolds with bounded curvatures and diameters
    • K. Fukaya, A boundary of the set of the Riemannian manifolds with bounded curvatures and diameters, J. Differential Geom. 28 (1988) 1-21.
    • (1988) J. Differential Geom , vol.28 , pp. 1-21
    • Fukaya, K.1
  • 14
    • 0000518663 scopus 로고
    • The fundamental groups of almost nonnegatively curved mamfold.S
    • K. Fukaya and T. Yamaguchi, The fundamental groups of almost nonnegatively curved m, a, mfold,.s, Ann. of Math. 136 (1992) 253-333.
    • (1992) Ann. of Math , vol.136 , pp. 253-333
    • Fukaya, K.1    Yamaguchi, T.2
  • 15
    • 51249162326 scopus 로고
    • Isometry groups of singular spaces
    • K. Fukaya and T. Yamaguchi, Isometry groups of singular spaces, Math. Z. 216 (1994) 31-44.
    • (1994) Math. Z , vol.216 , pp. 31-44
    • Fukaya, K.1    Yamaguchi, T.2
  • 16
    • 0011318950 scopus 로고
    • A generalization of Berger’s rigidity theorem for positively curved manifolds
    • D. Gromoll & K. Grove, A generalization of Berger’s rigidity theorem for positively curved manifolds, Ann. Sci. École Norm. Sup. 20 (1987) 227-239.
    • (1987) Ann. Sci. École Norm. Sup , vol.20 , pp. 227-239
    • Gromoll, D.1    Grove, K.2
  • 17
    • 0038495726 scopus 로고
    • Structures métriques pour les variétés riemanniennes
    • CéDIC, Paris, (Éds. J. Lafontaine and P. Pansu)
    • M. Gromov, Structures métriques pour les variétés riemanniennes, Textes Math. (Math. Texts), Vol. 1, CéDIC, Paris, 1981, (Éds. J. Lafontaine and P. Pansu).
    • (1981) Textes Math. (Math. Texts) , vol.1
    • Gromov, M.1
  • 19
    • 0007378451 scopus 로고
    • A radius sphere theorem, Invent
    • K. Grove & P. Petersen, A radius sphere theorem, Invent. Math. 112 (1993) 577-583.
    • (1993) Math , vol.112 , pp. 577-583
    • Grove, K.1    Petersen, P.2
  • 20
    • 0000976022 scopus 로고
    • Geometric finiteness theorems via controlled topology
    • K. Grove, P. Petersen & J.-Y. Wu, Geometric finiteness theorems via controlled topology, Invent. Math. 99 (1990) 205-213
    • (1990) Invent. Math , vol.99 , pp. 205-213
    • Grove, K.1    Petersen, P.2    Wu, J.-Y.3
  • 21
    • 0010717327 scopus 로고
    • Geometric finiteness theorems via controlled topology
    • K. Grove, P. Petersen & J.-Y. Wu, Geometric finiteness theorems via controlled topology Érratum in Invent. Math. 104 (1991) 221-222.
    • (1991) Érratum in Invent. Math , vol.104 , pp. 221-222
    • Grove, K.1    Petersen, P.2    Wu, J.-Y.3
  • 22
    • 84972555746 scopus 로고
    • Four-manifolds with positive curvature operator
    • R. S. Hamilton, Four-manifolds with positive curvature operator, J. Differential Geom. 24 (1986) 153-179.
    • (1986) J. Differential Geom , vol.24 , pp. 153-179
    • Hamilton, R.S.1
  • 23
    • 0003309519 scopus 로고
    • 3-manifolds
    • Princeton Univ. Press, Princeton
    • J. Hempel, 3-manifolds, Ann. of Math. Stud., Vol. 86, Princeton Univ. Press, Princeton, 1976.
    • (1976) Ann. of Math. Stud , vol.86
    • Hempel, J.1
  • 24
    • 0001446871 scopus 로고
    • A compactification of a manifold with asymptotically nonnegative curvature
    • A. Kasue, A compactification of a manifold with asymptotically nonnegative curvature, Ann. Sci. École Norm. Sup. 21 (1988) 593-622.
    • (1988) Ann. Sci. École Norm. Sup , vol.21 , pp. 593-622
    • Kasue, A.1
  • 25
    • 84996717708 scopus 로고    scopus 로고
    • The Gaussian curvature of Alexandrov surfaces
    • Y. Machigashira, The Gaussian curvature of Alexandrov surfaces, J. Math. Soc. Japan 50 (1998) 859-878.
    • (1998) J. Math. Soc. Japan , vol.50 , pp. 859-878
    • Machigashira, Y.1
  • 26
    • 0003348894 scopus 로고
    • Seifert manifolds
    • Springer, Berlin, No. 291
    • P. Orlik, Seifert manifolds, Lecture Notes in Math., Springer, Berlin, No. 291, 1972.
    • (1972) Lecture Notes in Math
    • Orlik, P.1
  • 27
    • 84972500357 scopus 로고
    • The Riemannian structure of Alexandrov spaces
    • Y. Otsu & T. Shioya, The Riemannian structure of Alexandrov spaces, J. Differ-ential Geom. 39 (1994) 629-658.
    • (1994) J. Differ-Ential Geom , vol.39 , pp. 629-658
    • Otsu, Y.1    Shioya, T.2
  • 29
    • 0013532801 scopus 로고
    • Elements of Morse theory on Alexandrov spaces
    • G. Perelman, Elements of Morse theory on Alexandrov spaces, St. Petersburg Math. J. 5 (1994) 207-214.
    • (1994) St. Petersburg Math. J , vol.5 , pp. 207-214
    • Perelman, G.1
  • 30
    • 0039618553 scopus 로고
    • Collapsing with no proper extremal subsets, Comparison geometry
    • Berke-ley, CA, Cambridge Univ. Press, Cam-bridge
    • G. Perelman, Collapsing with no proper extremal subsets, Comparison geometry (Berke-ley, CA, 1993-1994), Math. Sci. Res. Inst. Publ., Cambridge Univ. Press, Cam-bridge, Vol. 30, 1997, 149-155.
    • (1993) Math. Sci. Res. Inst. Publ , vol.30 , pp. 149-155
    • Perelman, G.1
  • 31
    • 0038292676 scopus 로고
    • The geometries of 3-manifolds
    • P. Scott, The geometries of 3-manifolds, Bull. London Math. Soc. 15 (1983) 401-487.
    • (1983) Bull. London Math. Soc , vol.15 , pp. 401-487
    • Scott, P.1
  • 32
    • 0041044884 scopus 로고
    • Lecture Notes Ser., No. 8, Res. Institute of Math., Global Anal. Res. Center, Seoul National Univ., Seoul, Korea
    • K. Shiohama, An introduction to the geometry of Alexandrov spaces, Lecture Notes Ser., No. 8, Res. Institute of Math., Global Anal. Res. Center, Seoul National Univ., Seoul, Korea, 1992, 151-742.
    • (1992) An Introduction to the Geometry of Alexandrov Spaces , pp. 151-742
    • Shiohama, K.1
  • 33
    • 0039618558 scopus 로고
    • Behavior of distant maximal geodesics in finitely connected complete 2-dimensional Riemannian manifolds
    • T. Shioya, Behavior of distant maximal geodesics in finitely connected complete 2-dimensional Riemannian manifolds, Mem. Amer. Math. Soc. 108 (1994) No. 517, x+73.
    • (1994) Mem. Amer. Math. Soc , vol.108 , Issue.517 , pp. 73
    • Shioya, T.1
  • 34
    • 51249162132 scopus 로고
    • Mass of rays in Alexandrov spaces of nonnegative curvature
    • T. Shioya, Mass of rays in Alexandrov spaces of nonnegative curvature, Comment. Math. Helv. 69 (1994) 208-228.
    • (1994) Comment. Math. Helv , vol.69 , pp. 208-228
    • Shioya, T.1
  • 35
    • 51249189714 scopus 로고
    • Deformation of homeomorphisms on stratified sets. I, II
    • L. C. Siebenmann, Deformation of homeomorphisms on stratified sets. I, II, Comment. Math. Helv. 47 (1972) 123-163.
    • (1972) Comment. Math. Helv , vol.47 , pp. 123-163
    • Siebenmann, L.C.1
  • 37
    • 0003967369 scopus 로고    scopus 로고
    • Three-dimensional geometry and topology
    • Princeton Univ. Press, Princeton, Edited by Silvio Levy
    • W. P. Thurston, Three-dimensional geometry and topology. Vol. 1, Princeton Math. Ser., Princeton Univ. Press, Princeton, Vol. 35, 1997, Edited by Silvio Levy.
    • (1997) Princeton Math. Ser , vol.1-35
    • Thurston, W.P.1
  • 38
    • 0004270735 scopus 로고
    • fifth ed., Publish or Perish, Inc., Houston, Tex
    • J. Wolf, Spaces of constant curvature, fifth ed., Publish or Perish, Inc., Houston, Tex., 1984.
    • (1984) Spaces of Constant Curvature
    • Wolf, J.1
  • 39
    • 0000269280 scopus 로고
    • And pinching under a lower curvature bound
    • T. Yamaguchi, Collapsing and pinching under a lower curvature bound, Ann. of Math. 133 (1991) 317-357.
    • (1991) Ann. of Math , vol.133 , pp. 317-357
    • Yamaguchi, T.1
  • 40
    • 0001164183 scopus 로고    scopus 로고
    • A convergence theorem in the geometry of Alexandrov spaces
    • (Luminy, 1992), Semin. Congr, Soc. Math. France, Paris
    • T. Yamaguchi, A convergence theorem in the geometry of Alexandrov spaces, Actes de la Table Ronde de Géometrié Difébrentielle (Luminy, 1992), Semin. Congr., Vol. 1, Soc. Math. France, Paris, 1996, 601-642.
    • (1996) Actes De La Table Ronde De Geometrie Difíbrentielle , vol.1 , pp. 601-642
    • Yamaguchi, T.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.