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ISSN 1088-9485(e) ISSN 0273-0979(p)

Aspects of global Riemannian geometry

Author(s): Peter Petersen
Journal: Bull. Amer. Math. Soc. 36 (1999), 297-344.
MSC (1991): Primary 53C20
Posted: May 24, 1999
MathSciNet review: 1698926
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Abstract: In this article we survey some of the developments in Riemannian geometry. We place special emphasis on explaining the relationship between curvature and topology for Riemannian manifolds with lower curvature bounds.

References:

1.: U. Abresch and D. Gromoll, On complete manifolds with nonnegative Ricci curvature, J. Amer. Math. Soc. 3 (1990) 355-374. MR 91a:53071
2.: C.B. Allendoerfer and A. Weil, The Gauss-Bonnet theorem for Riemannian polyhedra, Trans. Amer. Math. Soc. 53 (1943) 101-129. MR 4:169e
3.: M.T. Anderson, Short geodesics and gravitational instantons, J. Differential Geom. 31 (1990) 265-275. MR 91b:53040
4.: M.T. Anderson, Metrics of positive Ricci curvature with large diameter, Manu. Math. 68 (1990) 405-415. MR 91g:53045
5.: M.T. Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds, Invent. Math. 102 (1990) 429-445. MR 92c:53024
6.: M.T. Anderson and J. Cheeger, Diffeomorphism finiteness for manifolds with Ricci curvature and $L^{n/2}$ curvature bounded, Geom. and Funct. Anal. 1 (1991), 231-251. MR 92h:53052
7.: W. Ballmann, Lectures on spaces of non-positive curvature, Basel: Birkhäuser, 1995. MR 97a:53053
8.: W. Ballmann, V. Schroeder and M. Gromov, Manifolds of non-positive curvature, Boston: Birkhäuser, 1985. MR 87h:53050
9.: P. Bérard, From vanishing theorems to estimating theorems: the Bochner technique revisited, Bull. Amer. Math. Soc. (N.S.) 19 (1988), 371-406. MR 89i:58152
10.: M. Berger, Riemannian geometry during the second half of the twentieth century, Jahresber. Deutsch. Math.-Verein. 100 (1998), no. 2, 45-208. CMP 98:16
11.: M. Berger, Les variétés riemanniennes à courbure positive, Bull. Soc. Math. Belg. 10 (1958) 88-104. See also Bull. Soc. Math. France 87 (1959), 285-292; C. R. Acad. Sci Paris 247 (1958), 1165-1168. MR 23:A1338; MR 20:2754
12.: M. Berger, Le variétés riemanniennes 1/4-pincées, C. R. Acad. Sci. Paris 250 (1960) 442-444. MR 22:1865
13.: M. Berger, Le variétés riemanniennes 1/4-pincées, Ann. Scuole Norm. Sup. Pisa 14 (1960), 161-170. MR 25:3478
14.: M. Berger, On the diameter of some Riemannian manifolds, preprint, Berkeley, 1962.
15.: A.L. Besse, Einstein Manifolds, Berlin-Heidelberg: Springer-Verlag, 1987. MR 88f:53087
16.: A.L. Besse, Manifolds all of whose geodesics are closed, Berlin-Heidelberg: Springer-Verlag, 1978. MR 80c:53044
17.: R.L. Bishop and R.J. Crittenden, Geometry of Manifolds, New York: Academic Press, 1964. MR 29:6401
18.: W. Blaschke, Vorlesungen über Differentialgeometrie, Berlin-Heidelberg: Springer Verlag, first edition in 1921. MR 7:391g
19.: S. Bochner, Vector fields and Ricci curvature, Bull. Amer. Math. Soc. 52 (1946), 776-797. MR 8:230a
20.: O. Bonnet, Sur quelques propriétés des lignes géodésiques, C. R. Acad. Sci. Paris 40 (1855), 1311-1313.
21.: E. Calabi, An extension of E. Hopf's maximum principle with an application to Riemannian geometry, Duke Math. J. 25 (1958), 45-56. MR 19,1056e
22.: E. Cartan, Geometry of Riemannian spaces, Boston: Math. Sci. Press, 1983. First French edition in 1925 and second French edition 1951. MR 85m:53001
23.: I. Chavel, Riemannian Geometry-A Modern Introduction, New York: Cambridge University Press, 1993. MR 95j:53001
24.: J. Cheeger, Comparison and finiteness theorems for Riemannian manifolds, Ph.D. thesis, Princeton University, 1967.
25.: J. Cheeger, Finiteness theorems for Riemannian manifolds, Amer. J. Math. 92 (1970), 61-74. MR 41:7697
26.: J. Cheeger and T.H. Colding, Lower bounds on Ricci curvature and almost rigidity of warped products, Ann. Math. 144 (1996), 189-237. MR 97h:53038
27.: J. Cheeger and T.H. Colding, On the structure of spaces with Ricci curvature bounded below I, J. Differential Geom. 46 (1997), 406-480. MR 98k:53044
28.: J. Cheeger and D. Ebin, Comparison theorems in Riemannian geometry, North-Holland, 1975. MR 56:16538
29.: J. Cheeger, Critical points of distance functions and applications to geometry, Geometric Topology: Recent developments, Lecture Notes in Math., 1504, Berlin-Heidelberg: Springer-Verlag, 1991. MR 94a:53075
30.: J. Cheeger and D. Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differential Geometry 6 (1971) 119-128. MR 46:2597
31.: J. Cheeger and D. Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann. Math. 96 (1972), 413-443. MR 46:8121
32.: J. Cheeger, K. Fukaya, and M. Gromov, Nilpotent structures and invariant metrics on collapsed manifolds, J. Amer. Math. Soc. 5 (1992), 372-372. MR 93a:53036
33.: S.S. Chern, ed., Global Differential Geometry, 2nd edition, MAA Studies 27, Washington: Mathematical Association of America, 1989. MR 90d:53003
34.: B. Chow and D. Yang, Rigidity of nonnegatively curved compact quaternionic-Kähler manifolds, J. Differential Geom. 29 (1989), 361-372. MR 91f:53065
35.: S. Cohn-Vossen, Kützeste Wege und Totalkrümming auf Flächen, Comp. Math. 2 (1935), 69-133.
36.: T.H. Colding, Ricci curvature and volume convergence, Ann. Math. 145 (1997), 477-501. MR 98d:53050
37.: D. DeTurk and J. Kazdan, Some regularity theorems in Riemannian geometry, Ann. Sci. École Norm. Sup. (4) 14 (1981) 249-260. MR 83f:53018
38.: P.A.M. Dirac, General Theory of Relativity, First published by Wiley in 1975, reprinted by Princeton University Press 1996. MR 97a:83002
39.: P. Dombrowski, 150 years after Gauss' ``Disquisitiones generales circa superficies curvas'', Asterique 62, 1979. MR 80g:01013
40.: W. Dyck, Beitrage zur Analysis Situs I. Ein- und zwei-dimensionale Mannifaltigkeiten, Math. Ann. 32 (1888), 457-512.
41.: P.B. Eberlein, Geometry of nonpositively curved manifolds, Chicago Lectures in Mathematics, Chicago University Press, 1996. MR 98h:53002
42.: J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc. 10 (1978), 1-68. MR 82b:58033
43.: J.-H. Eschenburg, Local convexity and nonnegative curvature-Gromov's proof of the sphere theorem, Invent. Math. 84 (1986) 507-522. MR 87j:53080
44.: J.-H. Eschenburg and E. Heintze, An elementary proof of the Cheeger-Gromoll splitting theorem, Ann. Global Anal. and Geom. 2 (1984) 141-151. MR 86h:53042
45.: K. Fukaya, Hausdorff convergence of Riemannian manifolds and its applications, Advanced Studies in Pure Math. 18-I (1990) Recent topics in Differential and Analytic Geometry pp. 143-238. MR 92k:53076
46.: S. Gallot, Isoperimetric inequalities based on integral norms of Ricci curvature, Astérisque, 157-158 (1988), 191-216. MR 90a:58179
47.: S. Gallot, D. Hulin and J. Lafontaine, Riemannian Geometry, Berlin-Heidelberg: Springer-Verlag, 1987. MR 88k:53001
48.: S. Gallot and D. Meyer, Opérateur de courbure et laplacien des formes différentielles d'une variété riemannienne, J. Math. Pures Appl. 54 (1975), 259-284. MR 56:13128
49.: W.-D. Geyer, ed., Jahresbericht der Deutschen Mathematiker-Vereinigung: Jubiläumstagung, 100 Jahre DMV, Bremen 1990, Stuttgart : B.G. Teubner, 1992.
50.: G. Gibbons and S. Hawking, Gravitational multi-instantons, Phys. Lett. B 78 (1978) 430-432.
51.: D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Berlin-Heidelberg: Springer-Verlag, 1983. MR 86c:35035
52.: R. Greene and H.-H. Wu, Lipschitz convergence of Riemannian manifolds, Pac. J. Math. 131 (1988), 119-141. MR 90j:53061
53.: R. Greene and S.T. Yau, eds., Differential Geometry, Proc. Symp. Pure Math. 54, vol. 3, Amer. Math. Soc. (1993). MR 94a:00012
54.: D. Gromoll, W. Klingenberg, and W. Meyer, Riemansche Geometrie im Grossen, Lecture Notes in Mathematics, 55, Berlin-Heidelberg: Springer-Verlag, 1968 and 1975. MR 51:1651
55.: D. Gromoll and W. Meyer, On complete open manifolds of positive curvature, Ann. Math. 90 (1969), 74-90. MR 40:854
56.: M. Gromov, Almost flat manifolds, J. Differential Geom. 13 (1978), 231-241. MR 80h:53041
57.: M. Gromov, Manifolds of negative curvature, J. Differential Geom. 13 (1978), 223-230. MR 80h:53040
58.: M. Gromov, Curvature, diameter, and Betti numbers, Comment. Math. Helv. 56 (1981), 179-195. MR 82k:53062
59.: M. Gromov, Structures métriques pour les variétés riemanniennes (J. Lafontaine and P. Pansu, eds.) Paris: Cedic/Fernand Nathan, 1981. MR 85e:53051
60.: M. Gromov and W. Thurston, Pinching constants for hyperbolic manifolds, Invent. Math. 89 (1987) 1-12. MR 88e:53058
61.: K. Grove, H. Karcher, and E. Ruh, Group actions and curvature, Invent. Math. 23 (1974), 31-48. MR 52:6609
62.: K. Grove and P. Petersen, eds., Comparison Geometry, MSRI publications vol. 30, New York: Cambridge University Press, 1997. MR 97m:53001
63.: K. Grove and P. Petersen, Bounding homotopy types by geometry, Ann. Math. (1988), 195-206. MR 90a:53044
64.: K. Grove, P. Petersen, and J.-Y. Wu, Geometric finiteness theorems via controlled topology, Invent. Math. 99 (1990) 205-213, Erratum Invent. Math. 104 (1991) 221-222.MR 90k:53075; MR 92b:53065
65.: K. Grove and K. Shiohama, A generalized sphere theorem, Ann. Math. (1977), 201-211. MR 58:18268
66.: J. Hadamard, Les surfaces à courbures opposées, J. Math. Pures Appl. 4 (1889), 27-73.
67.: R.S. Hamilton, The formation of singularities in the Ricci flow, Surveys in Differential Geometry, Vol. II, International Press (1995), 7-136. MR 97e:53075
68.: R.S. Hamilton, Four-manifolds with positive isotropic curvature, Comm. Anal. Geom. 5 (1997), 1-92. MR 99e:53049
69.: S.W. Hawking and G.F.R Ellis, The large scale structure of space-time, Cambridge University Press, 1973. MR 54:12154
70.: E. Heintze and H. Karcher, A general comparison theorem with applications to volume estimates for submanifolds, Ann. Sci. École Norm. Sup. 11 (1978), 451-470. MR 80i:53026
71.: W.V.D. Hodge, The existence theorem for harmonic integrals, Proc. London Math. Soc. 41 (1936), 483-496.
72.: H. Hopf, Zum Clifford-Kleinschen Raumproblem, Math. Ann. 95 (1926), 313-339.
73.: H. Hopf, Über die Curvatura integra geschlossener Hyperflächen, Math. Ann. 95 (1926), 340-367.
74.: H. Hopf and W. Rinow, Über den Begriff der vollständigen differential-geometrischen Flächen, Comment. Math. Helv. 3 (1931), 209-225.
75.: J. Jost and H. Karcher, Geometrische Methoden zur Gewinnung von a-priori-Schranken für harmonische Abbildungen, Manuscripta Math. 40 (1982) 27-77. MR 84e:58023
76.: A. Kasue, A convergence theorem for Riemannian manifolds and some applications, Nagoya Math. J. 114 (1989), 21-51. MR 90g:53053
77.: P. Kirby and L. Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, Ann. Math. Stud. 88, Princeton Univ. Press (1977). MR 58:31082
78.: W. Klingenberg, Contributions to Riemannian geometry in the large, Ann. Math. 69 (1959), 654-666. MR 21:4445
79.: W. Klingenberg, Über Riemannsche Mannigfaltigheiten mit positiver Krümmung, Comment. Math. Helv. 35 (1961), 47-54. MR 25:2559
80.: W. Klingenberg, Riemannian Geometry, de Gruyter Studies in Mathematics, 1, Berlin, 1982. MR 84j:53001
81.: L. Kronecker, Über Systeme von Funktionen mehrer Variablen, Monatsberichte Berlin Akad. (1869), 159-193 and 688-698.
82.: H.B. Lawson Jr. and M.-L. Michelsohn, Spin Geometry, Princeton: Princeton University Press, 1989. MR 91g:53001
83.: C. Lanczos, Ein vereinfachendes Koordinatensystem für die Einsteinschen Gravitationsgleichungen, Phys. Z. 23 (1922) 537-539.
84.: J.M. Lee, Riemannian Manifolds, Graduate Texts in Mathematics, 176, Berlin-Heidelberg: Springer Verlag (1997). MR 98d:53001
85.: T. Levi-Civita, The absolute differential calculus, London and Glasgow: Blackie and Son, first Italian ed. 1923, English ed. in 1926, and revised ed. in 1961.
86.: A. Lichnerowicz, Géométrie des groupes de transformations, Travaux et Recherches Mathématiques, III. Dunod, Paris, 1958. MR 23:A1329
87.: A. Lichnerowicz, Propagateurs et Commutateurs en relativité générale, Inst. Hautes Études Sci. Publ. Math. No. 10 (1961) 293-343. MR 28:967
88.: A. Lichnerowicz, Spineurs harmoniques, C. R. Acad. Sci. Paris 257 (1963), 7-9. MR 27:6218
89.: H. v. Mangoldt, Über diejenigen Punkte aur positiv gekrummten Flächen, welche die Eigenschaft haben, dass die von ihnen ausgehenden geodätischen Linien nie aufhören, kürzeste Linien zu sien, J. Reine Angew. Math. 91 (1881), 23-52.
90.: M.J. Micallef and J.D. Moore, Minimal 2-spheres and the topology of manifolds with positive curvature on totally isotropic 2-planes, Ann. of Math. 127 (1988), 199-227. MR 89e:53088
91.: M.J. Micallef and M.Y. Wang, Metrics with nonnegative isotropic curvature, Duke Math. J. 72 (1993), 649-672. MR 94k:53052
92.: J.W. Milnor, Morse Theory, Princeton University Press, Princeton, N.J., 1963. MR 29:634
93.: C.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation, W. H. Freeman and Co., San Francisco, Calif., 1973. MR 54:6869
94.: N. Mok, The uniformization theorem for compact Kähler manifolds of non-negative holomorphic bi-sectional curvature, J. Diff. Geom. 27 (1988), 179-214. MR 89d:53115
95.: J.W. Morgan, The Seiberg-Witten equations and applications to the topology of smooth four-manifolds, Princeton Univ. Press, Princeton, N.J., 1996. MR 97d:57042
96.: M. Morse, Relations between the critical points of a real analytic function of n independent variables, Trans. Amer. Math. Soc. 27 (1925) 345-396.
97.: M. Morse, The calculus of variations in the large, AMS Coll. Publ. 18, New York, 1934.
98.: S.B. Myers, Riemannian manifolds in the large, Duke Math. J. 1 (1935), 39-49.
99.: S.B. Myers, Riemannian manifolds with positive mean curvature, Duke Math. J. 8 (1941) 401-404. MR 3,18f
100.: S.B. Myers and N. Steenrod, The group of isometries of a Riemannian manifold, Ann. Math. 40 (1939), 400-416.
101.: P. Nabonnand, Contribution à l'histoire de la théorie des géodésiques au XIX $^{e}$ siècle, Rev. Histoire Math. 1 (1995) 159-200. MR 96j:01027
102.: I.G. Nikolaev, Parallel translation and smoothness of the metric of spaces of bounded curvature, Dokl. Akad. Nauk SSSR 250 (1980), 1056-1058. MR 81d:53052
103.: B. O'Neill, Semi-Riemannian Geometry, Pure and Applied Mathematics, 103, Academic Press, New York-London, 1983. MR 85f:53002
104.: Y. Otsu, On manifolds of positive Ricci curvature with large diameters, Math. Z. 206 (1991) 255-264. MR 91m:53033
105.: M. Özaydin and G. Walschap, Vector bundles with no soul, Proc. Amer. Math. Soc 120 (1994) 565-567. MR 94d:53057
106.: R. Penrose, Techniques of differential topology in relativity, CBMS-NSF Regional Conference Series in Applied Mathematics 7, 1973. MR 57:8942
107.: G. Perel'man, Proof of the soul conjecture of Cheeger and Gromoll, J. Diff. Geom. 40 (1994), 209-212. MR 95d:53037
108.: G. Perel'man, Alexandrov's spaces with curvatures bounded from below II, preprint.
109.: G. Perel'man, Manifolds of positive Ricci curvature with almost maximal volume, J. Amer. Math. Soc. 7 (1994) 299-305. MR 94f:53077
110.: S. Peters, Cheeger's finiteness theorem for diffeomorphism classes of manifolds, J. Reine Angew. Math. 349 (1984) 77-82. MR 85j:53046
111.: S. Peters, Convergence of Riemannian manifolds, Compositio. Math. 62 (1987), 3-16. MR 88i:53076
112.: P. Petersen, Riemannian geometry, Graduate Texts in Mathematics, 171, Springer-Verlag, New York, 1998. MR 98m:53001
113.: P. Petersen, S. Shteingold and G. Wei, Comparison geometry with integral curvature bounds, Geom. Funct. Anal. 7 (1997), 1011-1030. MR 99c:53022
114.: P. Petersen and C. Sprouse, Integral curvature bounds, distance estimates and applications, J. Diff. Geom. 50 (1998), 269-298.
115.: P. Petersen and G. Wei, Relative volume comparison with integral Ricci curvature bounds, Geom. Funct. Anal. 7 (1997) 1031-1045. MR 99c:53023
116.: P. Petersen and G. Wei, Geometry and analysis on manifolds with integral Ricci curvature bounds, preprint from UCLA. To appear in Trans. Amer. Math. Soc.
117.: P. Petersen, G. Wei, and R. Ye, Controlled geometry via smoothing, to appear in Comm. Math. Helv.
118.: A. Preissmann, Quelques propriétés globales des espaces de Riemann, Comment. Math. Helv. 15 (1943), 175-216. MR 6:20g
119.: H.E. Rauch, A contribution to differential geometry in the large, Ann. Math. 54 (1951), 38-55. MR 13:159b
120.: G. de Rham, Differentiable Manifolds, Berlin-Heidelberg: Springer-Verlag, 1984. MR 85m:58005
121.: B. Riemann, Über die Hypothesen welche der Geometrie zu Grunde liegen, Habilitationschrift 1854; Gött. Abh. 13, 1868; Gesamm. Werke, 2nd ed. Leipzig, 1892. Edited with comments by H. Weyl, Berlin, 1919 and revised in 1923.
122.: X. Rong, The almost cyclicity of the fundamental groups of positively curved manifolds, Invent. Math. 126 (1996) 47-64. And Positive curvature, local and global symmetry, and fundamental groups, preprint, Rutgers, New Brunswick. MR 97g:53045
123.: X. Rong, A Bochner Theorem and Applications, Duke Math. J. 91 (1998), 381-392. MR 99e:53043
124.: E. Ruh, Almost flat manifolds, J. Diff. Geom. 17 (1982), 1-14. MR 84a:53047
125.: I.J. Schoenberg, Some applications of the calculus of variations to Riemannian geometry, Ann. Math. 33 (1932) 485-495.
126.: K. Shiohama, T. Sakai and T. Sunada, Eds., Curvature and Topology of Riemannian Manifolds, Lecture Notes in Mathematics 1201, Berlin-Heidelberg: Springer-Verlag, 1986. MR 87i:53001
127.: M. Spivak, A Comprehensive Introduction to Differential Geometry, vols. I-V, Wilmington: Publish or Perish, 1979. MR 82g:53003a; MR 82g:53003b; MR 82g:53003c; MR 82g:53003d; MR 82g:53003e
128.: J.J. Stoker, Differential Geometry, New York: Wiley-Interscience, 1989. MR 90f:53002
129.: S. Stolz, Simply connected manifolds with positive scalar curvature, Ann. Math. 136 (1992), 511-540. MR 93i:57033
130.: J.L. Synge, The first and second variations of the length-integral in Riemann spaces, Proc. London Math. Soc. 25 (1926), 247-264.
131.: J.L. Synge, On the neighborhood of a geodesic in Riemannian space, Duke Math. J. 1 (1935), 527-537.
132.: J.L. Synge, On the connectivity of spaces of positive curvature, Quart. J. Math. 7 (1936), 316-320.
133.: S. Tachibana, A theorem on Riemannian manifolds of positive curvature operator, Proc. Japan Acad. 50 (1974), 301-302. MR 51:1667
134.: V.A. Toponogov, Riemannian spaces having their curvature bounded below by a positive number, Uspehi Mat. Nauk 14 (1959), 87-130; Dokl. Akad. Nauk SSSR 120 (1958), 719-721. MR 21:2278; MR 20:6139
135.: V.A. Toponogov, Dependence between curvature and topological structure of Riemann spaces of even dimension, Soviet Math. Dokl. 1 (1961), 943-945. MR 25:3476
136.: V.A. Toponogov, The metric structure of Riemannian spaces of nonnegative curvature which contain straight lines, Sibirsk. Mat. Z. 5 (1964), 1358-1369; Dokl. Akad. Nauk. SSSR 127 (1959), 977-979. MR 32:3017; MR 21:7520
137.: F.W. Warner, Foundations of Differentiable Manifolds and Lie Groups, New York: Springer-Verlag, 1983. MR 84k:58001
138.: A. Weinstein, On the homotopy type of positively pinched manifolds, Archiv Math. 18 (1967), 523-524. MR 36:3376
139.: R. Weitzenbock, Invariantentheorie, Groningen: Nordhoff, 1923.
140.: F.H. Wilhelm, On intermediate Ricci curvature and the fundamental group, Illinois J. of Math. 41 (1997), 488-494. MR 98h:53065
141.: H.H. Wu, The Bochner Technique in Differential Geometry, Mathematical Reports vol. 3, part 2, London: Harwood Academic Publishers, 1988.MR 91h:58031

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