A panoramic glimpse of manifolds with sectional curvature bounded from below

Grove, K.

doi:10.1090/spmj/1479

A panoramic glimpse of manifolds with sectional curvature bounded from below
HTML articles powered by AMS MathViewer

by K. Grove

St. Petersburg Math. J. 29 (2018), 3-31

DOI: https://doi.org/10.1090/spmj/1479

Published electronically: December 27, 2017

PDF | Request permission

Abstract:

Rather than providing a comprehensive survey on manifolds curved from below, the paper is aimed at exhibiting and discussing some of the main ideas and tools that have been developed over decades. For the same reason, only a relatively small sample of results is presented to illustrate this development and in doing this, simplicity is emphasized over generality. In the same vein, at most a glimpse of an idea or strategy of a proof is given.

References

Marcos M. Alexandrino, On polar foliations and the fundamental group, Results Math. 60 (2011), no. 1-4, 213–223. MR 2836894, DOI 10.1007/s00025-011-0171-4
Marcos M. Alexandrino and Dirk Töben, Singular Riemannian foliations on simply connected spaces, Differential Geom. Appl. 24 (2006), no. 4, 383–397. MR 2231053, DOI 10.1016/j.difgeo.2005.12.005
Simon Aloff and Nolan R. Wallach, An infinite family of distinct $7$-manifolds admitting positively curved Riemannian structures, Bull. Amer. Math. Soc. 81 (1975), 93–97. MR 370624, DOI 10.1090/S0002-9904-1975-13649-4
Manuel Amann and Lee Kennard, Topological properties of positively curved manifolds with symmetry, Geom. Funct. Anal. 24 (2014), no. 5, 1377–1405. MR 3261629, DOI 10.1007/s00039-014-0300-9
Ya. V. Bazaĭkin, On a family of $13$-dimensional closed Riemannian manifolds of positive curvature, Sibirsk. Mat. Zh. 37 (1996), no. 6, 1219–1237, ii (Russian, with Russian summary); English transl., Siberian Math. J. 37 (1996), no. 6, 1068–1085. MR 1440379, DOI 10.1007/BF02106732
Ya. V. Bazaĭkin, A manifold with positive sectional curvature and fundamental group $Z_3\oplus Z_3$, Sibirsk. Mat. Zh. 40 (1999), no. 5, 994–996, i (Russian, with Russian summary); English transl., Siberian Math. J. 40 (1999), no. 5, 834–836. MR 1726845, DOI 10.1007/BF02674713
Igor Belegradek and Vitali Kapovitch, Obstructions to nonnegative curvature and rational homotopy theory, J. Amer. Math. Soc. 16 (2003), no. 2, 259–284. MR 1949160, DOI 10.1090/S0894-0347-02-00418-6
L. Berard-Bergery, Les variétés riemanniennes homogènes simplement connexes de dimension impaire à courbure strictement positive, J. Math. Pures Appl. (9) 55 (1976), no. 1, 47–67 (French). MR 417987
V. N. Berestovskiĭ, Spaces with bounded curvature and distance geometry, Sibirsk. Mat. Zh. 27 (1986), no. 1, 11–25, 197 (Russian). MR 847410
V. N. Berestovskii and Luis Guijarro, A metric characterization of Riemannian submersions, Ann. Global Anal. Geom. 18 (2000), no. 6, 577–588. MR 1800594, DOI 10.1023/A:1006683922481
M. Berger, Les variétés riemanniennes homogènes normales simplement connexes à courbure strictement positive, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 15 (1961), 179–246 (French). MR 133083
M. Berger and R. Bott, Sur les variétés à courbure strictement positive, Topology 1 (1962), 301–311 (French). MR 146776, DOI 10.1016/0040-9383(62)90017-4
Renato G. Bettiol and Ricardo A. E. Mendes, Strongly nonnegative curvature, Math. Ann. 368 (2017), no. 3-4, 971–986. MR 3673642, DOI 10.1007/s00208-016-1457-3
Keith Burns and Gabriel P. Paternain, On the growth of the number of geodesics joining two points, International Conference on Dynamical Systems (Montevideo, 1995) Pitman Res. Notes Math. Ser., vol. 362, Longman, Harlow, 1996, pp. 7–20. MR 1460794
Ju. D. Burago, Three-dimensional open Riemannian spaces of nonnegative curvature, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 66 (1976), 103–113, 207 (Russian, with English summary). Studies in Topology, II. MR 0445426
Dmitri Burago, Yuri Burago, and Sergei Ivanov, A course in metric geometry, Graduate Studies in Mathematics, vol. 33, American Mathematical Society, Providence, RI, 2001. MR 1835418, DOI 10.1090/gsm/033
Yu. Burago, M. Gromov, and G. Perel′man, A. D. Aleksandrov spaces with curvatures bounded below, Uspekhi Mat. Nauk 47 (1992), no. 2(284), 3–51, 222 (Russian, with Russian summary); English transl., Russian Math. Surveys 47 (1992), no. 2, 1–58. MR 1185284, DOI 10.1070/RM1992v047n02ABEH000877
Ju. D. Burago and V. A. Zalgaller, Convex sets in Riemannian spaces of nonnegative curvature, Uspehi Mat. Nauk 32 (1977), no. 3(195), 3–56, 247 (Russian). MR 0464119
Jeff Cheeger, Finiteness theorems for Riemannian manifolds, Amer. J. Math. 92 (1970), 61–74. MR 263092, DOI 10.2307/2373498
Jeff Cheeger, Some examples of manifolds of nonnegative curvature, J. Differential Geometry 8 (1973), 623–628. MR 341334
Jeff Cheeger and David G. Ebin, Comparison theorems in Riemannian geometry, North-Holland Mathematical Library, Vol. 9, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. MR 0458335
Jeff Cheeger, Kenji Fukaya, and Mikhael Gromov, Nilpotent structures and invariant metrics on collapsed manifolds, J. Amer. Math. Soc. 5 (1992), no. 2, 327–372. MR 1126118, DOI 10.1090/S0894-0347-1992-1126118-X
Jeff Cheeger and Detlef Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann. of Math. (2) 96 (1972), 413–443. MR 309010, DOI 10.2307/1970819
Tobias H. Colding, Spaces with Ricci curvature bounds, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), 1998, pp. 299–308. MR 1648080
Owen Dearricott, A 7-manifold with positive curvature, Duke Math. J. 158 (2011), no. 2, 307–346. MR 2805071, DOI 10.1215/00127094-1334022
S. V. Dyatlov, Preservation of the positivity of sectional curvature in the case of factorization with respect to some nonfree actions, Mat. Tr. 10 (2007), no. 2, 62–91 (Russian, with Russian summary). MR 2382417
J.-H. Eschenburg, New examples of manifolds with strictly positive curvature, Invent. Math. 66 (1982), no. 3, 469–480. MR 662603, DOI 10.1007/BF01389224
Jost-Hinrich Eschenburg, Freie isometrische Aktionen auf kompakten Lie-Gruppen mit positiv gekrümmten Orbiträumen, Schriftenreihe des Mathematischen Instituts der Universität Münster, 2. Serie [Series of the Mathematical Institute of the University of Münster, Series 2], vol. 32, Universität Münster, Mathematisches Institut, Münster, 1984 (German). MR 758252
C. Escher and C. Searle, Non-negative curvature and torus actions, arXiv:1506.08685.
Fuquan Fang and Karsten Grove, Reflection groups in non-negative curvature, J. Differential Geom. 102 (2016), no. 2, 179–205. MR 3454545
—, Decomposible polar action in nonegative curvature (in preparation).
F. Fang, K. Grove, and G. Thorbergsson, Tits geometry and positive curvature, Acta. Math. (to appear).
Fuquan Fang, Karsten Grove, and Gudlaugur Thorbergsson, Rank three geometry and positive curvature, Comm. Anal. Geom. 24 (2016), no. 3, 487–520. MR 3521316, DOI 10.4310/CAG.2016.v24.n3.a3
F. Fang and X. Rong, Positive pinching, volume and second Betti number, Geom. Funct. Anal. 9 (1999), no. 4, 641–674. MR 1719590, DOI 10.1007/s000390050098
Fuquan Fang and Xiaochun Rong, The second twisted Betti number and the convergence of collapsing Riemannian manifolds, Invent. Math. 150 (2002), no. 1, 61–109. MR 1930882, DOI 10.1007/s00222-002-0230-2
Fuquan Fang and Xiaochun Rong, Homeomorphism classification of positively curved manifolds with almost maximal symmetry rank, Math. Ann. 332 (2005), no. 1, 81–101. MR 2139252, DOI 10.1007/s00208-004-0618-y
Kenji Fukaya and Takao Yamaguchi, The fundamental groups of almost non-negatively curved manifolds, Ann. of Math. (2) 136 (1992), no. 2, 253–333. MR 1185120, DOI 10.2307/2946606
Fernando Galaz-Garcia, Nonnegatively curved fixed point homogeneous manifolds in low dimensions, Geom. Dedicata 157 (2012), 367–396. MR 2893494, DOI 10.1007/s10711-011-9615-y
Fernando Galaz-Garcia and Martin Kerin, Cohomogeneity-two torus actions on non-negatively curved manifolds of low dimension, Math. Z. 276 (2014), no. 1-2, 133–152. MR 3150196, DOI 10.1007/s00209-013-1190-5
F. Galaz-Garcia, M. Kerin, and M. Radeschi, Torus actions on rationally-elliptic manifolds, Preprint, arXiv:1511.08383 (2015).
Fernando Galaz-Garcia and Catherine Searle, Low-dimensional manifolds with non-negative curvature and maximal symmetry rank, Proc. Amer. Math. Soc. 139 (2011), no. 7, 2559–2564. MR 2784821, DOI 10.1090/S0002-9939-2010-10655-X
Fernando Galaz-Garcia and Catherine Searle, Nonnegatively curved 5-manifolds with almost maximal symmetry rank, Geom. Topol. 18 (2014), no. 3, 1397–1435. MR 3228455, DOI 10.2140/gt.2014.18.1397
Sebastian Goette, Adiabatic limits of Seifert fibrations, Dedekind sums, and the diffeomorphism type of certain 7-manifolds, J. Eur. Math. Soc. (JEMS) 16 (2014), no. 12, 2499–2555. MR 3293803, DOI 10.4171/JEMS/492
S. Goette, M. Kerin, and K. Shankar, Highly connected $7$-manifolds and non-negative sectional curvature (in preparation).
David González-Álvaro, Nonnegative curvature on stable bundles over compact rank one symmetric spaces, Adv. Math. 307 (2017), 53–71. MR 3590513, DOI 10.1016/j.aim.2016.10.038
Francisco J. Gozzi, Low dimensional polar actions, Geom. Dedicata 175 (2015), 219–247. MR 3323638, DOI 10.1007/s10711-014-0037-5
Alfred Gray, Pseudo-Riemannian almost product manifolds and submersions, J. Math. Mech. 16 (1967), 715–737. MR 0205184
R. E. Greene and H. Wu, $C^{\infty }$ approximations of convex, subharmonic, and plurisubharmonic functions, Ann. Sci. École Norm. Sup. (4) 12 (1979), no. 1, 47–84. MR 532376
Detlef Gromoll and Karsten Grove, A generalization of Berger’s rigidity theorem for positively curved manifolds, Ann. Sci. École Norm. Sup. (4) 20 (1987), no. 2, 227–239. MR 911756
Detlef Gromoll and Karsten Grove, The low-dimensional metric foliations of Euclidean spheres, J. Differential Geom. 28 (1988), no. 1, 143–156. MR 950559
Michael Gromov, Curvature, diameter and Betti numbers, Comment. Math. Helv. 56 (1981), no. 2, 179–195. MR 630949, DOI 10.1007/BF02566208
Mikhael Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53–73. MR 623534
Karsten Grove, Geometry of, and via, symmetries, Conformal, Riemannian and Lagrangian geometry (Knoxville, TN, 2000) Univ. Lecture Ser., vol. 27, Amer. Math. Soc., Providence, RI, 2002, pp. 31–53. MR 1922721, DOI 10.1090/ulect/027/02
Karsten Grove, Critical point theory for distance functions, Differential geometry: Riemannian geometry (Los Angeles, CA, 1990) Proc. Sympos. Pure Math., vol. 54, Amer. Math. Soc., Providence, RI, 1993, pp. 357–385. MR 1216630, DOI 10.1090/pspum/054.3/1216630
Karsten Grove, Finiteness theorems in Riemannian geometry, Explorations in complex and Riemannian geometry, Contemp. Math., vol. 332, Amer. Math. Soc., Providence, RI, 2003, pp. 101–120. MR 2018334, DOI 10.1090/conm/332/05931
Karsten Grove and Stephen Halperin, Contributions of rational homotopy theory to global problems in geometry, Inst. Hautes Études Sci. Publ. Math. 56 (1982), 171–177 (1983). MR 686045
Karsten Grove and Steen Markvorsen, New extremal problems for the Riemannian recognition program via Alexandrov geometry, J. Amer. Math. Soc. 8 (1995), no. 1, 1–28. MR 1276824, DOI 10.1090/S0894-0347-1995-1276824-4
Karsten Grove and Peter Petersen V, Bounding homotopy types by geometry, Ann. of Math. (2) 128 (1988), no. 1, 195–206. MR 951512, DOI 10.2307/1971439
Karsten Grove and Peter Petersen, Manifolds near the boundary of existence, J. Differential Geom. 33 (1991), no. 2, 379–394. MR 1094462
Karsten Grove and Peter Petersen V, Volume comparison à la Aleksandrov, Acta Math. 169 (1992), no. 1-2, 131–151. MR 1179015, DOI 10.1007/BF02392759
Karsten Grove, Peter Petersen V, and Jyh Yang Wu, Geometric finiteness theorems via controlled topology, Invent. Math. 99 (1990), no. 1, 205–213. MR 1029396, DOI 10.1007/BF01234418
Karsten Grove and Catherine Searle, Positively curved manifolds with maximal symmetry-rank, J. Pure Appl. Algebra 91 (1994), no. 1-3, 137–142. MR 1255926, DOI 10.1016/0022-4049(94)90138-4
Karsten Grove and Catherine Searle, Differential topological restrictions curvature and symmetry, J. Differential Geom. 47 (1997), no. 3, 530–559. MR 1617636
Karsten Grove and Krishnan Shankar, Rank two fundamental groups of positively curved manifolds, J. Geom. Anal. 10 (2000), no. 4, 679–682. MR 1817780, DOI 10.1007/BF02921991
Karsten Grove and Katsuhiro Shiohama, A generalized sphere theorem, Ann. of Math. (2) 106 (1977), no. 2, 201–211. MR 500705, DOI 10.2307/1971164
Karsten Grove, Luigi Verdiani, Burkhard Wilking, and Wolfgang Ziller, Non-negative curvature obstructions in cohomogeneity one and the Kervaire spheres, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 5 (2006), no. 2, 159–170. MR 2244696
Karsten Grove, Luigi Verdiani, and Wolfgang Ziller, An exotic $T_1\Bbb S^4$ with positive curvature, Geom. Funct. Anal. 21 (2011), no. 3, 499–524. MR 2810857, DOI 10.1007/s00039-011-0117-8
Karsten Grove and Frederick Wilhelm, Hard and soft packing radius theorems, Ann. of Math. (2) 142 (1995), no. 2, 213–237. MR 1343322, DOI 10.2307/2118635
Karsten Grove and Frederick Wilhelm, Metric constraints on exotic spheres via Alexandrov geometry, J. Reine Angew. Math. 487 (1997), 201–217. MR 1454266
Karsten Grove and Burkhard Wilking, A knot characterization and 1-connected nonnegatively curved 4-manifolds with circle symmetry, Geom. Topol. 18 (2014), no. 5, 3091–3110. MR 3285230, DOI 10.2140/gt.2014.18.3091
Karsten Grove, Burkhard Wilking, and Wolfgang Ziller, Positively curved cohomogeneity one manifolds and 3-Sasakian geometry, J. Differential Geom. 78 (2008), no. 1, 33–111. MR 2406265
Karsten Grove and Wolfgang Ziller, Curvature and symmetry of Milnor spheres, Ann. of Math. (2) 152 (2000), no. 1, 331–367. MR 1792298, DOI 10.2307/2661385
Karsten Grove and Wolfgang Ziller, Lifting group actions and nonnegative curvature, Trans. Amer. Math. Soc. 363 (2011), no. 6, 2865–2890. MR 2775790, DOI 10.1090/S0002-9947-2011-05272-4
Karsten Grove and Wolfgang Ziller, Polar manifolds and actions, J. Fixed Point Theory Appl. 11 (2012), no. 2, 279–313. MR 3000673, DOI 10.1007/s11784-012-0087-y
Luis Guijarro, On the metric structure of open manifolds with nonnegative curvature, Pacific J. Math. 196 (2000), no. 2, 429–444. MR 1800586, DOI 10.2140/pjm.2000.196.429
Luis Guijarro, Improving the metric in an open manifold with nonnegative curvature, Proc. Amer. Math. Soc. 126 (1998), no. 5, 1541–1545. MR 1443388, DOI 10.1090/S0002-9939-98-04287-7
Luis Guijarro and Vitali Kapovitch, Restrictions on the geometry at infinity of nonnegatively curved manifolds, Duke Math. J. 78 (1995), no. 2, 257–276. MR 1333500, DOI 10.1215/S0012-7094-95-07811-9
John Harvey and Catherine Searle, Orientation and symmetries of Alexandrov spaces with applications in positive curvature, J. Geom. Anal. 27 (2017), no. 2, 1636–1666. MR 3625167, DOI 10.1007/s12220-016-9734-7
Richard S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geometry 17 (1982), no. 2, 255–306. MR 664497
Allen E. Hatcher, A proof of the Smale conjecture, $\textrm {Diff}(S^{3})\simeq \textrm {O}(4)$, Ann. of Math. (2) 117 (1983), no. 3, 553–607. MR 701256, DOI 10.2307/2007035
C. He and P. Rajan, Fake $13$-projective spaces with cohomogeneity one actions, arXiv: 1601.03723.
B. Herzog and F. Wilhelm, The sub-index of critical points of distance functions, arXiv: 1409.0132.
Wu-Yi Hsiang and Bruce Kleiner, On the topology of positively curved $4$-manifolds with symmetry, J. Differential Geom. 29 (1989), no. 3, 615–621. MR 992332
Vitali Kapovitch, Perelman’s stability theorem, Surveys in differential geometry. Vol. XI, Surv. Differ. Geom., vol. 11, Int. Press, Somerville, MA, 2007, pp. 103–136. MR 2408265, DOI 10.4310/SDG.2006.v11.n1.a5
V. Kapovitch, Regularity of limits of noncollapsing sequences of manifolds, Geom. Funct. Anal. 12 (2002), no. 1, 121–137. MR 1904560, DOI 10.1007/s00039-002-8240-1
Vitali Kapovitch, Anton Petrunin, and Wilderich Tuschmann, Nilpotency, almost nonnegative curvature, and the gradient flow on Alexandrov spaces, Ann. of Math. (2) 171 (2010), no. 1, 343–373. MR 2630041, DOI 10.4007/annals.2010.171.343
Lee Kennard, On the Hopf conjecture with symmetry, Geom. Topol. 17 (2013), no. 1, 563–593. MR 3039770, DOI 10.2140/gt.2013.17.563
Lee Kennard, Positively curved Riemannian metrics with logarithmic symmetry rank bounds, Comment. Math. Helv. 89 (2014), no. 4, 937–962. MR 3284301, DOI 10.4171/CMH/340
B. Kleiner, Non-negatively curved $4$-manifolds with symmetry, PhD thesis, U.C. Berkeley, 1989.
Soon Kyu Kim, Dennis McGavran, and Jingyal Pak, Torus group actions on simply connected manifolds, Pacific J. Math. 53 (1974), 435–444. MR 368051
L. Kennard and B. Wilking, The topology of fixed point components of isometric $T^5$ actions on positively curved manifolds, in preparation.
Barrett O’Neill, The fundamental equations of a submersion, Michigan Math. J. 13 (1966), 459–469. MR 200865
Murad Özaydin and Gerard Walschap, Vector bundles with no soul, Proc. Amer. Math. Soc. 120 (1994), no. 2, 565–567. MR 1162091, DOI 10.1090/S0002-9939-1994-1162091-X
Richard S. Palais and Chuu-Lian Terng, A general theory of canonical forms, Trans. Amer. Math. Soc. 300 (1987), no. 2, 771–789. MR 876478, DOI 10.1090/S0002-9947-1987-0876478-4
G. Perelman, Proof of the soul conjecture of Cheeger and Gromoll, J. Differential Geom. 40 (1994), no. 1, 209–212. MR 1285534
G. Ya. Perel′man, Elements of Morse theory on Aleksandrov spaces, Algebra i Analiz 5 (1993), no. 1, 232–241 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 5 (1994), no. 1, 205–213. MR 1220498
G. Perelman, Spaces with curvature bounded below, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 517–525. MR 1403952
G. Perelman, Spaces with curvature bounded below, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994) Birkhäuser, Basel, 1995, pp. 517–525. MR 1403952
Peter Petersen, Riemannian geometry, Graduate Texts in Mathematics, vol. 171, Springer-Verlag, New York, 1998. MR 1480173, DOI 10.1007/978-1-4757-6434-5
Peter Petersen, Variations on a theme of Synge, Explorations in complex and Riemannian geometry, Contemp. Math., vol. 332, Amer. Math. Soc., Providence, RI, 2003, pp. 241–251. MR 2018343, DOI 10.1090/conm/332/05940
P. Petersen and F. Wilhelm, An exotic sphere with positive sectional curvature, arXiv: 0805.0812.
Anton Petrunin, Semiconcave functions in Alexandrov’s geometry, Surveys in differential geometry. Vol. XI, Surv. Differ. Geom., vol. 11, Int. Press, Somerville, MA, 2007, pp. 137–201. MR 2408266, DOI 10.4310/SDG.2006.v11.n1.a6
A. Petrunin and W. Tuschmann, Diffeomorphism finiteness, positive pinching, and second homotopy, Geom. Funct. Anal. 9 (1999), no. 4, 736–774. MR 1719602, DOI 10.1007/s000390050101
Conrad Plaut, Metric spaces of curvature $\geq k$, Handbook of geometric topology, North-Holland, Amsterdam, 2002, pp. 819–898. MR 1886682
Conrad Plaut, Spaces of Wald curvature bounded below, J. Geom. Anal. 6 (1996), no. 1, 113–134. MR 1402389, DOI 10.1007/BF02921569
Curtis Pro, Michael Sill, and Frederick Wilhelm, The diffeomorphism type of manifolds with almost maximal volume, Comm. Anal. Geom. 25 (2017), no. 1, 243–267. MR 3663318, DOI 10.4310/CAG.2017.v25.n1.a8
C. Pro and F. Wilhelm, Diffeomorphism stability and codimension four, arXiv:1606.01828.
A. Rigas, Geodesic spheres as generators of the homotopy groups of $\textrm {O}$, $B\textrm {O}$, J. Differential Geometry 13 (1978), no. 4, 527–545 (1979). MR 570216
Xiaochun Rong, Collapsed manifolds with bounded sectional curvature and applications, Surveys in differential geometry. Vol. XI, Surv. Differ. Geom., vol. 11, Int. Press, Somerville, MA, 2007, pp. 1–23. MR 2408262, DOI 10.4310/SDG.2006.v11.n1.a2
Xiaochun Rong, Positive curvature, local and global symmetry, and fundamental groups, Amer. J. Math. 121 (1999), no. 5, 931–943. MR 1713297
Xiaochun Rong, On fundamental groups of positively curved manifolds with local torus actions, Asian J. Math. 9 (2005), no. 4, 545–559. MR 2216245, DOI 10.4310/AJM.2005.v9.n4.a6
Catherine Searle and Frederick Wilhelm, How to lift positive Ricci curvature, Geom. Topol. 19 (2015), no. 3, 1409–1475. MR 3352240, DOI 10.2140/gt.2015.19.1409
Catherine Searle and DaGang Yang, On the topology of non-negatively curved simply connected $4$-manifolds with continuous symmetry, Duke Math. J. 74 (1994), no. 2, 547–556. MR 1272983, DOI 10.1215/S0012-7094-94-07419-X
Krishnan Shankar, On the fundamental groups of positively curved manifolds, J. Differential Geom. 49 (1998), no. 1, 179–182. MR 1642117
V. A. Šarafutdinov, The Pogorelov-Klingenberg theorem for manifolds that are homeomorphic to $\textbf {R}^{n}$, Sibirsk. Mat. Ž. 18 (1977), no. 4, 915–925, 958 (Russian). MR 0487896
Takashi Shioya and Takao Yamaguchi, Volume collapsed three-manifolds with a lower curvature bound, Math. Ann. 333 (2005), no. 1, 131–155. MR 2169831, DOI 10.1007/s00208-005-0667-x
Takashi Shioya and Takao Yamaguchi, Collapsing three-manifolds under a lower curvature bound, J. Differential Geom. 56 (2000), no. 1, 1–66. MR 1863020
Lorenz J. Schwachhöfer and Wilderich Tuschmann, Metrics of positive Ricci curvature on quotient spaces, Math. Ann. 330 (2004), no. 1, 59–91. MR 2091679, DOI 10.1007/s00208-004-0538-x
W. Spindeler, $S^1$-actions on $4$-manifolds and fixed point homogeneous manifolds of non-negative curvature, Thesis Münster, arXiv:1510.01548.
J. Szenthe, Orthogonally transversal submanifolds and the generalizations of the Weyl group, Period. Math. Hungar. 15 (1984), no. 4, 281–299. MR 782429, DOI 10.1007/BF02454161
Luigi Verdiani, Cohomogeneity one Riemannian manifolds of even dimension with strictly positive curvature. I, Math. Z. 241 (2002), no. 2, 329–339. MR 1935489, DOI 10.1007/s002090200417
Luigi Verdiani, Cohomogeneity one manifolds of even dimension with strictly positive sectional curvature, J. Differential Geom. 68 (2004), no. 1, 31–72. MR 2152908
Luigi Verdiani and Wolfgang Ziller, Concavity and rigidity in non-negative curvature, J. Differential Geom. 97 (2014), no. 2, 349–375. MR 3263509
A. Wald, Begrundung einer koordinatenlosen Differentialgeometrie der Flachen, Ergeb. Math. Colloq. 7 (1935), 24–46.
Nolan R. Wallach, Compact homogeneous Riemannian manifolds with strictly positive curvature, Ann. of Math. (2) 96 (1972), 277–295. MR 307122, DOI 10.2307/1970789
Alan Weinstein, A fixed point theorem for positively curved manifolds, J. Math. Mech. 18 (1968/1969), 149–153. MR 0227894, DOI 10.1512/iumj.1969.18.18016
Michael Wiemeler, Torus manifolds and non-negative curvature, J. Lond. Math. Soc. (2) 91 (2015), no. 3, 667–692. MR 3355120, DOI 10.1112/jlms/jdv008
Burkhard Wilking, Index parity of closed geodesics and rigidity of Hopf fibrations, Invent. Math. 144 (2001), no. 2, 281–295. MR 1826371, DOI 10.1007/PL00005801
Burkhard Wilking, The normal homogeneous space $(\textrm {SU}(3)\times \textrm {SO}(3))/\textrm {U}^\bullet (2)$ has positive sectional curvature, Proc. Amer. Math. Soc. 127 (1999), no. 4, 1191–1194. MR 1469441, DOI 10.1090/S0002-9939-99-04613-4
Burkhard Wilking, Torus actions on manifolds of positive sectional curvature, Acta Math. 191 (2003), no. 2, 259–297. MR 2051400, DOI 10.1007/BF02392966
Burkhard Wilking, Positively curved manifolds with symmetry, Ann. of Math. (2) 163 (2006), no. 2, 607–668. MR 2199227, DOI 10.4007/annals.2006.163.607
Burkhard Wilking, A duality theorem for Riemannian foliations in nonnegative sectional curvature, Geom. Funct. Anal. 17 (2007), no. 4, 1297–1320. MR 2373019, DOI 10.1007/s00039-007-0620-0
Burkhard Wilking, Nonnegatively and positively curved manifolds, Surveys in differential geometry. Vol. XI, Surv. Differ. Geom., vol. 11, Int. Press, Somerville, MA, 2007, pp. 25–62. MR 2408263, DOI 10.4310/SDG.2006.v11.n1.a3
B. Wilking and W. Ziller, Revisiting homogeneous spaces with positive curvature, J. Reine Ange. Math. (to appear); arXiv:1503.06256.
Takao Yamaguchi, Collapsing and pinching under a lower curvature bound, Ann. of Math. (2) 133 (1991), no. 2, 317–357. MR 1097241, DOI 10.2307/2944340
Takao Yamaguchi, Collapsing Riemannian 4-manifolds, Sūgaku 52 (2000), no. 2, 172–186 (Japanese). MR 1761517
J. Yeager, Rational ellipticity in cohomogeneity two, PhD thesis, Univ. Maryland, 2012; arXiv:1309.5509.
Wolfgang Ziller, Examples of Riemannian manifolds with non-negative sectional curvature, Surveys in differential geometry. Vol. XI, Surv. Differ. Geom., vol. 11, Int. Press, Somerville, MA, 2007, pp. 63–102. MR 2408264, DOI 10.4310/SDG.2006.v11.n1.a4