A panoramic glimpse of manifolds with sectional curvature bounded from below

K. Grove

Algebra i analiz, Tome 29 (2017) no. 1, pp. 7-48.

Résumé

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.

Détail

Keywords: lower curvature bound, Toponogov distance comparison theorem, Bishop–Gromov volume comparison theorem, Gromov–Hausdorff topology.

Bibliographie

[1] Alexandrino M., “On polar foliations and fundamental group”, Results Math., 60:1–4 (2011), 213–223 | DOI | MR | Zbl

[2] Alexandrino M., Töben D., “Singular Riemannian foliations on simply connected spaces”, Differential Geom. Appl., 24:4 (2006), 383–397 | DOI | MR | Zbl

[3] Aloff S., Wallach N., “An infinite family of $7$-manifolds admitting positively curved Riemannian structures”, Bull. Amer. Math. Soc., 81 (1975), 93–97 | DOI | MR | Zbl

[4] Amann M., Kennard L., “Topological properties of positively curved manifolds with symmetry”, Geom. Funct. Anal., 24:5 (2014), 1377–1405 | DOI | MR

[5] Bazaikin Ya. V., “Ob odnom semeistve $13$-mernykh zamknutykh rimanovykh mnogoobrazii polozhitelnoi krivizny”, Sib. mat. zh., 37:6 (1996), 1219–1237 | MR | Zbl

[6] Bazaikin Ya. V., “Mnogoobrazie polozhitelnoi sektsionnoi krivizny s fundamentalnoi gruppoi $Z_3\oplus Z_3$”, Sib. mat. zh., 40:5 (1999), 994–996 | MR | Zbl

[7] Belegradek I., Kapovitch V., “Obstructions to non-negative curvature and rational homotopy theory”, J. Amer. Math. Soc., 16:2 (2003), 259–284 | DOI | MR | Zbl

[8] Bérard Bergery L., “Les variétés Riemanniennes homogènes simplement connexes de dimension impaire à courbure strictement positive”, J. Math. Pure Appl., 55:1 (1976), 47–68 | MR | Zbl

[9] Berestovskii V. N., “Prostranstva s ogranichennoi kriviznoi i distantsionnaya geometriya”, Sib. mat. zh., 27:1 (1986), 11–25 | MR

[10] Berestovskii V., Guijarro L., “A metric characterization of Riemannian submersions”, Ann. Global Anal. Geom., 18:6 (2000), 577–588 | DOI | MR

[11] Berger M., “Les varietes Riemanniennes homogenes normales simplement connexes a courbure strictment positive”, Ann. Scuola Norm. Sup. Pisa (3), 15 (1961), 191–240 | MR

[12] Berger M., Bott R., “Sur les variétés á courbure strictement positive”, Topology, 1 (1962), 301–311 | DOI | MR | Zbl

[13] Bettiol R., Mendes R., “Strongly non-negative curvature”, Math. Ann. (to appear)

[14] Burns K., Paternain G., “On the growth of the number of geodesics joining two points”, Internat. Conf. Dynamical Systems (Montevideo, 1995), Pitman Res. Notes Math. Ser., 362, Longman, Harlow, 1996, 7–20 | MR | Zbl

[15] Burago Yu. D., “O trekhmernykh otkrytykh rimanovykh prostranstvakh neotritsatelnoi krivizny”, Zap. nauch. semin. LOMI, 66, 1976, 103–113 | MR | Zbl

[16] Burago D., Burago Y., Ivanov S., A course in metric geometry, Grad. Stud. Math., 33, Amer. Math. Soc., Providence, RI, 2001 | DOI | MR | Zbl

[17] Burago Yu. D., Gromov M. L., Perelman G. Ya., “Prostranstva A. D. Aleksandrova s ogranichennymi snizu kriviznami”, Uspekhi mat. nauk, 47:2 (1992), 3–51 | MR | Zbl

[18] Burago Yu. D., Zalgaller V. A., “Vypuklye mnozhestva v rimanovykh prostranstvakh neotritsatelnoi krivizny”, Uspekhi mat. nauk, 32:3 (1977), 3–55 | MR | Zbl

[19] Cheeger J., “Finiteness theorems for Riemannian manifolds”, Amer. J. Math., 92 (1970), 61–74 | DOI | MR | Zbl

[20] Cheeger J., “Some examples of manifolds of non-negative curvature”, J. Differential Geom., 8 (1973), 623–628 | DOI | MR | Zbl

[21] Cheeger J., Ebin D., Comparison theorems in Riemannian geometry, North-Holland Math. Library, 9, North-Holland Publ. Co., Amsterdam–Oxford; American Elsevier Publ. Co., Inc., New York, 1975 | MR | Zbl

[22] Cheeger J., Fukaya K., Gromov M., “Nilpotent structures and invariant metrics on collapsed manifolds”, J. Amer. Math. Soc., 5:2 (1992), 327–372 | DOI | MR | Zbl

[23] Cheeger J., Gromoll D., “On the structure of complete manifolds of non-negative curvature”, Ann. of Math. (2), 96 (1972), 413–443 | DOI | MR | Zbl

[24] Colding T. H., “Spaces with Ricci curvature bounds”, Proc. of the Internat. Congress of Math. (Berlin, 1998), v. II, Doc. Math., Extra vol. II, 1998, 299–308 | MR | Zbl

[25] Dearricott O., “A $7$-manifold with positive curvature”, Duke Math. J., 158:2 (2011), 307–346 | DOI | MR | Zbl

[26] Dyatlov S. V., “Sokhranenie polozhitelnosti sektsionnoi krivizny pri faktorizatsii po nekotorym nesvobodnym deistviyam”, Mat. tr., 10:2 (2007), 62–91 | MR | Zbl

[27] Eschenburg J. H., “New examples of manifolds with strictly positive curvature”, Inv. Math., 66:3 (1982), 469–480 | DOI | MR | Zbl

[28] Eschenburg J. H., Freie isometrische Aktionen auf kompakten Lie-Gruppen mit positiv gekrümmten Orbiträumen, Schriftenr. Math. Inst. Univ. Münster, 32, Münster, 1984 | MR | Zbl

[29] Escher C., Searle C., Non-negative curvature and torus actions, arXiv: 1506.08685

[30] Fang F., Grove K., “Reflection groups in non-negative curvature”, J. Differential Geom., 102:2 (2016), 179–205 | DOI | MR | Zbl

[31] Fang F., Grove K., Decomposible polar action in nonegative curvature, in preparation

[32] Fang F., Grove K., Thorbergsson G., “Tits geometry and positive curvature”, Acta. Math. (to appear)

[33] Fang F., Grove K., Thorbergsson G., “Rank three geometry and positive curvature”, Comm. Anal. Geom., 24:3 (2016), 471–520 | DOI | MR

[34] Fang F., Rong X., “Positive pinching, volume and second Betti number”, Geom. Funct. Anal., 9:4 (1999), 641–674 | DOI | MR | Zbl

[35] Fang F., Rong X., “The second twisted Betti number and the convergence of collapsing Riemannian manifolds”, Invent. Math., 150:1 (2002), 61–109 | DOI | MR | Zbl

[36] Fang F., Rong X., “Homeomorphism classification of positively curved manifolds with almost maximal symmetry rank”, Math. Ann., 332:1 (2005), 81–101 | DOI | MR | Zbl

[37] Fukaya K., Yamaguchi T., “The fundamental groups of almost non-negatively curved manifolds”, Ann. of Math. (2), 136:2 (1992), 253–333 | DOI | MR | Zbl

[38] Galaz-Garcia F., “Non-negatively curved fixed point homogeneous manifolds in low dimensions”, Geom. Dedicata, 157 (2012), 367–396 | DOI | MR | Zbl

[39] Galaz-Garcia F., Kerin M., “Cohomogeneity-two torus actions on non-negatively curved manifolds of low dimension”, Math. Z., 276:1–2 (2014), 133–152 | DOI | MR | Zbl

[40] Galaz-Garcia F., Kerin M., Radeschi M., Torus actions on rationally-elliptic manifolds, Preprint, 2015, arXiv: 1511.08383

[41] Galaz-Garcia F., Searle C., “Low-dimensional manifolds with non-negative curvature and maximal symmetry rank”, Proc. Amer. Math. Soc., 139:7 (2011), 2559–2564 | DOI | MR | Zbl

[42] Galaz-Garcia F., Searle C., “Non-negatively curved $5$-manifolds with almost maximal symmetry rank”, Geom. Topol., 18:3 (2014), 1397–1435 | DOI | MR | Zbl

[43] Goette S., “Adiabatic limits of Seifert fibrations, Dedekind sums, and the diffeomorphism type of certain $7$-manifolds”, J. Eur. Math. Soc. (JEMS), 16:12 (2014), 2499–2555 | DOI | MR | Zbl

[44] Goette S., Kerin M., Shankar K., Highly connected $7$-manifolds and non-negative sectional curvature, in preparation

[45] González-Álvaro D., “Nonnegative curvature on stable bundles over compact rank one symmetric spaces”, Advances in Math., 307 (2017), 53–71 ; arXiv: 1506.03972 | DOI | MR | Zbl

[46] Gozzi F. J., “Low dimensional polar actions”, Geom. Dedicata, 175:1 (2015), 219–247 | DOI | MR | Zbl

[47] Gray A., “Pseudo-Riemannian almost product manifolds and submersions”, J. Math. Mech., 16 (1967), 715–737 | MR | Zbl

[48] Greene R. E., Wu H., “$C^\infty$ approximations of convex, subharmonic, and plurisubharmonic functions”, Ann. Sci. École Norm. Sup. (4), 12:1 (1979), 47–84 | DOI | MR

[49] Gromoll D., Grove K., “A generalization of Berger's rigidity theorem for positively curved manifolds”, Ann. Sci. École Norm. Sup. (4), 20:2 (1987), 227–239 | DOI | MR | Zbl

[50] Gromoll D., Grove K., “The low-dimensional metric foliations of Euclidean spheres”, J. Differential Geom., 28:1 (1988), 143–156 | DOI | MR | Zbl

[51] Gromov M., “Curvature, diameter and Betti numbers”, Comment. Math. Helv., 56:2 (1981), 179–195 | DOI | MR | Zbl

[52] Gromov M., “Groups of polynomial growth and expanding maps”, Inst. Hautes Études Sci. Publ. Math., 53 (1981), 53–73 | DOI | MR

[53] Grove K., “Geometry of, and via, Symmetries”, Univ. Lecture Ser., 27, Amer. Math. Soc., Providence, RI, 2002, 31–53 | DOI | MR

[54] Grove K., “Critical point theory for distance functions”, Differential Geometry: Riemannian Geometry (Los Angeles, CA, 1990), Proc. Sympos. Pure Math., 54, Amer. Math. Soc., Providence, RI, 1993, 357–385 | DOI | MR | Zbl

[55] Grove K., “Finiteness theorems in Riemannian geometry”, Explorations in Complex and Riemannian Geometry, Contemp. Math., 332, Amer. Math. Soc., Providence, RI, 2003, 101–120 | DOI | MR | Zbl

[56] Grove K., Halperin S., “Contributions of rational homotopy theory to global problems in geometry”, Inst. Hautes Études Sci. Publ. Math., 56 (1982), 171–177 | DOI | MR

[57] Grove K., Markvorsen S., “New extremal problems for the Riemannian recognition program via Alexandrov geometry”, J. Amer. Math. Soc., 8:1 (1995), 1–28 | DOI | MR | Zbl

[58] Grove K., Petersen P., “Bounding homotopy types by geometry”, Ann. of Math. (2), 128:1 (1988), 195–206 | DOI | MR | Zbl

[59] Grove K., Petersen P., “Manifolds near the boundary of existence”, J. Differential Geom., 33:2 (1991), 379–394 | DOI | MR | Zbl

[60] Grove K., Petersen P., “Volume comparison à la Aleksandrov”, Acta Math., 169:1–2 (1992), 131–151 | DOI | MR | Zbl

[61] Grove K., Petersen P., Wu J. Y., “Geometric finiteness theorems via controlled topology”, Invent. Math., 99:1 (1990), 205–213 ; Invent. Math., 104:1 (1991), 221–222 | DOI | MR | Zbl | DOI | MR | Zbl

[62] Grove K., Searle C., “Positively curved manifolds with maximal symmetry-rank”, J. Pure Appl. Algebra, 91:1–3 (1994), 137–142 | DOI | MR | Zbl

[63] Grove K., Searle C., “Differential topological restrictions by curvature and symmetry”, J. Differential Geom., 47:3 (1997), 530–559 | DOI | MR | Zbl

[64] Grove K., Shankar K., “Rank two fundamental groups of positively curved manifolds”, J. Geom. Anal., 10:4 (2000), 679–682 | DOI | MR | Zbl

[65] Grove K., Shiohama K., “A generalized sphere theorem”, Ann. of Math. (2), 106:2 (1977), 201–211 | DOI | MR | Zbl

[66] Grove K., Verdiani L., Wilking B., Ziller W., “Non-negative curvature obstructions in cohomogeneity one and the Kervaire spheres”, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 5:2 (2006), 159–170 | MR | Zbl

[67] Grove K., Verdiani L., Ziller W., “An exotic $T_1\mathbb S^4$ with positive curvature”, Geom. Funct. Anal., 21:3 (2011), 499–524 | DOI | MR | Zbl

[68] Grove K., Wilhelm F., “Hard and soft packing radius theorems”, Ann. of Math. (2), 142 (1995), 213–237 | DOI | MR | Zbl

[69] Grove K., Wilhelm F., “Metric constraints on exotic spheres via Alexandrov geometry”, J. Reine Angew. Math., 487 (1997), 201–217 | MR | Zbl

[70] Grove K., Wilking B., “A knot characterization and $1$-connected non-negatively curved $4$-manifolds with circle symmetry”, Geom. Topol., 18:5 (2014), 3091–3110 | DOI | MR | Zbl

[71] Grove K., Wilking B., Ziller W., “Positively curved cohomogeneity one manifolds and $3$-Sasakian geometry”, J. Differential Geom., 78:1 (2008), 33–111 | DOI | MR | Zbl

[72] Grove K., Ziller W., “Curvature and symmetry of Milnor spheres”, Ann. of Math. (2), 152:1 (2000), 331–367 | DOI | MR | Zbl

[73] Grove K., Ziller W., “Lifting group actions and non-negative curvature”, Trans. Amer. Math. Soc., 363:6 (2011), 2865–2890 | DOI | MR | Zbl

[74] Grove K., Ziller W., “Polar manifolds and actions”, J. Fixed Point Theory Appl., 11:2 (2012), 279–313 | DOI | MR | Zbl

[75] Guijarro L., “On the metric structure of open manifolds with non-negative curvature”, Pacific J. Math., 196:2 (2000), 429–444 | DOI | MR | Zbl

[76] Guijarro L., “Improving the metric in an open manifold with non-negative curvature”, Proc. Amer. Math. Soc., 126:5 (1998), 1541–1545 | DOI | MR | Zbl

[77] Guijarro L., Kapovitch V., “Restrictions on the geometry at infinity of non-negatively curved manifolds”, Duke Math. J., 78:2 (1995), 257–276 | DOI | MR | Zbl

[78] Harvey J., Searle C., “Orientation and symmetries of Alexandrov spaces with applications in positive curvature”, J. Geom. Anal., 27:2 (2017), 1636–1666. ; arXiv: 1209.1366 | DOI | MR

[79] Hamilton R. S., “Three-manifolds with positive Ricci curvature”, J. Differential Geom., 17:2 (1982), 255–306 | DOI | MR | Zbl

[80] Hatcher A., “A proof of a Smale conjecture, $Diff(S^3)\simeq\mathrm O(4)$”, Ann. of Math. (2), 117:3 (1983), 553–607 | DOI | MR | Zbl

[81] He C., Rajan P., Fake $13$-projective spaces with cohomogeneity one actions, arXiv: 1601.03723

[82] Herzog B., Wilhelm F., The sub-index of critical points of distance functions, arXiv: 1409.0132

[83] Hsiang W.-Y., Kleiner B., “On the topology of positively curved $4$-manifolds with symmetry”, J. Differential Geom., 29:3 (1989), 615–621 | DOI | MR | Zbl

[84] Kapovitch V., “Perelman's stability theorem”, Surveys in Differential Geometry, v. XI, Surv. Differ. Geom., 11, Int. Press, Somerville, MA, 2007, 103–136 | DOI | MR | Zbl

[85] Kapovitch V., “Regularity of limits of noncollapsing sequences of manifolds”, Geom. Funct. Anal., 12:1 (2002), 121–137 | DOI | MR | Zbl

[86] Kapovitch V., Petrunin A., Tuschmann W., “Nilpotency, almost non-negative curvature, and the gradient flow on Alexandrov spaces”, Ann. of Math. (2), 171:1 (2010), 343–373 | DOI | MR | Zbl

[87] Kennard L., “On the Hopf conjecture with symmetry”, Geom. Topol., 17:1 (2013), 563–593 | DOI | MR | Zbl

[88] Kennard L., “Positively curved Riemannian manifolds with logarithmic symmetry rank bounds”, Comment. Math. Helv., 89:4 (2014), 937–962 | DOI | MR | Zbl

[89] Kleiner B., Non-negatively curved $4$-manifolds with symmetry, PhD thesis, U. C. Berkeley, 1989

[90] Kim S., McGavran D., Pak J., “Torus group actions on simply connected manifolds”, Pacific J. Math., 53:2 (1974), 435–444 | DOI | MR | Zbl

[91] O'Neill B., “The fundamental equations of a submersion”, Michigan Math. J., 13 (1966), 459–469 | DOI | MR | Zbl

[92] Özaydin M., Walschap G., “Vector bundles with no soul”, Proc. Amer. Math. Soc., 120:2 (1994), 565–567 | DOI | MR | Zbl

[93] Palais R. S., Terng C.-L., “A general theory of canonical forms”, Trans. Amer. Math. Soc., 300:2 (1987), 771–789 | DOI | MR | Zbl

[94] Perelman G., “Proof of the soul conjecture of Cheeger and Gromoll”, J. Differential Geom., 40:1 (1994), 209–212 | DOI | MR | Zbl

[95] Perelman G. Ya., “Nachala teorii Morsa na prostranstvakh Aleksandrova”, Algebra i analiz, 5:1 (1993), 232–241 | MR | Zbl

[96] Perelman G., Alexandrov spaces with curvature bounded below. II, Preprint

[97] Perelman G., “Spaces with curvature bounded below”, Proc. of the Internat. Congress of Math. (Zürich, 1994), v. I, II, Birkhäuser, Basel, 1995, 517–525 | MR | Zbl

[98] Petersen P., Riemannian geometry, Grad. Texts in Math., 171, Springer-Verlag, New York, 1998 | DOI | MR | Zbl

[99] Petersen P., “Variations on a theme of Synge”, Explorations in Complex and Riemannian Geometry, Contemp. Math., 332, Amer. Math. Soc., Providence, RI, 2003, 241–251 | DOI | MR | Zbl

[100] Petersen P., Wilhelm F., An exotic sphere with positive sectional curvature, arXiv: 0805.0812

[101] Petrunin A., “Semiconcave functions in Alexandrov's geometry”, Surveys in Differential Geometry, v. XI, Surv. Differ. Geom., 11, Int. Press, Somerville, MA, 2007, 137–201 | DOI | MR | Zbl

[102] Petrunin A., Tuschmann W., “Diffeomorphism finiteness, positive pinching, and second homotopy”, Geom. Funct. Anal., 9:4 (1999), 736–774 | DOI | MR | Zbl

[103] Plaut C., “Metric spaces of curvature $\ge k$”, Handbook of geometric topology, North-Holland, Amsterdam, 2002, 819–898 | MR

[104] Plaut C., “Spaces of Wald curvature bounded below”, J. Geom. Anal., 6:1 (1996), 113–134 | DOI | MR | Zbl

[105] Pro C., Sill M., Wilhelm F., “The Diffeomorphism Type of Manifolds with Almost Maximal Volume”, Comm. Anal. Geom. (to appear)

[106] Pro C., Wilhelm F., Diffeomorphism stability and codimension four, arXiv: 1606.01828

[107] Rigas A., “Geodesic spheres as generators of the homotopy groups of $O$, $BO$”, J. Differential Geom., 13:4 (1978), 527–545 | DOI | MR | Zbl

[108] Rong X., “Collapsed manifolds with bounded sectional curvature and applications”, Surveys in Differential Geometry, v. XI, Surv. Differ. Geom., 11, Int. Press, Somerville, MA, 2007, 1–23 | DOI | MR | Zbl

[109] Rong X., “Positive curvature, local and global symmetry, and fundamental groups”, Amer. J. Math., 121:5 (1999), 931–943 | DOI | MR | Zbl

[110] Rong X., “On fundamental groups of positively curved manifolds with local torus actions”, Asian J. Math., 9:4 (2005), 545–560 | DOI | MR | Zbl

[111] Searle C., Wilhelm F., “How to lift positive Ricci curvature”, Geom. Topol., 19:3 (2015), 1409–1475 | DOI | MR | Zbl

[112] Searle C., DaGang Yang, “On the topology of non-negatively curved simply connected $4$-manifolds with continuous symmetry”, Duke Math. J., 74:2 (1994), 547–556 | DOI | MR | Zbl

[113] Shankar S., “On the fundamental groups of positively curved manifolds”, J. Differential Geom., 49:1 (1998), 179–182 | DOI | MR | Zbl

[114] Sharafutdinov V., “Teorema Pogorelova–Klingenberga dlya mnogoobrazii gomeomorfnykh $\mathbb R^n$”, Sib. mat. zh., 18:4 (1977), 915–925 | MR | Zbl

[115] Shioya T., Yamaguchi T., “Volume collapsed three-manifolds with a lower curvature bound”, Math. Ann., 333:1 (2005), 131–155 | DOI | MR | Zbl

[116] Shioya T., Yamaguchi T., “Collapsing three-manifolds under a lower curvature bound”, J. Differential Geom., 56:1 (2000), 1–66 | DOI | MR | Zbl

[117] Schwachhöfer L., Tuschmann W., “Metrics of positive Ricci curvature on quotient spaces”, Math. Ann., 330:1 (2004), 59–91 | DOI | MR | Zbl

[118] Spindeler W., $S^1$-actions on $4$-manifolds and fixed point homogeneous manifolds of non-negative curvature, Thesis, Münster

[119] Szenthe J., “Orthogonally transversal submanifolds and the generalizations of the Weyl group”, Period. Math. Hungar., 15:4 (1984), 281–299 | DOI | MR | Zbl

[120] Verdiani L., “Cohomogeneity one Riemannian manifolds of even dimension with strictly positive sectional curvature. I”, Math. Z., 241:2 (2002), 329–339 | DOI | MR | Zbl

[121] Verdiani L., “Cohomogeneity one manifolds of even dimension with strictly positive sectional curvature”, J. Differential Geom., 68:1 (2004), 31–72 | DOI | MR | Zbl

[122] Verdiani L., Ziller W., “Concavity and rigidity in non-negative curvature”, J. Differential Geom., 97:2 (2014), 349–375 | DOI | MR | Zbl

[123] Wald A., “Begrundung einer koordinatenlosen Differentialgeometrie der Flachen”, Ergeb. Math. Colloq., 7 (1935), 24–46

[124] Wallach N., “Compact homogeneous Riemannian manifolds with strictly positive curvature”, Ann. of Math. (2), 96 (1972), 277–295 | DOI | MR | Zbl

[125] Weinstein A., “A fixed point theorem for positively curved manifolds”, J. Math. Mech., 18 (1968/1969), 149–153 | MR | Zbl

[126] Wiemeler M., “Torus manifolds and non-negative curvature”, J. London Math. Soc. (2), 91:3 (2015), 667–692 | DOI | MR | Zbl

[127] Wilking B., “Index parity of closed geodesics and rigidity of Hopf fibrations”, Invent. Math., 144:2 (2001), 281–295 | DOI | MR | Zbl

[128] Wilking B., “The normal homogeneous space $(SU(3)-SO(3))/U^.(2)$ has positive sectional curvature”, Proc. Amer. Math. Soc., 127:4 (1999), 1191–1194 | DOI | MR | Zbl

[129] Wilking B., “Torus actions on manifolds of positive sectional curvature”, Acta Math., 191:2 (2003), 259–297 | DOI | MR | Zbl

[130] Wilking B., “Positively curved manifolds with symmetry”, Ann. of Math. (2), 163:2 (2006), 607–668 | DOI | MR | Zbl

[131] Wilking B., “A duality theorem for Riemannian foliations in non-negative sectional curvature”, Geom. Funct. Anal., 17:4 (2007), 1297–1320 | DOI | MR | Zbl

[132] Wilking B., “Non-negatively and Positively Curved Manifolds”, Surveys in Differential Geometry, v. XI, Surv. Differ. Geom., 11, Int. Press, Somerville, MA, 2007, 25–62 | DOI | MR | Zbl

[133] Wilking B., Ziller W., “Revisiting homogeneous spaces with positive curvature”, J. Reine Ange. Math. (to appear)

[134] Yamaguchi T., “Collapsing and pinching under a lower curvature bound”, Ann. of Math. (2), 133:2 (1991), 317–357 | DOI | MR | Zbl

[135] Yamaguchi T., “Collapsing Riemannian $4$-manifolds”, Sugaku, 52:2 (2000), 172–186 (Japanese) | MR

[136] Yeager J., Rational ellipticity in cohomogeneity two, PhD thesis, Univ. Maryland, 2012; arXiv: 1309.5509

[137] Ziller W., “Examples of Riemannian manifolds with non-negative sectional curvature”, Surveys in Differential Geometry, v. XI, Surv. Differ. Geom., 11, Int. Press, Somerville, MA, 2007, 63–102 | DOI | MR | Zbl