Almost nonnegative curvature operator and cohomology rings

Abstract

We give a survey of results on the construction of and obstructions to metrics of almost nonnegative curvature operator on closed manifolds and results on the cohomology rings of closed, simply-connected manifolds with a lower curvature and upper diameter bound. The latter is motivated by a question of Grove whether these condition imply finiteness of rational homotopy types. This question has answers by F. Fang–X. Rong, B. Totaro and recently A. Dessai and the present author.

Key words： Nonnegative curvature; homogeneous spaces; curvature operator; almost nonnegative curvature

References

1 Aloff S, Wallach N R. An infinite family of distinct 7-manifolds admitting positively curved Riemannian structures. Bull Amer Math Soc, 1975, 81: 93–97

2 Bérard P H.From vanishing theorems to estimating theorems: the Bochner technique revisited. Bull Amer Math Soc (NS), 1988, 19(2): 371–406

3 Berger M. Sur les groupes d’holonomie homog`ene des variétés àconnexion affine et des variétés riemanniennes. Bull Soc Math France, 1955, 83: 279–330

4 Böhm C, Wilking B. Manifolds with positive curvature operators are space forms. Ann of Math (2), 2008, 167(3): 1079–1097

5 Bourguignon J-P, Karcher H. Curvature operators: pinching estimates and geometric examples. Ann Sci Éc Norm Supér (4), 1978, 11(1): 71–92

6 Cao H D, Chow B. Compact Kähler manifolds with nonnegative curvature operator. Invent Math, 1986, 83(3): 553–556

7 Cheeger J, Gromoll D. On the structure of complete manifolds of nonnegative curvature. Ann of Math (2), 1972, 96: 413–443

8 Chow B, Lu P, Ni L. Hamilton’s Ricci flow. Graduate Studies in Mathematics, Vol 77. Providence/New York: Amer Math Soc/Science Press, 2006

9 Dessai A. Nonnegative curvature, low cohomogeneity and complex cohomology. Münster J Math (to appear), http://arxiv.org/abs/1503.08290

10 Fang, F Q, Rong X C. Curvature, diameter, homotopy groups, and cohomology rings. Duke Math J, 2001, 107(1): 135–158

11 Fukaya K, Yamaguchi T. The fundamental groups of almost non-negatively curved manifolds. Ann of Math (2), 1992, 136(2): 253–333

12 Gallot S, Meyer D. Opérateur de courbure et laplacien des formes différentielles d’une variété riemannienne. J Math Pures Appl (9), 1975, 54(3): 259–284

13 Gromov M. Almost flat manifolds. J Differential Geom, 1978, 13(2): 231–241

14 Gromov M. Curvature, diameter and Betti numbers. Comment Math Helv, 1981, 56(2): 179–195

15 Grove K. Critical point theory for distance functions. In: Differential Geometry: Riemannian geometry (Los Angeles, CA, 1990). Proc Sympos Pure Math, Vol 54. Providence: Amer Math Soc, 1993, 357–385

16 Grove K, Ziller W. Curvature and symmetry of Milnor spheres. Ann of Math (2), 2000, 152(1): 331–367

17 Hamilton R S. Four-manifolds with positive curvature operator. J Differential Geom, 1986, 24(2): 153–179

18 Herrmann M. Classification and characterization of rationally elliptic manifolds in low dimensions. Preprint, 2014, arXiv: 1409.8036

19 Herrmann M. Homogeneous spaces, curvature and cohomology. Differential Geom Appl (to appear)

20 Herrmann M, Sebastian D, Tuschmann W. Manifolds with almost nonnegative curvature operator and principal bundles. Ann Global Anal Geom, 2013, 44(4): 391–399

21 Hoelscher C A. On the homology of low-dimensional cohomogeneity one manifolds. Transform Groups, 2010, 15(1): 115–133

22 Huang H M. Some remarks on the pinching problems. Bull Inst Math Acad Sin, 1981, 9(2): 321–340

23 Kapovitch V, Petrunin A, Tuschmann W. Nilpotency, almost nonnegative curvature, and the gradient flow on Alexandrov spaces. Ann of Math (2), 2010, 171(1): 343–373

24 Klaus S. Einfach-zusammenhängende Kompakte Homogene Räume bis zur Dimension Neun. Diploma Thesis. Johannes Gutenberg Universität Mainz, 1988

25 Li P. On the Sobolev constant and the p-spectrum of a compact Riemannian manifold. Ann Sci Éc Norm Supér, 1980, 13(4): 451–468

26 Lott J. Collapsing with a lower bound on the curvature operator. Adv Math, 2014, 256: 291–317

27 Mercuri F, Noronha M H. On the topology of complete Riemannian manifolds with nonnegative curvature operator. Rend Sem Fac Sci Univ Cagliari, 1993, 63(2): 149–171

28 Micallef M J, Moore J D. Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes. Ann of Math (2), 1988, 127(1): 199–227

29 Mori S. Projective manifolds with ample tangent bundles. Ann of Math (2), 1979, 110(3): 593–606

30 Noronha M H. A splitting theorem for complete manifolds with nonnegative curvature operator. Proc Amer Math Soc, 1989, 105(4): 979–985

31 Petersen P. Riemannian Geometry. 2nd ed. Grad. Texts in Math, Vol 171. New York: Springer, 2006

32 Petrunin A, Tuschmann W. Diffeomorphism finiteness, positive pinching, and second homotopy. Geom Funct Anal, 1999, 9(4): 736–774

33 Püttmann T. Optimal pinching constants of odd dimensional homogeneous spaces. Invent Math, 1999, 138(3): 631–684

34 Schwachhöfer L J, Tuschmann W. Curvature and cohomogeneity one. Max Planck Institute for Mathematics in the Sciences, MIS-Preprint 62/2001, 2001, http://www.mis.mpg.de/de/publications/preprints/2001/prepr2001-62.html

35 Schwachhöfer L J, Tuschmann W. Metrics of positive Ricci curvature on quotient spaces. Math Ann, 2004, 330(1): 59–91

36 Sebastian D. Konstruktionen von Mannigfaltigkeiten mit fast nichtnegativem Krümmungsoperator. Dissertation. Karlsruhe Institute of Technology, 2011

37 Siu Y T, Yau S T. Compact Kähler manifolds of positive bisectional curvature. Invent Math, 1980, 59(2): 189–204

38 Stewart T E. Lifting group actions in fibre bundles. Ann of Math (2), 1961, 74: 192–198

39 Strake M. A splitting theorem for open nonnegatively curved manifolds. Manuscripta Math, 1988, 61(3): 315–325

40 Tachibana S. A theorem of Riemannian manifolds of positive curvature operator. Proc Japan Acad, 1974, 50: 301–302

41 Totaro B. Curvature, diameter, and quotient manifolds. Math Res Lett, 2003, 10(2-3): 191–203

42 Tuschmann W. Geometric diffeomorphism finiteness in low dimensions and homotopy group finiteness. Math Ann, 2002, 322(2): 413–420

43 Tuschmann W. Collapsing and almost nonnegative curvature. In: Global Differential Geometry. Springer Proc Math, Vol 17. Heidelberg: Springer, 2012, 93–106

44 Wilking B. On fundamental groups of manifolds of nonnegative curvature. Differential Geom Appl, 2000, 13(2): 129–165

45 Yim J-W. Space of souls in a complete open manifold of nonnegative curvature. J Differential Geom, 1990, 32(2): 429–455