SURVEY ARTICLE

Positive curvature, symmetry, and topology

  • Manuel AMANN
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  • Fakultät für Mathematik, Institut für Algebra und Geometrie, Karlsruher Institut für Technologie, Englerstraβe 2, 76131 Karlsruhe, Germany

Received date: 02 Feb 2016

Accepted date: 31 Jul 2016

Published date: 23 Sep 2016

Copyright

2016 Higher Education Press and Springer-Verlag Berlin Heidelberg

Abstract

We depict recent developments in the field of positive sectional curvature, mainly, but not exclusively, under the assumption of isometric torus actions. After an elaborate introduction to the field, we shall discuss various classification results, before we provide results on the computation of Euler characteristics. This will be the starting point for an examination of more involved invariants and further techniques. In particular, we shall discuss the Hopf conjectures, related decomposition results like the Wilhelm conjecture, results in differential topology and index theory as well as in rational homotopy theory, geometrically formal metrics in positive curvature and much more. The results we present will be discussed for arbitrary dimensions, but also specified to small dimensions. This survey article features mainly depictions of our own work interest in this area and cites results obtained in different collaborations; full statements and proofs can be found in the respective original research articles.

Cite this article

Manuel AMANN . Positive curvature, symmetry, and topology[J]. Frontiers of Mathematics in China, 2016 , 11(5) : 1099 -1122 . DOI: 10.1007/s11464-016-0580-5

1
Amann M. Non-formal homogeneous spaces. Math Z, 2013, 274(3-4): 1299–1325

DOI

2
Amann M, Kennard L. On a generalized conjecture of Hopf with symmetry. 2014, arXiv: 1402.7255

3
Amann M, Kennard L.Topological properties of positively curved manifolds with symmetry. Geom Funct Anal, 2014, 24(5): 1377–1405

DOI

4
Amann M, Kennard L.Positive curvature and rational ellipticity. Algebr Geom Topol, 2015, 15(4): 2269–2301

DOI

5
Amann M, Kennard L. Positive curvature and torus symmetry in small dimensions, I—Dimensions 10, 12, 14, and 16. 2015, arXiv: 1512.01302

6
Amann M, Kennard L. Positive curvature and torus symmetry in small dimensions, II. Preprint, 2016

7
Amann M, Kennard L. Generalizations of the Hopf conjecture with symmetry. Preprint, 2016

8
Amann M, Ziller W. Geometrically formal homogeneous metrics of positive curvature. J Geom Anal (to appear), arXiv: 1407.6201

9
Bär C. Geometrically formal 4-manifolds with nonnegative sectional curvature. 2012, arXiv: 1212.1325v2

10
Berard-Bergery L. Les variétés riemanniennes homogènes simplement connexes dedimension impaire à courbure strictement positive. J Math Pures Appl (9), 1976, 55(1): 47–67

12
Berger M. Trois remarques sur les variétés riemanniennes à courbure positive. C R Math Acad Sci Paris, 1966, 263: A76–A78

13
Breuillard E, Green B, Tao T.The structure of approximate groups. Publ Math Inst Hautes ′Etudes Sci, 2012, 116(1): 115–221

14
Chen X Y. Curvature and Riemannian Submersions. Ph D Thesis. Univ of Notre Dame, 2014

15
Conner P E. On the action of the circle group. Michigan Math J, 1957, 4: 241–247

DOI

16
Dearricott O. A 7-manifold with positive curvature. Duke Math J, 2011, 158(2): 307–346

DOI

17
Dessai A. Topology of positively curved 8-dimensional manifolds with symmetry. Pacific J Math, 2011, 249(1): 23–47

DOI

18
Fang F Q, Rong X C. Homeomorphism classification of positively curved manifolds with almost maximal symmetry rank. Math Ann, 2005, 332: 81–101

DOI

19
F′elix Y, Halperin S, Thomas J-C. Rational Homotopy Theory. Grad Texts in Math, Vol 205. New York: Springer-Verlag, 2001

20
Florit L A, Ziller W. Topological obstructions to fatness. Geom Topol, 2011, 15(2): 891–925

DOI

21
Frankel T. Manifolds with positive curvature. Pacific J Math, 1961, 11: 165–174

DOI

22
González D, Guijarro L. Soft restrictions on positively curved Riemannian submersions. 2015, arXiv: 1501.01813

23
Grosjean J-F, Nagy P-A. On the cohomology algebra of some classes of geometrically formal manifolds. Proc Lond Math Soc (3), 2009, 98(3): 607–630

24
Grove K, Halperin S. Dupin hypersurfaces, group actions and the double mapping cylinder. J Differential Geom, 1987, 26(3): 429–459

25
Grove K, Searle C. Positively curved manifolds with maximal symmetry-rank. J Pure Appl Algebra, 1994, 91(1-3): 137–142

DOI

26
Grove K, Verdiani L, Ziller W. An exotic T1S4 with positive curvature. Geom Funct Anal, 2011, 21(3): 499–524

DOI

27
Grove K, Wilking B. A knot characterization and 1-connected nonnegatively curved 4-manifolds with circle symmetry. Geom Topol, 2014, 18(5): 3091–3110

DOI

28
Hirzebruch F, Berger T, Jung R. Manifolds and Modular Forms. Aspects of Mathematics, E20. Braunschweig Friedr Vieweg & Sohn, 1992

DOI

29
Hirzebruch F, Slodowy P. Elliptic genera, involutions, and homogeneous spin manifolds. Geom Dedicata, 1990, 35(1-3): 309–343

DOI

30
Hsiang W Y, Kleiner B. On the topology of positively curved 4-manifolds with symmetry. J Differential Geom, 1989, 30: 615–621

31
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

32
Kapovitch V, Wilking B. Structure of fundamental groups of manifolds with Ricci curvature bounded below. 2012, arXiv: 1105.5955v2

33
Kapovitch V, Ziller W. Biquotients with singly generated rational cohomology. Geom Dedicata, 2004, 104: 149–160

DOI

34
Kennard L. On the Hopf conjecture with symmetry. Geom Topol, 2013, 17(1): 563–593

DOI

35
Kennard L.Positively curved Riemannian metrics with logarithmic symmetry rank bounds. Comment Math Helv, 2014, 89(4): 937–962

DOI

36
Kerin M. Some new examples with almost positive curvature. Geom Topol, 2011, 15(1): 217–260

DOI

37
Kobayashi S. Fixed points of isometries. Nagoya Math J, 1958, 13: 63–68

DOI

38
Kotschick D. On products of harmonic forms. Duke Math J, 2001, 107(3): 521–531

DOI

39
Kotschick D. Geometric formality and non-negative scalar curvature. 2012, arXiv: 1212.3317v4

40
Kotschick D, Terzić S. On formality of generalized symmetric spaces. Math Proc Cambridge Philos Soc, 2003, 134(3): 491–505

DOI

41
Kotschick D, Terzić S. Chern numbers and the geometry of partial flag manifolds. Comment Math Helv, 2009, 84(3): 587–616

DOI

42
Kotschick D, Terzić S.Geometric formality of homogeneous spaces and of biquotients. Pacific J Math, 2011, 249(1): 157–176

DOI

43
Kramer L. Homogeneous spaces, Tits buildings, and isoparametric hypersurfaces. Mem Amer Math Soc, Vol 158, No 752. Providence: Amer Math Soc, 2002

44
Ornea L, Pilca M. Remarks on the product of harmonic forms. Pacific J Math, 2011, 250(2): 353–363

DOI

45
Petersen P, Wilhelm F. An exotic sphere with positive sectional curvature. 2008, arXiv: 0805.0812v3

46
Prasad G, Yeung S-K. Arithmetic fake projective spaces and arithmetic fake Grassmannians. Amer J Math, 2009, 131(2): 379–407

DOI

47
Püttmann T, Searle C. The Hopf conjecture for manifolds with low cohomogeneity or high symmetry rank. author Proc Amer Math Soc, 2001, 130(1): 163–166

DOI

48
Rong X C. Positively curved manifolds with almost maximal symmetry rank. Geom Dedicata, 2002, 95(1): 157–182

DOI

49
Rong X C, Su X L. The Hopf conjecture for manifolds with abelian group actions. Commun Contemp Math, 2005, 7: 121–136

DOI

50
Su Z X. Rational analogs of projective planes. Algebr Geom Topol, 2014, 14: 421–438

DOI

51
Wallach N R. Compact homogeneous Riemannian manifolds with strictly positive curvature. Ann of Math (2), 1972, 96: 277–295

52
Walschap G. Soul-preserving submersions. Michigan Math J, 1994, 41(3): 609–617

DOI

53
Weisskopf N. Positive curvature and the elliptic genus. 2013, arXiv: 1305.5175v1

54
Wilking B. Torus actions on manifolds of positive sectional curvature. Acta Math, 2003, 191(2): 259–297

DOI

55
Ziller W. Examples of Riemannian manifolds with non-negative sectional curvature. 2007, arXiv: math/0701389v3

56
Ziller W. Riemannian manifolds with positive sectional curvature. 2012, arXiv: 1210.4102v1

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