Spaces and moduli spaces of Riemannian metrics

Abstract

These notes present and survey results about spaces and moduli spaces of complete Riemannian metrics with curvature bounds on open and closed manifolds, here focussing mainly on connectedness and disconnectedness properties. They also discuss several open problems and questions in the field.

Key words： Riemannian metrics; moduli spaces; sectional curvature; positive Ricci curvature; positive scalar curvature

References

1 Belegradek I, Farrell F T, Kapovitch V. Space of non-negatively curved manifolds and pseudoisotopies. J Differential Geom (to appear), arXiv: 1501.03475

2 Belegradek I, Hu J. Connectedness properties of the space of complete non-negatively curved planes. Math Ann, 2015, 362: 1273–1286

3 Belegradek I, Kwasik S, Schultz R. Moduli spaces of non-negative sectional curvature and non-unique souls. J Differential Geom, 2011, 89: 49–86

4 Botvinnik B, Ebert J, Randal-Williams O. Infinite loop spaces and positive scalar curvature. arXiv: 1411.7408

5 Botvinnik B, Gilkey P. The eta invariant and metrics of positive scalar curvature. Math Ann, 1995, 302(3): 507–517

6 Botvinnik B, Gilkey P. Metrics of positive scalar curvature on spherical space forms. Canad J Math, 1996, 48(1): 64–80

7 Botvinnik B, Hanke B, Schick T, Walsh M. Homotopy groups of the moduli space of metrics of positive scalar curvature. Geom Topol, 2010, 14: 2047–2076

8 Carr R. Construction of manifolds of positive scalar curvature. Trans Amer Math Soc, 1988, 307(1): 63–74

9 Cerf J. Sur les difféomorphismes de la sphère de dimension trois (Γ₄= 0). Lecture Notes in Math, Vol 53. Berlin: Springer-Verlag, 1968

10 Chen B L, Huang X T. Path-connectedness of the moduli spaces of metrics with positive isotropic curvature on four-manifolds. Math Ann (to appear)

11 Chernysh V. On the homotopy type of the space R⁺(M). Preprint, arXiv: GT/0405235

12 Codá Marques F. Deforming three-manifolds with positive scalar curvature. Ann of Math (2), 2012, 176(2): 815–863

13 Crowley D, Schick T. The Gromoll filtration, KO-characteristic classes and metrics of positive scalar curvature. Geom Topol, 2013, 17: 1773–1790

14 Dessai A, Klaus St, Tuschmann W. Nonconnected Moduli Spaces of Nonnegative Sectional Curvature Metrics on Simply Connected Manifolds. Preprint, 2016, arXiv: 1601.04877

15 Farrell F T, Ontaneda P. The Teichmüller space of pinched negatively curved metrics on a hyperbolic manifold is not contractible. Ann of Math (2), 2009, 170(1): 45–65

16 Farrell F T, Ontaneda P. The moduli space of negatively curved metrics of a hyperbolic manifold. J Topol, 2010, 3(3): 561–577

17 Farrell F T, Ontaneda P. On the topology of the space of negatively curved metrics. J Differential Geom, 2010, 86(2): 273–301

18 Gajer P. Riemannian metrics of positive scalar curvature on compact manifolds with boundary. Ann Global Anal Geom, 1987, 5(3): 179–191

19 Gromov M, Lawson H B Jr. Positive scalar curvature and the Dirac operator on complete Riemannian manifolds. Publ Math Inst Hautes Études Sci, 1983, 58: 83–196

20 Hanke B, Schick T, Steimle W. The space of metrics of positive scalar curvature. Publ Math Inst Hautes Études Sci, 2014, 120: 335–367

21 Hitchin N. Harmonic spinors. Adv Math, 1974, 14: 1–55

22 Kapovitch V, Petrunin A, Tuschmann W. Non-negative pinching, moduli spaces and bundles with infinitely many souls. J Differential Geom, 2005, 71(3): 365–383

23 Kreck M, Stolz S. Nonconnected moduli spaces of positive sectional curvature metrics. J Amer Math Soc, 1993, 6: 825–850

24 Lawson H B Jr, Michelsohn M-L. Spin Geometry. Princeton: Princeton Univ Press, 1989

25 Leichtnam E, Piazza P. On higher eta-invariants and metrics of positive scalar curvature. K-Theory, 2001, 24(4): 341–359

26 Lohkamp J. Curvature h-principles. Ann of Math, 1995, 142: 457–498

27 Piazza P, Schick Th. Groups with torsion, bordism and rho invariants. Pacific J Math, 2007, 232(2): 355–378

28 Piazza P, Schick Th. Rho classes, index theory and Stolz’ positive scalar curvature sequence. J Topology, 2014, 7(4): 965–1004

29 Rosenberg J, Stolz S. Metrics of positive scalar curvature and connections with surgery. In: Cappell S, Ranicki A, Rosenberg J, eds. Surveys on Surgery Theory, Vol 2. Ann of Math Stud, Vol 149. Princeton: Princeton Univ Press, 2001, 353–386

30 Ruberman D. Positive scalar curvature, diffeomorphisms and the Seiberg-Witten invariants. Geom Topol, 2001, 5: 895–924

31 Tuschmann W, Wraith D. Moduli Spaces of Riemannian Metrics. Oberwolfach Seminars, Vol 46. Basel: Birkhäuser, 2015

32 Walsh M. Metrics of Positive Scalar Curvature and Generalized Morse Functions, Part I. Mem Amer Math Soc, Vol 209, No 983. Providence: Amer Math Soc, 2011

33 Walsh M. Cobordism invariance of the homotopy type of the space of positive scalar curvature metrics. Proc Amer Math Soc, 2013, 141(7): 2475–2484

34 Walsh M. H-spaces, loop spaces and the space of positive scalar curvature metrics on the sphere. Geom Topol, 2014, 18(4): 2189–2243

35 Wang M, Ziller W. Einstein metrics on principal torus bundles. J Differential Geom, 1990, 31(1): 215–248

36 Weyl H. Über die Bestimmung einer geschlossenen konvexen Fläche durch ihr Linienelement. Vierteljahrsschr Naturforsch Ges Zür, 1916, 61: 40–72

37 Wraith D J. On the moduli space of positive Ricci curvature metrics on homotopy spheres. Geom Topol, 2011, 15: 1983–2015

38 Wraith D J. Non-negative versus positive scalar curvature. Preprint, 2016, arXiv: 1607.00657