Annals of Global Analysis and Geometry

, Volume 10, Issue 2, pp 103–123 | Cite as

Non-negative scalar curvature

  • Varghese Mathai
Article

Abstract

We study topological obstructions to the existence of Riemannian metrics of non-negative scalar curvature on almost spin manifolds using the Dirac operator, the Bochner technique, C* algebras and von Neumann algebras. We also derive some obstructions in terms of the eta invariants of Atiyah, Patodi and Singer. Next, we prove vanishing theorems for the Atiyah-Milnor genus. Finally, we derive obstructions to the existence of metrics of non-negative scalar curvature along the leaves of a leafwise non-amenable foliation on a spin manifold.

Key words

Scalar curvature eta invariants Novikov conjecture foliations 

MCS 1991

53C 58G 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Atiyah, M. F.: Elliptic operators, discrete groups and von Neumann algebras. Asterisque 32–33 (1976), 43–72.Google Scholar
  2. [2]
    Atiyah, M. F.: Vector fields on manifolds. Arbeitsgemeinschaft für Forschung des Landes Nordrhein-Westfalen; 200 (1970).Google Scholar
  3. [3]
    Atiyah, M. F.; Patodi, V. K.; Singer, I. M.: Spectral asymmetry and Riemannian geometry: Part I: Math. Proc. Cambridge Philos. Soc. 77 (1975), 43–69. Part II: Math. Proc. Cambridge Philos. Soc. 78 (1975), 405–432. Part III: Math. Proc. Cambridge Philos. Soc. 79 (1976), 71–99.Google Scholar
  4. [4]
    Atiyah, M. F.; Singer, I. M.: The index of elliptic operators. Part III: Ann. of Math. 87 (1968), 546–604. Part IV: Ann. of Math 93 (1971), 119–138. Part V: Ann. of Math. 93 (1971), 139–149.Google Scholar
  5. [5]
    Baum, P.; Connes, A.: Leafwise homotopy equivalence and rational Pontrjagin classes. In: Foliations. (Ed.: I. Temura). Adv. Stud. Pure Math. 5 (1985), 1–14.Google Scholar
  6. [6]
    Berard, P. H.: The Bochner technique revisited. Bull. Amer. Math. Soc. 19 (1988), 371–406.Google Scholar
  7. [7]
    Brooks, R.: Spectral geometry of foliations. Amer. J. Math. (1982), 1001–1012.Google Scholar
  8. [8]
    Brooks, R.: The fundamental group and the spectrum of the Laplacian. Comment. Math. Helv. 56 (1981), 581–598.Google Scholar
  9. [9]
    Brooks, R.: Amenabity and the spectrum of the Laplacian. Bull. Amer. Math. Soc. 6 (1982), 87–89.Google Scholar
  10. [10]
    Buser, P.: A remark on the isoperimetric constant. Ann. Sci. École Norm. Sup. 15 (1982), 213–230.Google Scholar
  11. [11]
    Cohen, J.: Von Neumann dimension and the homology of covering spaces. Quart. J. Math. Oxford 30 (1979), 133–142.Google Scholar
  12. [12]
    Connes, A.: Cyclic cohomology and the transverse fundamental class of a foliation. In: Geometric methods in operator algebras. (Eds.: H. Araki; E.G. Effros). Pitman Res. Notes Math. Ser. 123 (1986), 52–144.Google Scholar
  13. [13]
    Connes, A.: A survey of foliations and operator algebras. Proc. Sympos. Pure Math. 38 (1982), 521–630.Google Scholar
  14. [14]
    Connes, A.; Skandalis, G.: The longitudinal index theorem for foliations. Publ. Res. Inst. Math. Sci. Kyoto Univ. 20 (1984), 1139–1183.Google Scholar
  15. [15]
    Dixmier, J.: Von Neumann algebras. North-Holland Mathematical Library, Amsterdam, 27 (1981).Google Scholar
  16. [16]
    Duminy, G.: L'invariant de Godbillon-Vey d'un feulletage se localise dans les feuilles ressort. Preprint, 1982.Google Scholar
  17. [17]
    Gromov, M., Lawson, B.: Positive scalar curvature and the Dirac operator on complete Riemannian manifolds. Inst. Hautes Études Sci. Publ. Math. 58 (1983), 83–196.Google Scholar
  18. [18]
    Hitchin, N.: Harmonic spinors. Adv. in Math. 14 (1974), 1–55.Google Scholar
  19. [19]
    Hurder, S.: The Godbillon measure of amenable foliations. J. Differential Geom. 23 (1986), 347–365.Google Scholar
  20. [20]
    Kasparov, G. G.: On the homotopy invariance of rational Pontrjagin numbers. Soviet Math. Dokl. 11 (1970), 235–238.Google Scholar
  21. [21]
    Lawson, H.B.; Michelsohn, M.L.: Clifford bundles, immersions of manifolds and the vector field problem. J. Differential Geom. 15 (1980), 237–267.Google Scholar
  22. [22]
    [22]Lichnerowicz, A.: Spineurs harmoniques. C. R. Acad. Sci., Paris, Ser. A-B 257 (1963), 7–9.Google Scholar
  23. [23]
    Lusztig, G.: Novikov's higher signature and families of elliptic operators. J. Differential Geom. 7 (1971), 229–256.Google Scholar
  24. [24]
    Mathai, V.: Positive scalar curvature and reduced eta invariants. Preprint, 1987.Google Scholar
  25. [25]
    McKean, H.: An upper bound to the spectrum of the Laplacian on a manifold of negative curvature. J. Differential Geom. 4 (1970), 359–366.Google Scholar
  26. [26]
    Meyer, D.: Une inegalité de géométrie Hilbertienne et ses applications à la géométrie Riemannienne. C. R. Acad. Sci., Paris, Sér.I 295 (1982), 467–469.Google Scholar
  27. [27]
    Milnor, J.: Remarks concerning spin manifolds. In: Differential and Combinatorial Topology: Symposium in honour of Marston Morse. (Ed.: St. S. Cairns). Princeton, N.J.: University Press (1965), 55–62.Google Scholar
  28. [28]
    Miyazaki, T.: Simply connected spin manifolds with positive scalar curvature. Proc. Amer. Math. Soc. 93 (1985), 730–734.Google Scholar
  29. [29]
    Ono, K.: The scalar curvature and the spectrum of the Laplacian on spin manifolds. Math. Ann. 281 (1988), 163–168.Google Scholar
  30. [30]
    Rosenberg, J.: C *-algebras, positive scalar curvature, and the Novikov conjecture. Part I: Inst. Hautes Études Sci. Publ. Math. 58 (1983), 197–212. Part III: Topology 5 (1986) 3, 319–336.Google Scholar
  31. [31]
    Schoen, R.; Yau, S. T.: On the structure of manifolds with positive scalar curvature. Manuscripta Math. 28 (1979), 159–183.Google Scholar
  32. [32]
    Singer, I.M.: Some remarks on operator theory and index theory. Lecture Notes Math. 575 (1977), 128–137.Google Scholar
  33. [33]
    Sunada, T.: Unitary representations of fundamental groups and the spectrum of twisted Laplacians. Topology 28 (1989)2, 125–132.Google Scholar
  34. [34]
    Thurston, W.: Non-cobordant foliations on S 3. Bull. Amer. Math. Soc. 78 (1972), 511–514.Google Scholar

Copyright information

© Kluwer Academic Publishers 1992

Authors and Affiliations

  • Varghese Mathai
    • 1
  1. 1.Department of Pure MathematicsThe University of AdelaideSouth Australia

Personalised recommendations