Annals of Global Analysis and Geometry

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

Non-negative scalar curvature

  • Varghese Mathai


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 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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