Positive scalar curvature on foliations

Abstract

We generalize classical theorems due to Lichnerowicz and Hitchin on the existence of Riemannian metrics of positive scalar curvature on spin manifolds to the case of foliated spin manifolds. As a consequence, we show that there is no foliation of positive leafwise scalar curvature on any torus, which generalizes the famous theorem of Schoen-Yau and Gromov-Lawson on the nonexistence of metrics of positive scalar curvature on torus to the case of foliations. Moreover, our method, which is partly inspired by the analytic localization techniques of Bismut-Lebeau, also applies to give a new proof of the celebrated Connes vanishing theorem without using noncommutative geometry.

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      fjournal = {Kexue Tongbao. Chinese Science Bulletin},
      title = {Sub-signature operators and a local index theorem for them},
      year = {1996},
      pages = {294--295},
      }
  • [Z01] Go to document W. Zhang, Lectures on Chern-Weil Theory and Witten Deformations, River Edge, NJ: World Scientific Publishing Co., 2001, vol. 4.
    @BOOK{Z01, mrkey = {1864735},
      author = {Zhang, Weiping},
      mrclass = {58J20 (53C05 58J37)},
      series = {Nankai Tracts in Math.},
      isbn = {981-02-4686-2},
      address = {River Edge, NJ},
      publisher = {World Scientific Publishing Co.},
      doi = {10.1142/9789812386588},
      zblnumber = {0993.58014},
      volume = {4},
      mrnumber = {1864735},
      mrreviewer = {Steven Rosenberg},
      title = {Lectures on {C}hern-{W}eil Theory and {W}itten Deformations},
      year = {2001},
      pages = {xii+117},
      }
  • [Z04] Go to document W. Zhang, "Sub-signature operators, $\eta$-invariants and a Riemann-Roch theorem for flat vector bundles," Chinese Ann. Math. Ser. B, vol. 25, iss. 1, pp. 7-36, 2004.
    @ARTICLE{Z04, mrkey = {2033948},
      number = {1},
      issn = {0252-9599},
      author = {Zhang, Weiping},
      mrclass = {58J20 (58J28 58J30 58J35)},
      doi = {10.1142/S0252959904000032},
      journal = {Chinese Ann. Math. Ser. B},
      zblnumber = {1044.58028},
      volume = {25},
      mrnumber = {2033948},
      fjournal = {Chinese Annals of Mathematics. Series B},
      mrreviewer = {Steven Rosenberg},
      title = {Sub-signature operators, {$\eta$}-invariants and a {R}iemann-{R}och theorem for flat vector bundles},
      year = {2004},
      pages = {7--36},
      }
  • [MS88] Go to document R. J. Zimmer, "Appendix C: Positive scalar curvature along the leaves," in Global Analysis on Foliated Spaces by \sc C. C. Moore and \sc C. Schochet, New York: Springer-Verlag, 1988, vol. 9, p. vi.
    @INCOLLECTION{MS88, mrkey = {0918974},
      author = {Zimmer, R. J.},
      mrclass = {58G12 (46L99)},
      series = {Math. Sci. Res. Instit. Publ.},
      address = {New York},
      isbn = {0-387-96664-1},
      publisher = {Springer-Verlag},
      doi = {10.1007/978-1-4613-9592-8},
      volume = {9},
      mrnumber = {0918974},
      booktitle = {Global Analysis on Foliated Spaces {\rm by {\sc C. C. Moore} and {\sc C. Schochet}}},
      mrreviewer = {Thierry Fack},
      title = {Appendix {C}: {P}ositive scalar curvature along the leaves},
      year = {1988},
      pages = {vi+337},
      zblnumber = {0648.58034},
     }

Authors

Weiping Zhang

Chern Institute of Mathematics & LPMC, Nankai University, Tianjin, P. R. China and Center for Applied Mathematics, Tianjin University, Tianjin, P. R. China