Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Criteria for selfadjointness of the Dirac operator on pseudomanifolds


Author: Arthur W. Chou
Journal: Proc. Amer. Math. Soc. 106 (1989), 1107-1116
MSC: Primary 58G25; Secondary 35Q20, 47F05, 58G10, 58G30
DOI: https://doi.org/10.1090/S0002-9939-1989-0975634-1
MathSciNet review: 975634
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the Dirac operator on pseudomanifolds with piecewise constant curvature metric. The criteria for the self-adjointness of the Dirac operator are obtained and a vanishing theorem is proved. At the end we make some comments on the index theorem and the $ \widehat A$-genus.


References [Enhancements On Off] (What's this?)

  • [1] M. F. Atiyah, R. Bott and A. A. Shapiro, Clifford modules, Topology 3 (supp. 1) (1964), 3-38. MR 0167985 (29:5250)
  • [2] M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian geometry. I, Math. Proc. Camb. Philos. Soc. 77 (1975), 43-69. MR 0397797 (53:1655a)
  • [3] M. F. Atiyah and I. M. Singer, The index of elliptic operator. III, Ann. of Math. (2) 87 (1968), 546-604. MR 0236952 (38:5245)
  • [4] J.-M. Bismut and J. Cheeger, Family index for manifolds with boundary; superconnections and cones (preprint), April 1988.
  • [5] A. Borel et al., Intersection cohomology, Birkhäuser, Boston, 1984.
  • [6] J. Brüning and R. Seeley, Regular singular asymptotics, Adv. Math. 58 (1985), 133-148. MR 814748 (87b:41032)
  • [7] -, The resolvent expansion for second order regular singular operators, J. Funct. Anal. 73 (1987), 369-429. MR 899656 (88g:35151)
  • [8] -, An index theorem for first order regular singular operators, Amer. J. Math. 110 (1988), 659-714. MR 955293 (89k:58271)
  • [9] J. Cheeger, On the spectral geometry of spaces with cone-like singularities, Proc. Nat. Acad. Sci. U.S.A. 76 (1979), 2103-2106. (A revised version is in preprint (1980).) MR 530173 (80k:58098)
  • [10] -, On the Hodge theory of riemannian pseudomanifolds, Proc. Sympos. Pure Math., vol. 36, Amer. Math. Soc. Providence, R. I., 1980, 91-145. MR 573430 (83a:58081)
  • [11] -, Spectral geometry of singular spaces, J. Differential Geometry 18 (1983), 575-657. MR 730920 (85d:58083)
  • [12] -, A vanishing theorem for piecewise constant curvature space, in Curvature and Topology of Riemannian Manifolds, Lecture Notes in Math. vol. 1201, Springer-Verlag Berlin-Heidelberg 1986, pp. 33-40. MR 859575 (88a:58203)
  • [13] -, $ \eta $-invariant, the adiabatic approximation and conical singularities, J. Differential Geometry 26 (1987), 175-221. MR 892036 (89c:58123)
  • [14] A. Chou, The Dirac operator on spaces with conical singularities and positive scalar curvatures, Trans. Amer. Math. Soc. 289 (1985), 1-40. MR 779050 (86i:58124)
  • [15] P. Gerbert and R. Jackiw, Classical and quantum scattering on a spinning cone (preprint), July 1988.
  • [16] M. Goresky an R. MacPherson, Intersection homology theory, Topology 19 (1980), 135-162. MR 572580 (82b:57010)
  • [17] N. Hitchin, Harmonic spinors, Advances in Math. 14 (1974), 1-55. MR 0358873 (50:11332)
  • [18] H. B. Lawson and M. L. Michelson, The geometry of spinors (manuscript).
  • [19] A. Lichnerowicz, Spineurs harmoniques, C R. Acad. Sci. Paris Ser. A-B 257 (1963), 7-9. MR 0156292 (27:6218)
  • [20] J. Milnor, Spin structure on manifolds, L'Enseignement Math. 9 (1963), 198-203. MR 0157388 (28:622)
  • [21] -, Remarks concerning spin manifolds, Differential and Combinatorial Topology, Princeton Univ. Press, N. J., 1964, pp. 55-62. MR 0180978 (31:5208)
  • [22] E. Witten, Global gravitational anomalies, Comm. Math. Phys. 100 (1985), 197-229. MR 804460 (87k:58282)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 58G25, 35Q20, 47F05, 58G10, 58G30

Retrieve articles in all journals with MSC: 58G25, 35Q20, 47F05, 58G10, 58G30


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0975634-1
Keywords: Dirac operators, singular spaces, pseudomanifolds
Article copyright: © Copyright 1989 American Mathematical Society

American Mathematical Society