Criteria for selfadjointness of the Dirac operator on pseudomanifolds

Chou, Arthur W.

doi:10.1090/S0002-9939-1989-0975634-1

Criteria for selfadjointness of the Dirac operator on pseudomanifolds
HTML articles powered by AMS MathViewer

by Arthur W. Chou

Proc. Amer. Math. Soc. 106 (1989), 1107-1116

DOI: https://doi.org/10.1090/S0002-9939-1989-0975634-1

PDF | Request permission

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

M. F. Atiyah, R. Bott, and A. Shapiro, Clifford modules, Topology 3 (1964), no. suppl, suppl. 1, 3–38. MR 167985, DOI 10.1016/0040-9383(64)90003-5
M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian geometry. I, Math. Proc. Cambridge Philos. Soc. 77 (1975), 43–69. MR 397797, DOI 10.1017/S0305004100049410
M. F. Atiyah and I. M. Singer, The index of elliptic operators. III, Ann. of Math. (2) 87 (1968), 546–604. MR 236952, DOI 10.2307/1970717

Family index for manifolds with boundary; superconnections and cones

Intersection cohomology

Jochen Brüning and Robert Seeley, Regular singular asymptotics, Adv. in Math. 58 (1985), no. 2, 133–148. MR 814748, DOI 10.1016/0001-8708(85)90114-8
Jochen Brüning and Robert Seeley, The resolvent expansion for second order regular singular operators, J. Funct. Anal. 73 (1987), no. 2, 369–429. MR 899656, DOI 10.1016/0022-1236(87)90073-5
Jochen Brüning and Robert Seeley, An index theorem for first order regular singular operators, Amer. J. Math. 110 (1988), no. 4, 659–714. MR 955293, DOI 10.2307/2374646
Jeff Cheeger, On the spectral geometry of spaces with cone-like singularities, Proc. Nat. Acad. Sci. U.S.A. 76 (1979), no. 5, 2103–2106. MR 530173, DOI 10.1073/pnas.76.5.2103
Jeff Cheeger, On the Hodge theory of Riemannian pseudomanifolds, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 91–146. MR 573430
Jeff Cheeger, Spectral geometry of singular Riemannian spaces, J. Differential Geom. 18 (1983), no. 4, 575–657 (1984). MR 730920
Jeff Cheeger, A vanishing theorem for piecewise constant curvature spaces, Curvature and topology of Riemannian manifolds (Katata, 1985) Lecture Notes in Math., vol. 1201, Springer, Berlin, 1986, pp. 33–40. MR 859575, DOI 10.1007/BFb0075646
Jeff Cheeger, $\eta$-invariants, the adiabatic approximation and conical singularities. I. The adiabatic approximation, J. Differential Geom. 26 (1987), no. 1, 175–221. MR 892036
Arthur Weichung Chou, The Dirac operator on spaces with conical singularities and positive scalar curvatures, Trans. Amer. Math. Soc. 289 (1985), no. 1, 1–40. MR 779050, DOI 10.1090/S0002-9947-1985-0779050-8

Classical and quantum scattering on a spinning cone

Mark Goresky and Robert MacPherson, Intersection homology theory, Topology 19 (1980), no. 2, 135–162. MR 572580, DOI 10.1016/0040-9383(80)90003-8
Nigel Hitchin, Harmonic spinors, Advances in Math. 14 (1974), 1–55. MR 358873, DOI 10.1016/0001-8708(74)90021-8

The geometry of spinors

André Lichnerowicz, Spineurs harmoniques, C. R. Acad. Sci. Paris 257 (1963), 7–9 (French). MR 156292
J. Milnor, Spin structures on manifolds, Enseign. Math. (2) 9 (1963), 198–203. MR 157388
John W. Milnor, Remarks concerning spin manifolds, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N.J., 1965, pp. 55–62. MR 0180978
Edward Witten, Global gravitational anomalies, Comm. Math. Phys. 100 (1985), no. 2, 197–229. MR 804460, DOI 10.1007/BF01212448