Geometric invariants for Seifert fibred $3$-manifolds
HTML articles powered by AMS MathViewer
- by Ming Qing Ouyang PDF
- Trans. Amer. Math. Soc. 346 (1994), 641-659 Request permission
Abstract:
In this paper, we obtain a formula for the $\eta$-invariant of the signature operator for some circle bundles over Riemannian $2$-orbifolds. We then apply it to Seifert fibred $3$-manifolds endowed with one of the six Seifert geometries. By using a relation between the Chern-Simons invariant and the $\eta$-invariant, we also derive some elementary formulae for the Chern-Simons invariant of these manifolds. As applications, we show that some families of these manifolds cannot be conformally immersed into the Euclidean space ${{\mathbf {E}}^4}$.References
- M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian geometry. I, Math. Proc. Cambridge Philos. Soc. 77 (1975), 43β69. MR 397797, DOI 10.1017/S0305004100049410
- M. F. Atiyah, V. K. Patodi, and I. M. Singer, Spectral asymmetry and Riemannian geometry. I, Math. Proc. Cambridge Philos. Soc. 77 (1975), 43β69. MR 397797, DOI 10.1017/S0305004100049410
- Shiing Shen Chern and James Simons, Characteristic forms and geometric invariants, Ann. of Math. (2) 99 (1974), 48β69. MR 353327, DOI 10.2307/1971013
- Peter B. Gilkey, The boundary integrand in the formula for the signature and Euler characteristic of a Riemannian manifold with boundary, Advances in Math. 15 (1975), 334β360. MR 368084, DOI 10.1016/0001-8708(75)90141-3
- James L. Heitsch and H. Blaine Lawson Jr., Transgressions, Chern-Simons invariants and the classical groups, J. Differential Geometry 9 (1974), 423β434. MR 353328
- Morris W. Hirsch, Immersions of manifolds, Trans. Amer. Math. Soc. 93 (1959), 242β276. MR 119214, DOI 10.1090/S0002-9947-1959-0119214-4
- F. Hirzebruch and D. Zagier, The Atiyah-Singer theorem and elementary number theory, Mathematics Lecture Series, No. 3, Publish or Perish, Inc., Boston, Mass., 1974. MR 0650832
- Tetsuro Kawasaki, The signature theorem for $V$-manifolds, Topology 17 (1978), no.Β 1, 75β83. MR 474432, DOI 10.1016/0040-9383(78)90013-7
- Tetsuro Kawasaki, The index of elliptic operators over $V$-manifolds, Nagoya Math. J. 84 (1981), 135β157. MR 641150
- Mutsumi Komuro, On Atiyah-Patodi-Singer $\eta$-invariant for $S^{1}$-bundles over Riemann surfaces, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1984), no.Β 3, 525β548. MR 731516 W. Neumann and M. Jankins, Seifert manifolds, Lecture notes, Brandeis Univ., 1981.
- IchirΓ΄ Satake, The Gauss-Bonnet theorem for $V$-manifolds, J. Math. Soc. Japan 9 (1957), 464β492. MR 95520, DOI 10.2969/jmsj/00940464
- Peter Scott, The geometries of $3$-manifolds, Bull. London Math. Soc. 15 (1983), no.Β 5, 401β487. MR 705527, DOI 10.1112/blms/15.5.401
- JosΓ© Seade and Brian Steer, A note on the eta function for quotients of $\textrm {PSL}_2(\textbf {R})$ by co-compact Fuchsian groups, Topology 26 (1987), no.Β 1, 79β91. MR 880510, DOI 10.1016/0040-9383(87)90023-1 W. Thurston, The geometry and topology of $3$-manifolds, Lecture notes, Princeton Univ., 1978.
- William P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no.Β 3, 357β381. MR 648524, DOI 10.1090/S0273-0979-1982-15003-0
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 346 (1994), 641-659
- MSC: Primary 55R55; Secondary 55R10, 57N10, 57R20
- DOI: https://doi.org/10.1090/S0002-9947-1994-1257644-9
- MathSciNet review: 1257644