Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
권호 표지
DOI QR코드

DOI QR Code

THE ZETA-DETERMINANTS OF LAPLACIANS ON THE MOBIUS BAND AND KLEIN BOTTLE

  • Yoonweon Lee (Department of Mathematics Education, Inha University)
  • Received : 2024.01.22
  • Accepted : 2024.07.18
  • Published : 2024.12.20
Download PDF
⟨ Previous Next ⟩

Abstract

We compute the zeta-determinants of the scalar Laplacians defined on the Möbius band and Klein bottle when the flat metrics are given. We consider the difference between these zeta-determinants and those of the product manifolds, and use the BFK-gluing formula to compute the difference. The zeta-determinants of product manifolds are well known and this computes the zeta-determinants on the Möbius band and Klein bottle. We finally show that the zeta-determinant on the Klein bottle satisfies the BFK-gluing formula.

Keywords

Acknowledgement

This work was supported by INHA UNIVERSITY Research Grant.

References

  1. G. Bazzoni, J. C. Marrero, and J. Oprea, A splitting theorem for compact Vaisman manifolds, Rend. Semin. Mat. Univ. Politec. Torino 74 (2016), no. 1, 21-29.
  2. G. Bazzoni and J. Oprea, On the structure of co-Kahler manifolds, Geom. Dedicata 170 (2014), 71-85.
  3. D. E. Blair, The theory of quasi-Sasakian structures, J. Diff. Geom. 1 (1967), 331-345.
  4. D. Burghelea, L. Friedlander, and T. Kappeler, Mayer-Vietoris type formula for determinants of elliptic differential operators, J. of Funct. Anal. 107 (1992), 34-66.
  5. G. Carron, Determinant relatif et fonction Xi, Amer. J. Math. 124 (2002), 307-352.
  6. J. Cheeger, Analytic torsion and the heat equation, Ann. Math. (2) 109 (1979), 259-322.
  7. E. Elizalde, Ten Physical Applications of Spectral Zeta Functions, 2nd Ed. Springer, 2012.
  8. R. Forman, Functional Determinants and geometry, Invent. Math. 88 (1987), 447-493.
  9. K. Furutani and S. de Gosson, Determinant of Laplacians on Heisenberg manifolds, J. Math. Phys. 48 (2003), 438-479.
  10. P. B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, 2nd Edition, CRC Press, Inc., 1994.
  11. S. Hawking, Zeta function regularization of path integrals in curved space-time, Comm. Math. Phys. 55 (1977), 133-148.
  12. K. Kirsten and Y. Lee, The Burghelea-Friedlander-Kappeler-gluing formula for zeta-determinants on a warped product manifold and a product manifold, J. Math. Phys. 58 (2015), no. 12, 123501-1-19.
  13. K. Kirsten and Y. Lee, The BFK-gluing formula and relative determinants on manifolds with cusps, J. Geom. Phys. 117 (2017), 197-213.
  14. K. Kirsten and Y. Lee, The BFK-gluing formula and the curvature tensors on a 2-dimensional compact hypersurface, J. Spectr. Theory 10 (2020), 1007-1051.
  15. K. Kirsten and Y. Lee, The zeta-determinant of the Dirichlet-to-Neumann operator on the Steklov Problem on forms, to appear in Ann. Global Anal. Geom.; arXiv:2404.14562.
  16. K. Kirsten and F. L. Williams Edited, A Window into Zeta and Modular Physics, Cambridge Univ. Press, 2010.
  17. Y. Lee, The zeta-determinants and analytic torsion of a metric mapping torus, to appear in Kodai Math. J.; arXiv:2407.06609.
  18. H. Li, Topology of co-symplectric/co-Kahler manifolds, Asian J. Math. 12 (2008), no 4, 527-544.
  19. W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and theorems for special functions of mathematical physics. Springer-Verlag, Berlin.Heidelberg, 1966.
  20. W. Muller, Analytic torsion and R-torsion for Riemannian manifolds, Adv. in Math. 28 (1978), 233-305.
  21. J. Muller and W. Muller, Regularized determinants of Laplace type operators, analytic surgery and relative determinants, Duke. Math. J. 133 (2006), 259-312.
  22. D. B. Ray and I. M. Singer, R-torsion and the Laplacian on Riemannian manifolds, Adv. in Math. 7 (1971), 145-210.
  23. A. Voros, Spectral functions, special functions and Selberg zeta function, Comm. Math. Phys. 110 (1987), 439-465.