TY - JOUR

T1 - Zeta regularized determinant of the Laplacian for classes of spherical space forms

AU - Bauer, W.

AU - Furutani, K.

N1 - Funding Information: Both authors are partially supported by the Grant-in-aid Scientific Research (C) (No. 17540202), Japan Society for the Promotion of Science. Copyright: Copyright 2007 Elsevier B.V., All rights reserved.

PY - 2007/9/22

Y1 - 2007/9/22

N2 - We derive the spectral zeta function in terms of certain Dirichlet series for a variety of spherical space forms MG. Extending results in [C. Nash, D. O'Connor, Determinants of Laplacians on lens spaces, J. Math. Phys. 36 (3) (1995) 1462-1505] the zeta-regularized determinant of the Laplacian on MG is obtained explicitly from these formulas. In particular, our method applies to manifolds of dimension higher than 3 and it includes the case where G arises from the dihedral group of order 2 m. As a crucial ingredient in our analysis we determine the dimension of eigenspaces of the Laplacian in form of some combinatorial quantities for various infinite classes of manifolds from the explicit form of the generating function in [A. Ikeda, On the spectrum of a Riemannian manifold of positive constant curvature, Osaka J. Math. 17 (1980) 75-93].

AB - We derive the spectral zeta function in terms of certain Dirichlet series for a variety of spherical space forms MG. Extending results in [C. Nash, D. O'Connor, Determinants of Laplacians on lens spaces, J. Math. Phys. 36 (3) (1995) 1462-1505] the zeta-regularized determinant of the Laplacian on MG is obtained explicitly from these formulas. In particular, our method applies to manifolds of dimension higher than 3 and it includes the case where G arises from the dihedral group of order 2 m. As a crucial ingredient in our analysis we determine the dimension of eigenspaces of the Laplacian in form of some combinatorial quantities for various infinite classes of manifolds from the explicit form of the generating function in [A. Ikeda, On the spectrum of a Riemannian manifold of positive constant curvature, Osaka J. Math. 17 (1980) 75-93].

KW - Generating function

KW - Hurwitz zeta function

KW - Lens space

KW - Spectral zeta function

KW - Spherical space form

KW - Zeta regularized determinant

UR - http://www.scopus.com/inward/record.url?scp=37349081959&partnerID=8YFLogxK

U2 - 10.1016/j.geomphys.2007.09.007

DO - 10.1016/j.geomphys.2007.09.007

M3 - Article

AN - SCOPUS:37349081959

VL - 58

SP - 64

EP - 88

JO - Journal of geometry and physics

JF - Journal of geometry and physics

SN - 0393-0440

IS - 1

ER -