Ray-singer Zeta Functions for Compact Flat Manifolds

A compact orientable flat manifold M is expressed as M = R/Γ with a torsion free discrete subgroup of the group of orientation preserving motions of R. There is a natural one-to-one correspondence between the set of conjugacy classes [γ], γ ∈ Γ, and the set of free homotopy classes of maps of S into M . We denote by M[γ] the set of closed geodesics c : S −→ M belonging to the homotopy class [γ]. The space M[γ] equipped with compact open topology is a compact connected manifold, and the map M[γ] −→ M defined by c �→ c(0) is an immersion which induces a flat metric on M[γ] (see [8], [9]). The fundamental group of M[γ] is isomorphic to the centralizer Γγ of γ. We set lγ = length of c ∈ M[γ], which depends only on the class [γ]. The following proposition is a straightforward generalization of the trace formula established in [7].