Eta Invariants and Conformal Immersions

Abstract

The purpose of this paper is to give necessary conditions of the existence of a global conformal immersion of M in an appropriate Euclidean space in terms of the eta invariants and to give examples by applying these results. Our main results are Theorem 3. 10 and Theorem 3. 12. In their paper [1], Atiyah, Patodi and Singer defined a real valued spectral invariant of ]\I which is called the eta invariant of M. Throughout this paper ??(M) denotes the eta invariant of M. r/(M) can be calculated in some cases. Let M be a regular covering space over M with finite covering group. We assume that the orientations and the Riemannian metrics of M and M are compatible by the covering projection. Moreover we assume that M admits an orientation-reversing isometry. Then, the formula in [7] enable us to calculate i](M) with respect to any 'Riemannian metric of M. In Section 1, we recall the notion of the eta invariant and give an example of the calculation. On the one hand, in his paper [11], Simons defined a singular JR/Z-cochain on M which is called the-5-character or the Chern-Simons invariant. The S-character is an obstruction to the existence of a global conformal immersion of M in an appropriate Euclidean space. In many cases, however, it is difficult to calculate the S-characters and they seem to be not calculated with respect to non-standard metrics. In Section 2, we recall the notion of the S-character. In Section 3, by connecting the ^-character and the eta invariant, we give necessary conditions of the existence of a global conformal

Computations and Applications of η Invariants

“…These η-invariants are closely related to the ξ-invariant of M and the Γ-equivariant η-invariants of S 3 , see section 4.a. Seade [76] and Tsuboi [79] compute η-invariants for certain spherical space forms as average over equivariant η-invariants, see also [7]. Degeratu extends these computations to orbifold quotients in [33] and exhibits a relation with the Molien series.…”