About the Eta-Invariants of Berger Spheres

TL;DR

This paper calculates eta-invariants for Berger spheres, providing explicit formulas for Dirac and signature invariants using special Riemannian metrics and multiplicative Pontryagin forms.

Contribution

It introduces a method to compute eta-invariants for Berger metrics on spheres via integral formulas involving Pontryagin forms.

Findings

Explicit formulas for eta-invariants of Berger spheres

Connection between Pontryagin forms and eta-invariants

Generating functions for Dirac and signature eta-invariants

Abstract

The integral of the top dimensional term of the multiplicative sequence of Pontryagin forms associated to an even formal power series is calculated for special Riemannian metrics on the unit ball of a hermitean vector space. Using this result we calculate the generating function of the reduced Dirac and signature eta-invariants for the family of Berger metrics on the odd dimensional spheres.