  We show that the R/Z part of the analytically defined eta invariant of
Atiyah-Patodi-Singer for a Dirac operator on an odd dimensional closed spin
manifold can be expressed purely geometrically through a stable Chern-Simons
current on a higher dimensional sphere. As a preliminary application, we
discuss the relation with the Atiyah-Patodi-Singer R/Z index theorem for
unitary flat vector bundles, and prove an R refinement in the case where the
Dirac operator is replaced by the Signature operator. We also extend the above
discussion to the case of eta invariants associated to Hermitian vector bundles
with non-unitary connection, which are constructed by using a trick due to
Lott.
