  In this note we study the problem of conformally flat structures bounding
conformally flat structures and show that the eta invariants give obstructions.
These lead us to the definition of an abelian group, the conformal cobordism
group, which classifies the conformally flat structures according to whether
they bound (higher dimensional) conformally flat structures in a conformally
invariant way. The eta invariant gives rise to a homomorphism from this group
to the circle group, which can be highly nontrivial. It remains an interesting
question of how to compute this group.
