Recently I've been stuck by a concrete problem. I'll try to make it more general.
Suppose $M$ is a simply connected spin manifold (with higher enough dimension), and $S^1$ acts on $M$ effectively. Then the action will induce a framing $F$ on the normal bundle of a principal orbit $C\cong S^1$. Note that $C$ also inherits a natural framing $F'$ from the spin structure of $M$. Is there any way to determine if they are the same?
