Framing of an embedded sphere
= Framing of an embedded sphere
A framing of an embedding $S^{k-1}\hookrightarrow N^{n-1}$ is a trivialization of its rank-$(n-k)$ <normal bundle>, equivalently a continuously varying ordered basis of normal vectors. After fixing one framing, homotopy classes of orientation-compatible framings form a torsor for $[S^{k-1},SO(n-k)]$.