On framed manifolds

Question

Let $M$ be an $n$-dimensional compact manifold without boundary sitting in $\mathbb R^{n+k}$. We call $M$ framed if at every point $x \in M$ there exist linearly independent vectors $v_1, \dots, v_k$ spanning the normal space at $x$ and varying smoothly with $x$. An example would be the $2$-sphere in $\mathbb R^3$. Now consider the Klein bottle. It seems fairly clear that the obvious framing of normals to it (in $\mathbb R^3$) cannot be made smooth: at one point the normal will change its direction by 180 degrees because the Klein bottle is not orientable. However it is not too clear to me whether there cannot be any smoothly varying normal vectors for any $k\neq 1$. More generally, I am wondering about how to prove non-existence of framings of arbitrary smooth closed manifolds. How to prove a given smooth closed manifolds is not framable?

Answer 1

A framed manifold in $\mathbb R^{n+k}$ is necessarily orientable. Consider the canonical volume form $\omega_0 = dx^1\wedge \dots \wedge dx^{n+k}$. If $v_1, dots, v_k$ is a framing, then a orientation form on $M$ can be defined by $$\omega(w_1, \dots, w_n) := \omega_0(v_1, \dots, v_k, w_1, \dots, w_n).$$ In codimension one this is not only necessary but also sufficient. In higher codimension, there are additional obstructions I think.