TITLE: Klein bottle in $\mathbb{R}^4$ does not have a couple of normal vector fields
QUESTION [7 upvotes]: For an orientable 2-manifold in $\mathbb{R}^4$, it's somewhat obvious that if the manifold has a normal vector field then it has a pair of normal vector fields.
I am trying to understand why it is not true for the Klein bottle embedded in $\mathbb{R}^4$. It has a normal vector field, but I can't visualize how adding a second normal vector will cause a problem.

REPLY [2 votes]: Came up with a bit simpler solution. Let $e_1(x)$ be a tangent vector field on the Klein bottle $K$. Suppose that $n_1(x)$ and $n_2(x)$ - pair of linearly indenpendent normal vector fields. Then we can choose $e_2(x)$ in the $TK_x$ so that four vectors $(e_1(x), e_2(x), n_1(x), n_2(x))$ form a positively oriented basis in $\mathbb{R}^4$. This would give us orientation on the Klein bottle.