TITLE: When can one continuously prescribe a unit vector orthogonal to a given orthonormal system?
QUESTION [58 upvotes]: Let $1 \leq k < n$ be natural numbers.  Given orthonormal vectors $u_1,\dots,u_k$ in ${\bf R}^n$, one can always find an additional unit vector $v \in {\bf R}^n$ that is orthogonal to the preceding $k$.  My question is: under what conditions on $k,n$ is it possible to make $v$ depend continuously on $u_1,\dots,u_k$, as the tuple $(u_1,\dots,u_k)$ ranges over all possible orthonormal systems?  (For my application I actually want smooth dependence, but I think that a continuous map can be averaged out to be smooth without difficulty.)
When $k=n-1$ then one can just pick the unique unit normal to the span of the $u_1,\dots,u_k$ that is consistent with a chosen orientation on ${\bf R}^n$ (i.e., take wedge product and then Hodge dual, or just cross product in the $(k,n)=(2,3)$ case).  But I don't know what is going on in lower dimension.  Intuitively it seems to me that if $n$ is much larger than $k$ then the problem is so underdetermined that there should be no topological obstructions (such as that provided by the Borsuk-Ulam theorem), but I don't have the experience in algebraic topology to make this intuition precise.
It would suffice to exhibit a global section of the normal bundle of the (oriented) Grassmannian $Gr(k,n)$, though I don't know how to calculate the space of such sections.

REPLY [67 votes]: $\def\RR{\mathbb{R}}$ This problem was solved by 
Whitehead, G. W., Note on cross-sections in Stiefel manifolds, Comment. Math. Helv. 37, 239-240 (1963). ZBL0118.18702.
Such sections exist only in the cases $(k,n) = (1,2m)$, $(n-1, n)$, $(2,7)$ and $(3,8)$. 
All sections can be given by antisymmetric multilinear maps (and thus, in particular, can be taken to be smooth). The $(2,7)$ product is the seven dimensional cross product, which is octonion multiplication restricted to the octonions of trace $0$. 
The $(3,8)$ product was computed by 
Zvengrowski, P., A 3-fold vector product in $R^8$, Comment. Math. Helv. 40, 149-152 (1966). ZBL0134.38401
to be given by the formula 
$$X(a,b,c) = -a (\overline{b} c) + a (b \cdot c) - b (c \cdot a) + c (a \cdot b)$$
where $\cdot$ is dot product while multiplication with no symbol and $\overline{b }$ have their standard octonion meanings. Note that, if $(a,b,c)$ are orthogonal, the last $3$ terms are all $0$, so the expression simplifies to $- a (\overline{b} c)$; writing in the formula in the given manner has the advantage that $X(a,b,c)$ is antisymmetric in its arguments and perpendicular to the span of $a$, $b$ and $c$ for all $(a,b,c)$.