Orthonormal frame on $\mathbb{S}^3$ orthogonal to foliations [duplicate]

Question

Does there exist a smooth orthonormal frame $X_1,X_2,X_3$ on $\mathbb{S}^3$ such that the distribution spanned by $X_i$ and $X_j$ is integrable for all $1\leq i,j\leq 3$?

Answer 1

I assume that you are asking about the global problem of existence on the round $3$-sphere, as the local existence of such frame fields is well-known and goes by the name 'triply orthogonal systems' in the classical literature.

Because the problem is conformally invariant, the local problem is the same as finding such orthonormal frame fields in $\mathbb{R}^3$. There, the existence and uniqueness theorem can be stated as follows:

If $S\subset\mathbb{R}^3$ is an embedded smooth surface and $X_1,X_2,X_3$ are a smooth orthonormal frame field along $S$ with the property that each of the $X_i$ is nowhere tangent to $S$ along $S$, then there is an open neighborhood $U$ of $S$ in $\mathbb{R}^3$ on which the $X_i$ extend uniquely to $U$ as an orthonormal frame field such that the 2-plane fields $D_{ij}$ spanned by $X_i$ and $X_j$ for $i\not=j$ are integrable in $U$.

In the real-analytic category, this result was known to Darboux, but, in the smooth category, the first place I know of a proof is in a 1984 paper by Dennis DeTurck and Deane Yang, Existence of elastic deformations with prescribed principal strains and triply orthogonal systems, Duke Math. J. $\mathbf{51}$, 243–260.

Globally on $S^3$, as far as I'm aware, there is no known solution. It is interesting to note that by a 1964 result of Novikov, any codimension $1$ foliation of the $3$-sphere has a compact leaf that is a torus. This can be used to show that there is no real-analytic codimension $1$ foliation of the $3$-sphere. (See H. Blaine Lawson, Jr. The Qualitative Theory of Foliations, CBMS Regional Conference Series in Mathematics, Volume 27 (1977), AMS/CBMS.)

Thus, while there might be a global smooth solution on the $3$-sphere, there cannot be a global real-analytic one.