I'm stuck on a portion of Exercise 21, Chapter 2 in Petersen's Riemannian geometry text.
Fix a Riemannian manifold $(M,g).$ Suppose that I have two distributions $D^1$ and $D^2$ defined on $M.$ Suppose that these distributions are
- mutually orthogonal
- span the tangent space
- parallel (and hence integrable)
I want to conclude that $M$ splits locally as a product manifold.
I would like to apply the Frobenius theorem to the integrable distributions $D^1$ and $D^2.$ However, it seems to be that I cannot write $U=U_1xU_2$ with $TU_1 \subset D_1$ and $TU_2 \subset D_2$ unless I also know that the distributions commute (i.e. $[X,Y]=0, X\in D_1, Y \in D_2.$)
What am I doing wrong?