Integrable horizontal distribution

Let $\pi:E\to M$ be a surjective submersion and let $H\subset TE$ be a horizontal distribution .

Definition. The horizontal distribution $H$ is integrable if it is involutive: for any smooth vector fields $X,Y$ on $E$ taking values in $H$ (i.e. horizontal vector fields), their Lie bracket $[X,Y]$ also takes values in $H$.

By the Frobenius theorem, integrability is equivalent to the existence of a foliation of $E$ by immersed submanifolds whose tangent spaces equal $H$. For a horizontal distribution, such leaves are automatically transverse to the fibers, so locally each leaf projects diffeomorphically onto an open subset of $M$. In many geometric settings, non-integrability is measured by an appropriate notion of curvature (the “vertical part” of brackets of horizontal fields).

Examples

1. Product horizontals are integrable. On $E=M\times F$ with $H_{(x,f)}=T_xM\oplus\{0\}$, the horizontal leaves are the slices $M\times\{f\}$, so $H$ is integrable.
2. Flat principal connection gives integrable horizontals. On a principal bundle with a flat connection, the horizontal distribution is involutive; locally, horizontal leaves provide local “parallel” sections.
3. Non-example from curvature. For the Levi-Civita connection on the frame bundle of a curved Riemannian manifold (e.g. the round sphere), the induced horizontal distribution is typically not integrable; horizontal loops can produce nontrivial holonomy.