Horizontal distribution

Let $\pi:E\to M$ be a surjective submersion. For each $e\in E$, write $V_eE=\ker(d\pi_e)\subset T_eE$.

Definition. A horizontal distribution is a smooth assignment

of a constant-rank subspace such that for every $e\in E$,

“Smooth” means that locally there exist smooth vector fields on $E$ whose values span $H_eE$ at each point.

A horizontal distribution is the pointwise version of a horizontal subbundle ; the two viewpoints are equivalent. The distribution is called integrable precisely when it is involutive, in the sense of integrability of horizontals .

Examples

1. Constant horizontals on a product. On $M\times F$, taking $H_{(x,f)}E=T_xM\oplus\{0\}$ defines a horizontal distribution.
2. Transverse foliation. If $E$ is foliated by submanifolds that project locally diffeomorphically onto $M$ (for example, graphs of local sections), then their tangent spaces define a horizontal distribution.
3. From a connection. Any Ehresmann connection (or principal connection) determines a horizontal distribution by declaring $H_eE$ to be the horizontal subspace at $e$.