A smooth choice of horizontal tangent subspaces complementing the vertical spaces in a fiber bundle.
Let π:E→M be a surjective submersion. For each e∈E, write VeE=ker(dπe)⊂TeE.
Definition. A horizontal distribution is a smooth assignment
e⟼HeE⊂TeE
of a constant-rank subspace such that for every e∈E,
TeE=HeE⊕VeE.
“Smooth” means that locally there exist smooth vector fields on E whose values span HeE 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.
ExamplesOpen
Constant horizontals on a product. On M×F, taking H(x,f)E=TxM⊕{0} defines a horizontal distribution.
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.
From a connection. Any Ehresmann connection (or principal connection) determines a horizontal distribution by declaring HeE to be the horizontal subspace at e.
Definition. A (smooth) vector field on M is a smooth map X:M→TM such that π∘X=idM. Equivalently, X is a smooth section of the tangent bundle, assigning to each p∈M a tangent vector
Xp∈TpM
(where TpM is the tangent space at p) in a way that is smooth in local coordinates.
A vector field can also be viewed as a derivation on smooth functions: for each X and each f∈C∞(M), one obtains a smooth function X(f)∈C∞(M) defined by differentiating f in the direction X. Using the pairing between tangent and cotangent spaces (see the cotangent bundle), this can be written pointwise as
Definition. A horizontal subbundle is a smooth subbundle HE⊂TE such that for every e∈E,
TeE=HeE⊕VeE.
A choice of horizontal subbundle is exactly the same data as an Ehresmann connection. In particular, the restriction of dπe to HeE is an isomorphism HeE≅Tπ(e)M, which is what makes the horizontal lift of a tangent vector well-defined and unique.
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).