  Using the notion of Levi form of a smooth distribution, we discuss the local
and the global problem of existence of one horizontal section of a smooth
vector bundle endowed with a horizontal distribution. The analysis will lead to
the formulation of a "one-leaf" analogue of the classical Frobenius
integrability theorem in elementary differential geometry. Several applications
of the result will be discussed. First, we will give a characterization of
symmetric connections arising as Levi-Civita connections of semi-Riemannian
metric tensors. Second, we will prove a general version of the classical
Cartan-Ambrose-Hicks Theorem giving conditions on the existence of an affine
map with prescribed differential at one point between manifolds endowed with
connections.
