The Single-leaf Frobenius Theorem with Applications

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.