Piccione, Paolo; Tausk, Daniel V. The single-leaf Frobenius theorem with applications. (English) Zbl 1156.53306 Resen. Inst. Mat. Estat. Univ. São Paulo 6, No. 4, 337-381 (2005). Summary: 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. Cited in 2 Documents MSC: 53C05 Connections (general theory) 55R25 Sphere bundles and vector bundles in algebraic topology 58A30 Vector distributions (subbundles of the tangent bundles) Keywords:smooth distributions; Levi form; Frobenius theorem; affine connections; Levi-Civita connections; Cartan-Ambrose-Hicks theorem PDFBibTeX XMLCite \textit{P. Piccione} and \textit{D. V. Tausk}, Resen. Inst. Mat. Estat. Univ. São Paulo 6, No. 4, 337--381 (2005; Zbl 1156.53306) Full Text: arXiv