×

The single-leaf Frobenius theorem with applications. (English) Zbl 1156.53306

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.

MSC:

53C05 Connections (general theory)
55R25 Sphere bundles and vector bundles in algebraic topology
58A30 Vector distributions (subbundles of the tangent bundles)
PDFBibTeX XMLCite
Full Text: arXiv