The single-leaf Frobenius Theorem with Applications

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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.
39 pages, no figure

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Differential Geometry, 53B05; 53C05; 53C42; 55R25

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http://salesiana.dossiersoluciones.com/handle/123456789/103815

Consulte el texto completo en el siguiente enlace:

https://arxiv.org/abs/math/0510555
 http://arxiv.org/abs/math/0510555

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arXiv