# American Institute of Mathematical Sciences

December  2018, 10(4): 445-465. doi: 10.3934/jgm.2018017

## On some aspects of the geometry of non integrable distributions and applications

 Departamento de Matemáticas-UPC, C. J. Girona, 3, Edif. C-3, Campus Nord-UPC, E-08034-Barcelona, Spain

Received  July 2017 Revised  September 2018 Published  November 2018

Fund Project: We acknowledge the financial support of the "Ministerio de Ciencia e Innovación" (Spain) project MTM2014-54855-P and and the Catalan Government project 2017–SGR–932.

We consider a regular distribution $\mathcal{D}$ in a Riemannian manifold $(M, g)$. The Levi-Civita connection on $(M, g)$ together with the orthogonal projection allow to endow the space of sections of $\mathcal{D}$ with a natural covariant derivative, the intrinsic connection. Hence we have two different covariant derivatives for sections of $\mathcal{D}$, one directly with the connection in $(M, g)$ and the other one with this intrinsic connection. Their difference is the second fundamental form of $\mathcal{D}$ and we prove it is a significant tool to characterize the involutive and the totally geodesic distributions and to give a natural formulation of the equation of motion for mechanical systems with constraints. The two connections also give two different notions of curvature, curvature tensors and sectional curvatures, which are compared in this paper with the use of the second fundamental form.

Citation: Miguel-C. Muñoz-Lecanda. On some aspects of the geometry of non integrable distributions and applications. Journal of Geometric Mechanics, 2018, 10 (4) : 445-465. doi: 10.3934/jgm.2018017
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