# 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
##### References:

show all references

##### References:
 [1] Dawan Mustafa, Bernt Wennberg. Chaotic distributions for relativistic particles. Kinetic & Related Models, 2016, 9 (4) : 749-766. doi: 10.3934/krm.2016014 [2] Axel Heim, Vladimir Sidorenko, Uli Sorger. Computation of distributions and their moments in the trellis. Advances in Mathematics of Communications, 2008, 2 (4) : 373-391. doi: 10.3934/amc.2008.2.373 [3] Fengwei Li, Qin Yue, Fengmei Liu. The weight distributions of constacyclic codes. Advances in Mathematics of Communications, 2017, 11 (3) : 471-480. doi: 10.3934/amc.2017039 [4] Mark Pollicott. Closed geodesic distribution for manifolds of non-positive curvature. Discrete & Continuous Dynamical Systems - A, 1996, 2 (2) : 153-161. doi: 10.3934/dcds.1996.2.153 [5] H.T. Banks, Jimena L. Davis. Quantifying uncertainty in the estimation of probability distributions. Mathematical Biosciences & Engineering, 2008, 5 (4) : 647-667. doi: 10.3934/mbe.2008.5.647 [6] BronisŁaw Jakubczyk, Wojciech Kryński. Vector fields with distributions and invariants of ODEs. Journal of Geometric Mechanics, 2013, 5 (1) : 85-129. doi: 10.3934/jgm.2013.5.85 [7] Sari Lasanen. Non-Gaussian statistical inverse problems. Part I: Posterior distributions. Inverse Problems & Imaging, 2012, 6 (2) : 215-266. doi: 10.3934/ipi.2012.6.215 [8] Misha Bialy. On Totally integrable magnetic billiards on constant curvature surface. Electronic Research Announcements, 2012, 19: 112-119. doi: 10.3934/era.2012.19.112 [9] Thomas Hillen, Kevin J. Painter, Amanda C. Swan, Albert D. Murtha. Moments of von mises and fisher distributions and applications. Mathematical Biosciences & Engineering, 2017, 14 (3) : 673-694. doi: 10.3934/mbe.2017038 [10] Azmy S. Ackleh, Ben G. Fitzpatrick, Horst R. Thieme. Rate distributions and survival of the fittest: a formulation on the space of measures. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 917-928. doi: 10.3934/dcdsb.2005.5.917 [11] Georgy P. Karev. Dynamics of heterogeneous populations and communities and evolution of distributions. Conference Publications, 2005, 2005 (Special) : 487-496. doi: 10.3934/proc.2005.2005.487 [12] Eugen Mihailescu. Equilibrium measures, prehistories distributions and fractal dimensions for endomorphisms. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2485-2502. doi: 10.3934/dcds.2012.32.2485 [13] Francis Ribaud. Semilinear parabolic equations with distributions as initial data. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 305-316. doi: 10.3934/dcds.1997.3.305 [14] Livio Flaminio, Giovanni Forni, Federico Rodriguez Hertz. Invariant distributions for homogeneous flows and affine transformations. Journal of Modern Dynamics, 2016, 10: 33-79. doi: 10.3934/jmd.2016.10.33 [15] Slobodan N. Simić. Hölder forms and integrability of invariant distributions. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 669-685. doi: 10.3934/dcds.2009.25.669 [16] Genghong Lin, Jianshe Yu, Zhan Zhou, Qiwen Sun, Feng Jiao. Fluctuations of mRNA distributions in multiple pathway activated transcription. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1543-1568. doi: 10.3934/dcdsb.2018219 [17] Marco Zambon, Chenchang Zhu. Distributions and quotients on degree $1$ NQ-manifolds and Lie algebroids. Journal of Geometric Mechanics, 2012, 4 (4) : 469-485. doi: 10.3934/jgm.2012.4.469 [18] Todd Young. Asymptotic measures and distributions of Birkhoff averages with respect to Lebesgue measure. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 359-378. doi: 10.3934/dcds.2003.9.359 [19] Evgeny L. Korotyaev. Estimates for solutions of KDV on the phase space of periodic distributions in terms of action variables. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 219-225. doi: 10.3934/dcds.2011.30.219 [20] Jan Boman. Unique continuation of microlocally analytic distributions and injectivity theorems for the ray transform. Inverse Problems & Imaging, 2010, 4 (4) : 619-630. doi: 10.3934/ipi.2010.4.619

2018 Impact Factor: 0.525