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:
[1]

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M. Barbero-LiñánM. de LeónD. Martín de DiegoJ. C. Marrero and M. C. Muñoz-Lecanda, Kinematic reduction and the Hamilton-Jacobi equation, J. Geom. Mech., 4 (2012), 207-237.  doi: 10.3934/jgm.2012.4.207.  Google Scholar

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F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems. Modeling, Analysis, and Design for Simple Mechanical Control Systems, Texts in Applied Mathematics, 49. Springer-Verlag, New York, 2005. doi: 10.1007/978-1-4899-7276-7.  Google Scholar

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M. de León, A historical review on nonholonomic mechanics, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 106 (2012), 191-224.  doi: 10.1007/s13398-011-0046-2.  Google Scholar

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M. P. do Carmo, Riemannian Geometry, Birkhäuser, Berlin, 1992. doi: 10.1007/978-1-4757-2201-7.  Google Scholar

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O. Gil-Medrano, Geometric properties of some classes of Riemannian almost product manifolds, Rendiconti Circ. Mat. Palermo, 32 (1983), 315-329.  doi: 10.1007/BF02848536.  Google Scholar

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A. Gray, Pseudo-Riemannian almost product manifolds and submersions, J. Math. and Mech., 16 (1967), 715-737.   Google Scholar

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N. J. Hicks, Notes on Differential Geometry, Van Nostrand Mathematical Studies, No. 3 D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965.  Google Scholar

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M. H. Kobayashi and W. M. Oliva, Nonholonomic systems and the geometry of constraints, Qual. Theory Dyn. Syst., 5 (2004), 247-259.  doi: 10.1007/BF02972680.  Google Scholar

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I. Kupka and W. M. Oliva, The non-holonomic mechanics, J. Differential Equations, 169 (2001), 169-189.  doi: 10.1006/jdeq.2000.3897.  Google Scholar

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J. M. Lee, Introduction to Smooth Manifolds, Second edition. Graduate Texts in Mathematics, 218. Springer, New York, 2013.  Google Scholar

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J. M. Lee, Riemannian Manifolds. An Introduction to Curvature, Graduate Texts in Mathematics, 176. Springer-Verlag, New York, 1997. doi: 10.1007/b98852.  Google Scholar

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A. D. Lewis and R. M. Murray, Controllability of simple mechanical control systems, SIAM J. Control Optim., 35 (1997), 766-790.  doi: 10.1137/S0363012995287155.  Google Scholar

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A. D. Lewis, Affine connections and distributions with applications to nonholonomic mechanics, Rep. Math. Phys., 42 (1998), 135-164.  doi: 10.1016/S0034-4877(98)80008-6.  Google Scholar

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A. D. Lewis, Affine connections and distributions with applications to mechanics, Reports on Mathematical Physics, 42 (1998), 135-164.  doi: 10.1016/S0034-4877(98)80008-6.  Google Scholar

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A. D. Lewis, Simple mechanical control systems with constraints. Mechanics and nonlinear control systems, IEEE Trans. Automat. Control, 45 (2000), 1420-1436.  doi: 10.1109/9.871752.  Google Scholar

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H. Nijmeijer and A. van der Schaft, Nonlinear Dynamical Control Systems, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4757-2101-0.  Google Scholar

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B. O'Neill, Semi-Riemannian Geometry with Applications to Relativity, Pure and Applied Mathematics, 103. Academic Press, Inc., New York, 1983.  Google Scholar

[24]

G. Prince, Torsion and the second fundamental form for distributions, Commun. Math., 24 (2016), 23-28.  doi: 10.1515/cm-2016-0003.  Google Scholar

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B. L. Reinhart, The second fundamental form of a plane field, J. Diff. Geom., 12 (1977), 619-627.  doi: 10.4310/jdg/1214434230.  Google Scholar

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B. L. Reinhart, Differential Geometry of Foliations, Springer-Verlag, New York, 1983. doi: 10.1007/978-3-642-69015-0.  Google Scholar

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A. Solov'ev, Second fundamental form of a distribution, Math. Zametki, 31 (1982), 139–146; English transl.: Math. Notes, 31 (1982), 71–75.  Google Scholar

[28]

A. Solov'ev, Curvature of a distribution, Mat. Zametki, 35 (1984), 111–124; English transl.: Math. Notes., 5 (1984), 61–68.  Google Scholar

[29]

M. Spivak, A Comprehensive Introduction to Differential Geometry. Vol. III, Second edition. Publish or Perish Inc., Wilmington, Del., 1979.  Google Scholar

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J. L. Synge, On the geometry of dynamics, Phil. Trans. R. Soc., 226 (1926), 31-106.   Google Scholar

[31]

J. L. Synge, Geodesics in non-holonomic geometry, Mathematische Annalen, 99 (1928), 738-751.  doi: 10.1007/BF01459122.  Google Scholar

[32]

G. Terra and M. H. Kobayashi, On classical mechanical systems with non-linear constraints, J. Geom. Phys., 49 (2004), 385-417.  doi: 10.1016/j.geomphys.2003.08.005.  Google Scholar

[33]

G. Terra and M. H. Kobayashi, On the variational mechanics with non-linear constraints, J. Math. Pures Appl., 83 (2004), 629-671.  doi: 10.1016/S0021-7824(03)00069-2.  Google Scholar

[34]

G. Vranceanu, Sur les espaces non holonomes, C. R. Acad. Sci. Paris, 183 (1926), 852-854.   Google Scholar

show all references

References:
[1]

V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, Dynamical systems, Ⅲ, Encyclopaedia Math. Sci., 3, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-540-48926-9_1.  Google Scholar

[2]

M. Barbero-LiñánM. de LeónD. Martín de DiegoJ. C. Marrero and M. C. Muñoz-Lecanda, Kinematic reduction and the Hamilton-Jacobi equation, J. Geom. Mech., 4 (2012), 207-237.  doi: 10.3934/jgm.2012.4.207.  Google Scholar

[3]

M. Barbero-Liñán and A. D. Lewis, Geometric interpretations of the symmetric product in affine differential geometry and applications, Int. J. Geom. Methods Mod. Phys., 9 (2012), 1250073, 33 pp. doi: 10.1142/S0219887812500739.  Google Scholar

[4]

A. Bejancu and H. R. Farran, Foliations and Geometric Structures, Mathematics and Its Applications, 580. Springer-Verlag, Dordrecht, 2006.  Google Scholar

[5]

F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems. Modeling, Analysis, and Design for Simple Mechanical Control Systems, Texts in Applied Mathematics, 49. Springer-Verlag, New York, 2005. doi: 10.1007/978-1-4899-7276-7.  Google Scholar

[6]

J. Cheeger and D. G. Ebin, Comparison Theorems in Riemannian Geometry, North-Holland Mathematical Library, Vol. 9. North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975.  Google Scholar

[7]

L. Conlon, Differentiable Manifolds, Second edition. Birkhäuser Advanced Texts: Basler Lehrb her. Birkh ser Boston, Inc., Boston, MA, 2001. doi: 10.1007/978-0-8176-4767-4.  Google Scholar

[8]

P. E. Crouch, Geometric structures in systems theory, IEE Proceedings. D. Control Theory and Applications, 128 (1981), 242-252.  doi: 10.1049/ip-d.1981.0051.  Google Scholar

[9]

M. de León, A historical review on nonholonomic mechanics, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 106 (2012), 191-224.  doi: 10.1007/s13398-011-0046-2.  Google Scholar

[10]

M. P. do Carmo, Riemannian Geometry, Birkhäuser, Berlin, 1992. doi: 10.1007/978-1-4757-2201-7.  Google Scholar

[11]

O. Gil-Medrano, Geometric properties of some classes of Riemannian almost product manifolds, Rendiconti Circ. Mat. Palermo, 32 (1983), 315-329.  doi: 10.1007/BF02848536.  Google Scholar

[12]

A. Gray, Pseudo-Riemannian almost product manifolds and submersions, J. Math. and Mech., 16 (1967), 715-737.   Google Scholar

[13]

N. J. Hicks, Notes on Differential Geometry, Van Nostrand Mathematical Studies, No. 3 D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965.  Google Scholar

[14]

M. H. Kobayashi and W. M. Oliva, Nonholonomic systems and the geometry of constraints, Qual. Theory Dyn. Syst., 5 (2004), 247-259.  doi: 10.1007/BF02972680.  Google Scholar

[15]

I. Kupka and W. M. Oliva, The non-holonomic mechanics, J. Differential Equations, 169 (2001), 169-189.  doi: 10.1006/jdeq.2000.3897.  Google Scholar

[16]

J. M. Lee, Introduction to Smooth Manifolds, Second edition. Graduate Texts in Mathematics, 218. Springer, New York, 2013.  Google Scholar

[17]

J. M. Lee, Riemannian Manifolds. An Introduction to Curvature, Graduate Texts in Mathematics, 176. Springer-Verlag, New York, 1997. doi: 10.1007/b98852.  Google Scholar

[18]

A. D. Lewis and R. M. Murray, Controllability of simple mechanical control systems, SIAM J. Control Optim., 35 (1997), 766-790.  doi: 10.1137/S0363012995287155.  Google Scholar

[19]

A. D. Lewis, Affine connections and distributions with applications to nonholonomic mechanics, Rep. Math. Phys., 42 (1998), 135-164.  doi: 10.1016/S0034-4877(98)80008-6.  Google Scholar

[20]

A. D. Lewis, Affine connections and distributions with applications to mechanics, Reports on Mathematical Physics, 42 (1998), 135-164.  doi: 10.1016/S0034-4877(98)80008-6.  Google Scholar

[21]

A. D. Lewis, Simple mechanical control systems with constraints. Mechanics and nonlinear control systems, IEEE Trans. Automat. Control, 45 (2000), 1420-1436.  doi: 10.1109/9.871752.  Google Scholar

[22]

H. Nijmeijer and A. van der Schaft, Nonlinear Dynamical Control Systems, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4757-2101-0.  Google Scholar

[23]

B. O'Neill, Semi-Riemannian Geometry with Applications to Relativity, Pure and Applied Mathematics, 103. Academic Press, Inc., New York, 1983.  Google Scholar

[24]

G. Prince, Torsion and the second fundamental form for distributions, Commun. Math., 24 (2016), 23-28.  doi: 10.1515/cm-2016-0003.  Google Scholar

[25]

B. L. Reinhart, The second fundamental form of a plane field, J. Diff. Geom., 12 (1977), 619-627.  doi: 10.4310/jdg/1214434230.  Google Scholar

[26]

B. L. Reinhart, Differential Geometry of Foliations, Springer-Verlag, New York, 1983. doi: 10.1007/978-3-642-69015-0.  Google Scholar

[27]

A. Solov'ev, Second fundamental form of a distribution, Math. Zametki, 31 (1982), 139–146; English transl.: Math. Notes, 31 (1982), 71–75.  Google Scholar

[28]

A. Solov'ev, Curvature of a distribution, Mat. Zametki, 35 (1984), 111–124; English transl.: Math. Notes., 5 (1984), 61–68.  Google Scholar

[29]

M. Spivak, A Comprehensive Introduction to Differential Geometry. Vol. III, Second edition. Publish or Perish Inc., Wilmington, Del., 1979.  Google Scholar

[30]

J. L. Synge, On the geometry of dynamics, Phil. Trans. R. Soc., 226 (1926), 31-106.   Google Scholar

[31]

J. L. Synge, Geodesics in non-holonomic geometry, Mathematische Annalen, 99 (1928), 738-751.  doi: 10.1007/BF01459122.  Google Scholar

[32]

G. Terra and M. H. Kobayashi, On classical mechanical systems with non-linear constraints, J. Geom. Phys., 49 (2004), 385-417.  doi: 10.1016/j.geomphys.2003.08.005.  Google Scholar

[33]

G. Terra and M. H. Kobayashi, On the variational mechanics with non-linear constraints, J. Math. Pures Appl., 83 (2004), 629-671.  doi: 10.1016/S0021-7824(03)00069-2.  Google Scholar

[34]

G. Vranceanu, Sur les espaces non holonomes, C. R. Acad. Sci. Paris, 183 (1926), 852-854.   Google Scholar

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