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]

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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[12]

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

[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.

[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.

[15]

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

[16]

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

[17]

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

[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.

[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.

[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.

[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.

[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.

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[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.

[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.

[34]

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

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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[10]

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

[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.

[12]

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

[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.

[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.

[15]

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

[16]

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

[17]

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

[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.

[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.

[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.

[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.

[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.

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[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.

[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.

[34]

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

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