Abstract
The tangent bundle \(T^kM\) of order k, of a smooth Banach manifold M consists of all equivalent classes of curves that agree up to their accelerations of order k. For a Banach manifold M and a natural number k, first we determine a smooth manifold structure on \(T^kM\) which also offers a fiber bundle structure for \((\pi _k,T^kM,M)\). Then we introduce a particular lift of linear connections on M to geometrize \(T^kM\) as a vector bundle over M. More precisely based on this lifted nonlinear connection we prove that \(T^kM\) admits a vector bundle structure over M if and only if M is endowed with a linear connection. As a consequence, applying this vector bundle structure we lift Riemannian metrics and Lagrangians from M to \(T^kM\). In addition, using the projective limit techniques, we declare a generalized Fréchet vector bundle structure for \(T^\infty M\) over M.
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References
Aghasi, M., Dodson, C.T.J., Galanis, G.N., Suri, A.: Infinite dimensional second order ordinary differential equations via \(T^{2}M\). J. Nonlinear Anal. 67, 2829–2838 (2007)
Aghasi, M., Suri, A.: Splitting theorems for the double tangent bundles of Fréchet manifolds. Balkan J. Geom. Appl. 15(2), 1–13 (2010)
Averbuh, V.I., Smolyanov, O.G.: Differentiation theory in linear topological spaces, Uspehi Mat. Nauk 6. Russian Math. Surveys 6, 201–258 (1967)
Bucataru, I.: Linear connections for systems of higher order differential equations. Houston J. Math. 31(2), 315–332 (2005)
Bucataru, I.: Canonical semisprays for higher order Lagrange spaces. C. R. Acad. Sci. Paris Ser. I(345), 269–272 (2007)
Chernoff, P.R., Marsden, J.E.: Properties of Infinite Dimensional Hamiltonian Systems. Lecture Notes in Mathematics, vol. 421. Springer, New York (1974)
Crampin, M., Sarlet, W., Cantrijn, F.: Higher-order differential equations and higher-order Lagrangian mechanics. Math. Proc. Camb. Philos. Soc. 99(3), 565–587 (1986)
Dodson, C.T.J., Galanis, G.N.: Second order tangent bundles of infinite dimensional manifolds. J. Geom. Phys. 52, 127–136 (2004)
Galanis, G.N.: Projective limits of Banach vector bundles, Portugaliae Mathematica, vol. 55, Fasc. 1-1998, pp. 11–24
Galván, M.L.: Riemannian metrics on an infinite dimensional symplectic group. J. Math. Anal. Appl. 428, 1070–1084 (2015)
Hamilton, R.S.: The inverse function theorem of Nash and Moser. Bull. Am. Math. Soc. 7(1), 65–222 (1982)
Lang, S.: Fundumentals of Differential Geometry, Graduate Texts in Mathematics, vol. 191. Springer, Berlin (1999)
de León, M., Rodrigues, P.R.: Generalized Classical Mechanics and Field Theory, North-Holland Mathematics Studies, vol. 112. North-Holland Publishing, Amsterdam (1985)
Lloyd, J.W.: Higher order derivatives in topological linear spaces. J. Austral. Math. Soc. 25(Series A), 348–361 (1978)
Miron, R.: The Geometry of Higher Order Lagrange Spaces Applications to Mechanics and Physics. Kluwer Academic publishers, Dordrecht, Netherlands (1997)
Morimoto, A.: Liftings of tensor fields and connections to tangent bundles of higher order. Nagoya Math. J. 40, 99–120 (1970)
Popescu, M., Popescu, P.: Lagrangians and higher order tangent spaces. Balkan J. Geom. Appl. 15(1), 142–148 (2010)
Suri, A.: Geometry of the double tangent bundles of Banach manifodls. J. Geom. Phys. 74, 91–100 (2013)
Suri, A.: Isomorphism classes for higher order tangent bundles, Journal of Advences in Geometry (2014). arXiv:1412.7321 (to appear)
Suri, A.: Higher order frame bundles. Balkan J. Geom. Appl. 21(2), 102–117 (2016)
Vilms, J.: Connections on tangent bundles. J. Diff. Geom. 1, 235–243 (1967)
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Suri, A. Higher Order Tangent Bundles. Mediterr. J. Math. 14, 5 (2017). https://doi.org/10.1007/s00009-016-0812-7
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DOI: https://doi.org/10.1007/s00009-016-0812-7