Abstract
The tangent bundle TkM of order k of a smooth Banach manifold M consists of all equivalence classes of curves that agree up to their accelerations of order k. In previous work the author proved that TkM, 1 ≤ k ≤∞, admits a vector bundle structure on M if and only if M is endowed with a linear connection, or equivalently if a connection map on TkM is defined. This bundle structure depends heavily on the choice of the connection. In this paper we ask about the extent to which this vector bundle structure remains isomorphic. To this end we define the k-th order differential Tkg : TkM ⟶ TkN for a given differentiable map g between manifolds M and N. As we shall see, Tkg becomes a vector bundle morphism if the base manifolds are endowed with g-related connections. In particular, replacing a connection with a g-related one, where g : M ⟶ M is a diffeomorphism, one obtains invariant vector bundle structures. Finally, using immersions on Hilbert manifolds, convex combinations of connection maps and manifolds of Cr maps we offer three examples for our theory, showing its interaction with known problems such as the Sasaki lift of metrics.
References
[1] M. Aghasi, A. Suri, Splitting theorems for the double tangent bundles of Fréchet manifolds. Balkan J. Geom. Appl. 15 (2010), 1-13. MR2608533 Zbl 1218.58006Search in Google Scholar
[2] V. I. Averbuh, 0. G. Smoljanov, Differentiation theory in linear topological spaces. Uspehi Mat. Nauk22 (1967), 201-260. MR0223886 Zbl 0195.42601 Zbl 0175.14802Search in Google Scholar
[3] V. I. Averbuh, 0. G. Smoljanov, Different definitions of derivative in linear topological spaces. Uspehi Mat. Nauk23 (1968), 67-116. MR0246118 Zbl 0196.15702Search in Google Scholar
[4] I. Bucataru, Linear connections for systems of higher order differential equations. Houston J. Math. 31(2005), 315-332. MR2132839 Zbl 1078.58005Search in Google Scholar
[5] M. de León, P. R. Rodrigues, Generalized classical mechanics and field theory. North-Holland 1985. MR808964 Zbl 0581.58015Search in Google Scholar
[6] C. T. J. Dodson, G. N. Galanis, Second order tangent bundles of infinite dimensional manifolds. J. Geom. Phys. 52 (2004), 127-136. MR2088972 Zbl 1076.5800210.1016/j.geomphys.2004.02.005Search in Google Scholar
[7] C. T. J. Dodson, G. N. Galanis, E. Vassiliou, Isomorphism classes for Banach vector bundle structures of second tangents. Math. Proc. Cambridge Philos. Soc. 141 (2006), 489-496. MR2281411 Zbl 1109.5801110.1017/S0305004106009467Search in Google Scholar
[8] H. I. Elĭasson, Geometry of manifolds of maps. J. Differential Geometry1 (1967), 169-194. MR0226681 Zbl 0163.4390110.4310/jdg/1214427887Search in Google Scholar
[9] P. Flaschel, W. Klingenberg, Riemannsche Hilbertmannigfaltigkeiten. Periodische Geodätische. Mit einem Anhang von H. Karcher. Springer 1972. MR0341527 Zbl 0238.5800910.1007/BFb0080917Search in Google Scholar
[10] G. Galanis, Projective limits of Banach-Lie groups. Period. Math. Hungar. 32 (1996), 179-191. MR1407918 Zbl 0866.5800910.1007/BF02109787Search in Google Scholar
[11] G. Ν. Galanis, Projective limits of Banach vector bundles. Portugal. Math. 55 (1998), 11-24. MR1612319 Zbl 0904.58002Search in Google Scholar
[12] R. S. Hamilton, The inverse function theorem of Nash and Moser. Bull. Amer. Math. Soc. (N.S.)7 (1982), 65-222. MR656198 Zbl 0499.5800310.1090/S0273-0979-1982-15004-2Search in Google Scholar
[13] H. H. Keller, Differential calculus in locally convex spaces. Springer 1974. MR0440592 Zbl 0293.5800110.1007/BFb0070564Search in Google Scholar
[14] W. P. A. Klingen berg, Riemannian geometry, volume 1 of de Gruyter Studies in Mathematics. De Gruyter 1995. MR1330918 Zbl 0911.53022Search in Google Scholar
[15] 0. V. Kunakovskaya, On smooth partitions of unity on Banach manifolds. Izv. Vyssh. Uchebn. Zaved. Mat. (1997), no. 10, 51-58. MR1488002Search in Google Scholar
[16] S. Lang, Fundamentals of differential geometry. Springer 1999. MR1666820 Zbl 0932.5300110.1007/978-1-4612-0541-8Search in Google Scholar
[17] J. A. Leslie, On a differential structure for the group of diffeomorphisms. Topology6 (1967), 263-271. MR0210147 Zbl 0147.2360110.1016/0040-9383(67)90038-9Search in Google Scholar
[18] J. A. Leslie, Some Frobenius theorems in global analysis. J. Differential Geometry2 (1968), 279-297. MR0251750 Zbl 0169.5320110.4310/jdg/1214428441Search in Google Scholar
[19] J. W. Lloyd, Higher order derivatives in topological linear spaces. J. Austral. Math. Soc. Ser. A25 (1978), 348-361. MR0494213 Zbl 0405.5801010.1017/S1446788700021091Search in Google Scholar
[20] R. Miron, The geometry of higher-order Lagrange spaces, volume 82 of Fundamental Theories of Physics. Kluwer 1997. MR1437362 Zbl 0877.5300110.1007/978-94-017-3338-0Search in Google Scholar
[21] A. Morimoto, Liftings of tensor fields and connections to tangent bundles of higher order. Nagoya Math. J. 40 (1970), 99-120. MR0279719 Zbl 0208.5020110.1017/S002776300001388XSearch in Google Scholar
[22] H. H. Schaefer, M. P. Wolff, Topological vector spaces. Springer 1999. MR1741419 Zbl 0983.4600210.1007/978-1-4612-1468-7Search in Google Scholar
[23] L Schwartz, Cours d’analyse. 1, 2. Hermann, Paris 1981. MR756814 MR756815Search in Google Scholar
[24] A. Suri, Geometry of the double tangent bundles of Banach manifolds. J. Geom. Phys. 74 (2013), 91-100. MR3118575 Zbl 1282.5800310.1016/j.geomphys.2013.07.009Search in Google Scholar
[25] A. Suri, Higher order frame bundles. Balkan J. Geom. Appi. 21 (2016), 102-117. MR3511177Zbl 06655806Search in Google Scholar
[26] A. Suri, Higher Order Tangent Bundles. Mediterr. J. Math. 14 (2017), 14:5. MR358992110.1007/s00009-016-0812-7Search in Google Scholar
[27] A. Suri, M. Aghasi, Connections and second order differential equations on infinite dimensional manifolds. Int. Electron. J. Geom.6 (2013), 45-56. MR3125831 Zbl 1308.58003Search in Google Scholar
[28] E. Vassiliou, Transformations of linear connections. Period. Math. Hungar. 13 (1982), 289-308. MR698577 Zbl 0525.5304410.1007/BF01849241Search in Google Scholar
[29] J. Vilms, Connections on tangent bundles. J. Differential Geometry1 (1967), 235-243. MR0229168 Zbl 0162.5360310.4310/jdg/1214428091Search in Google Scholar
[30] K. Yano, S. Ishihara, Differential geometry of tangent bundles of order 2. Kōdai Math. Sem. Rep. 20 (1968), 318-354. MR0232300 Zbl 0167.1970310.2996/kmj/1138845701Search in Google Scholar
© 2017 Walter de Gruyter GmbH, Berlin/Boston
