Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Almost tangent and cotangent structures in the large


Authors: G. Thompson and U. Schwardmann
Journal: Trans. Amer. Math. Soc. 327 (1991), 313-328
MSC: Primary 53C15; Secondary 53C10
MathSciNet review: 1012509
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We examine some global properties of integrable almost tangent and cotangent manifolds. In particular, we extend several results which essentially characterize tangent and cotangent bundles as, respectively, regular almost tangent and cotangent structures.


References [Enhancements On Off] (What's this?)

  • [1] John K. Beem, Proper homothetic maps and fixed points, Lett. Math. Phys. 2 (1977/78), no. 4, 317–320. MR 501280, 10.1007/BF00419621
  • [2] Raoul Bott and Loring W. Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics, vol. 82, Springer-Verlag, New York-Berlin, 1982. MR 658304
  • [3] F. Brickell and R. S. Clark, Integrable almost tangent structures, J. Differential Geometry 9 (1974), 557–563. MR 0348666
  • [4] M. R. Bruckheimer, Doctoral thesis, University of Southampton, 1960.
  • [5] R. S. Clark and D. S. Goel, On the geometry of an almost tangent manifold, Tensor (N.S.) 24 (1972), 243–252. Commemoration volumes for Prof. Dr. Akitsugu Kawaguchi’s seventieth birthday, Vol. I. MR 0326613
  • [6] R. S. Clark and D. S. Goel, Almost cotangent manifolds, J. Differential Geometry 9 (1974), 109–122. MR 0365404
  • [7] M. Crampin, Tangent bundle geometry for Lagrangian dynamics, J. Phys. A 16 (1983), no. 16, 3755–3772. MR 727054
  • [8] M. Crampin and G. Thompson, Affine bundles and integrable almost tangent structures, Math. Proc. Cambridge Philos. Soc. 98 (1985), no. 1, 61–71. MR 789719, 10.1017/S0305004100063246
  • [9] M. Crampin and L. A. Ibort, Graded Lie algebras of derivations and Ehresmann connections, J. Math. Pures Appl. (9) 66 (1987), no. 1, 113–125. MR 884816
  • [10] Hermes A. Eliopoulos, Structures presque tangentes sur les variétés différentiables, C. R. Acad. Sci. Paris 255 (1962), 1563–1565 (French). MR 0142078
  • [11] Hermes A. Eliopoulos, On the general theory of differentiable manifolds with almost tangent structure, Canad. Math. Bull. 8 (1965), 721–748. MR 0193590
  • [12] Joseph Grifone, Structure presque-tangente et connexions. I, Ann. Inst. Fourier (Grenoble) 22 (1972), no. 1, 287–334 (French, with English summary). MR 0336636
  • [13] Joseph Grifone, Structure presque-tangente et connexions. II, Ann. Inst. Fourier (Grenoble) 22 (1972), no. 3, 291–338. (loose errata) (French, with English summary). MR 0341361
  • [14] Victor Guillemin and Shlomo Sternberg, Geometric asymptotics, American Mathematical Society, Providence, R.I., 1977. Mathematical Surveys, No. 14. MR 0516965
  • [15] Victor Guillemin and Shlomo Sternberg, Symplectic techniques in physics, Cambridge University Press, Cambridge, 1984. MR 770935
  • [16] Joseph Klein, Espaces variationnels et mécanique, Ann. Inst. Fourier (Grenoble) 12 (1962), 1–124 (French). MR 0215269
  • [17] -, Structures symplectiques ou $ J$-symplectiques homogènes sur l'espace tangent, Symposia Math. 14 (1974), 181-192.
  • [18] -, Geometry of sprays. Lagrangian case principle of least curvature, Proc. IUTAM-ISIMM Symposium on Modern Developments in Analytical Mechanics, Torino, 1982.
  • [19] Shoshichi Kobayashi, Transformation groups in differential geometry, Springer-Verlag, New York-Heidelberg, 1972. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 70. MR 0355886
  • [20] Detlef Laugwitz, Differential and Riemannian geometry, Translated by Fritz Steinhardt, Academic Press, New York-London, 1965. MR 0172184
  • [21] John W. Milnor and James D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. Annals of Mathematics Studies, No. 76. MR 0440554
  • [22] Shlomo Sternberg, Lectures on differential geometry, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0193578
  • [23] G. Thompson, Integrable almost cotangent structures and Legendrian bundles, Math. Proc. Cambridge Philos. Soc. 101 (1987), no. 1, 61–78. MR 877701, 10.1017/S030500410006641X
  • [24] Alan Weinstein, Lectures on symplectic manifolds, American Mathematical Society, Providence, R.I., 1977. Expository lectures from the CBMS Regional Conference held at the University of North Carolina, March 8–12, 1976; Regional Conference Series in Mathematics, No. 29. MR 0464312
  • [25] N. M. J. Woodhouse, Geometric quantization, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1992. Oxford Science Publications. MR 1183739

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 53C15, 53C10

Retrieve articles in all journals with MSC: 53C15, 53C10


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1991-1012509-X
Article copyright: © Copyright 1991 American Mathematical Society