The structure of Legendre foliations
Author:
MyungYull Pang
Journal:
Trans. Amer. Math. Soc. 320 (1990), 417455
MSC:
Primary 58F18; Secondary 57R30, 58F05
MathSciNet review:
1016808
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Abstract: The local and global structure of Legendre foliations of contact manifolds is analysed. The main invariant of a Legendre foliation is shown to be a quadratic form on the tangent bundle to the foliationthe fundamental quadratic form. The equivalence problem is solved in the case when the fundamental quadratic form is nondegenerate and a generalization of Chern's solution to the equivalence problem for Finsler manifolds is obtained. A normal form for Legendre foliations is given which is closely related to Weinstein's structure theorem for Lagrangian foliations. It is shown that every compact, simply connected leaf of a Legendre foliation is diffeomorphic to a sphere.
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 [DK]
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 [DPU]
 T. Duchamp, M. Pang and G. Uhlmann, Inverse scattering for metrics (work in progress).
 [G]
 R. B. Gardner, Differential geometric methods interfacing control theory, Differential Geometric Control Theory. (R. Millman, H. Sussman, Eds.), Progress in Math., 27, Birkhäuser, Boston, Mass., 1983, pp. 117180. MR 708501 (84k:58009)
 [KN]
 S. Kobayashi and K. Nomizu, Foundations of differential geometry, , Interscience, New York, pp. 5361.
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 N. Kamran and P. Olver, The equivalence problem for particle Lagrangians, J. Differential Equations (to appear). MR 1003250 (90g:58029)
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 F. Kamber, Ph. Tondeur, Foliated bundles and characteristic classes, Lecture Notes in Math., vol. 493, Springer, Berlin, Heidelberg, and New York, 1975. MR 0402773 (53:6587)
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 S. Sternberg, Lectures on differential geometry, PrenticeHall, Englwood Cliffs, N. J., 1964. MR 0193578 (33:1797)
 [W]
 A. Weinstein, Symplectic manifolds and their Lagrangian submanifolds, Adv. in Math. 6 (1971), 329346. MR 0286137 (44:3351)
 [Wo]
 J. A. Wolf, Spaces of constant curvature, Publish or Perish. MR 928600 (88k:53002)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199010168086
PII:
S 00029947(1990)10168086
Article copyright:
© Copyright 1990
American Mathematical Society
