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Transversally affine foliations

Published online by Cambridge University Press:  18 May 2009

P. M. D. Furness  and
E. Fédida
P. M. D. Furness
Affiliation:
Department of Mathematics, University of Southampton, Southampton, SO9 5NH
E. Fédida
Affiliation:
Institut de Recherche Mathematique Avancee, Université Louis Pasteur, 7, Rue Rene Descartes, 67084 Strasbourg Cedex
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Let ℱ be a smooth foliation of codimension p on a smooth manifold Mm. We can define ℱ by an atlas of coordinate charts (U, (x, y)), called leaf charts, where (x, y): U → Rm−p × Rp are coordinate functions for which the leaves of ℱ are given by y1 constant,…,yp constant, in U. Clearly, on the overlap of two such leaf charts (U, (x, y)) and (U′, (x′, y′)) we have a coordinate transformation of the form

If y′ is always affine in y, i.e.

where and Bi are constants, we shall say that ℱ is a transversally affine foliation. This notion is, in a sense, dual to that of affine foliation, see [2], in which x′ is affine in x and each leaf has an induced flat affine structure.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 17 , Issue 2 , July 1976 , pp. 106 - 111
Copyright
Copyright © Glasgow Mathematical Journal Trust 1976

References

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