Curvature of curvilinear 4-webs and pencils of one forms: Variation on a theorem of Poincaré, Mayrhofer and Reidemeister |
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Authors: | I Nakai |
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Institution: | (1) Department of Mathematics, Hokkaido University, Sapporo, Japan 060, e-mail: nakai@math.sci.hokudai.ac.jp, JP |
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Abstract: | A curvilinear d-web W = (F
1 , . . . , F
d
) is a configuration of d curvilinear foliations F
i
on a surface. When d = 3, Bott connections of the normal bundles of F
i
extend naturally to equal affine connection, which is called Chern connection. For 3 < d, this is the case if and only if the modulus of tangents to the leaves of F
i
at a point is constant. A d-web is associative if the modulus is constant and weakly associative if Chern connections of all 3-subwebs have equal curvature form. We give a geometric interpretation of the curvature form
in terms of fake billiard in §2, and prove that a weakly associative d-web is associative if Chern connections of triples of the members are non flat, and then the foliations are defined by members
of a pencil (projective linear family of dim 1) of 1-forms. This result completes the classification of weakly associative
4-webs initiated by Poincaré, Mayrhofer and Reidemeister for the flat case. In §4, we generalize the result for n + 2-webs of n-spaces.
Received: September 23, 1996 |
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Keywords: | , Web, Chern connection, Godbillon-Vey class, |
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