If F is a simple foliation defined on the manifold M by the projection π: M ➞ W onto a quotient manifold, then the “transverse geometry” is precisely the differential geometry of the quotient manifold. In general, in the case of an arbitrary foliation, there no longer exists a global quotient manifold. Nevertheless, the condition for a geometric structure to be “locally projectable along the leaves” still has a precise meaning. The study of structures that possess this property constitutes the transverse geometry of the foliation. These ideas have their origin in the works of C.Ehresmann [Eh] and A.Haefliger [Ha]2.
KeywordsPrincipal Bundle Transverse Field Riemannian Foliation Simple Foliation Projectable Connection
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