In Theorem 1, the C-ﬁbration is constructed simultaneously with the extension of the foliation F by gluing bifoliated manifolds.
A preparation theorem for codimension one foliations
Moreover, cM has a natural riemannian/ total ly geodesic bifoliation ( bF, bF ⊥ ), with respect to the pul l-back metric.
Singular Holonomy of Singular Riemannian Foliations with Sections
Let (F, F ⊥ ) be the bifoliation on cM deﬁned in Theorem 5.1.
Singular Holonomy of Singular Riemannian Foliations with Sections
It follows from Blumenthal and Hebda that the map Φ : eL × eΣ → deﬁned as ([β ], [γ ]) → [t → bH( ˆβ, ˆγ ) (t, t)], is a bifoliated diffeomorphism (i.e., foliated with respect to both pairs of foliations).
Singular Holonomy of Singular Riemannian Foliations with Sections
We conclude that Ψ : eL × eΣ → cM ([β ], [γ ]) → bH( ˆβ, ˆγ ) (1, 1) is a bifoliated universal covering map of cM .
Singular Holonomy of Singular Riemannian Foliations with Sections
The map Ψ is the universal covering map, and it is bifoliated with respect to the natural bifoliation of eL × eΣ and to ( cM ; bF, bF ⊥ ).
Singular Holonomy of Singular Riemannian Foliations with Sections
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