By Julian Lowell Coolidge

A radical creation to the speculation of algebraic airplane curves and their family members to varied fields of geometry and research. virtually completely constrained to the homes of the overall curve, and mainly employs algebraic technique. Geometric equipment are a lot hired, in spite of the fact that, particularly these related to the projective geometry of hyperspace. 1931 variation. 17 illustrations.

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**Example text**

Indeed, the holonomy cover of L can be identiﬁed with Gx = {γ ∈ G | s(γ) = x} with covering projection r : Gx → L, and then the representation Rx is given by Rx (f )ξ(γ) = γ1 ·γ2 =γ f (γ1 )ξ(γ2 ). This is well deﬁned because, for ﬁxed x and γ ∈ Gx , there are canonical identiﬁcations (D1/2 Gx )γ = D1/2 Mr(γ) , so that for γ1 · γ2 = γ, f (γ1 )ξ(γ2 ) is a density of order one with coeﬃcients in D1/2 Mr(γ1 ) = (D1/2 Gx )s(γ) , and thus the integral over {γ1 · γ2 = γ} is a half density on Gx . This is also well deﬁned in the non-Hausdorﬀ case because, for each γ, the ﬁber Gr(γ) is a proper Hausdorﬀ submanifold of G, hence the restriction f |Gr(γ) is smooth with compact support in the usual sense.

M. Torpe [176], and T. Fack and X. Wang [64] studied Reeb foliations, and one of the authors [17] studied related codimension one foliations. At other times the foliation has a structure related to that of a transformation group, and this is reﬂected in the foliation algebra. For instance, T. Natsume [140] studied foliations without holonomy, and H. Takai [166] studied Anosov foliations. 6. Quasi-invariant Currents This section discusses some aspects of the general concept of quasi-invariant measures for a foliated space.

X, By the cocycle property of the Radon-Nikodym derivatives, δi = δj on G(Ui ) ∩ G(Uj ). Furthermore, δi (γ1 · γ2 ) = δi (γ1 )δi (γ2 ) for every γ1 , γ2 ∈ G(Ui ). The idea now is to use the fact that every γ ∈ G can be written as a ﬁnite product γ = γ1 ·.