By Robert C. Gunning
The classical uniformization theorem for Riemann surfaces and its fresh extensions will be considered as introducing detailed pseudogroup constructions, affine or projective constructions, on Riemann surfaces. in reality, the extra buildings concerned might be regarded as neighborhood kinds of the uniformizations of Riemann surfaces. during this research, Robert Gunning discusses the corresponding pseudogroup constructions on higher-dimensional complicated manifolds, modeled at the conception as constructed for Riemann surfaces.
Originally released in 1978.
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Additional info for On Uniformization of Complex Manifolds: The Role of Connections (Mathematical Notes)
Sample text
0, 1 J1 J2 f be a nonsingular mapping JlJ2 hence that f is an affine mapping fi(z) = £,j aij z. + ai (72) for some constants aij, ai is just that the matrix (52) with A = 0 The condition that . {aij} is nonsingular. Igext for the defining group the associated pseudogroup consists of those nonsingular complex analytic mappings f and writing (z) = j2f(z) such that 6"(j2f(z)) = (1- 2) O(j2f(z)) = 0 and recalling (44) and (63) this readily reduces to the condition that ) ( z (73) _ E1 1J2 i( (z) aQ z j1 + Z.
Suppose therefore that the subgroup A C G2(n,1) associated to the subalgebra defined by (36) is integrable. The 3-jets of all mappings in compose an integrable subgroup A' C G3(n,T,) subalgebra £ ' C Q] 3(n,O) P = j2 AZ' , exist an element such that associated to A' and it follows from Lemma 1 that A' _ {81,A2,A3} e /Q, where , AI the subalgebra A by (36) ; and since T (A) A = j2A' ; the then has the property that ,t7 ' C A 1 . A = {81,A2} e A for all X. C ,V2(n,T) j3 There must describes it follows that (41) X j 3A3 + dp2(Xj3A2) - A2 E R2 for all j3 .
AJ-la o 3+l" ' aq+l p e Ua n... n o consists of a collection of . U", c+l .