By Werner Hildbert Greub
Read Online or Download Connections, Curvature, and Cohomology. Vol. 2: Lie Groups, Principal Bundles, and Characteristic Classes (Pure and Applied Mathematics Series; v. 47-II) PDF
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Additional info for Connections, Curvature, and Cohomology. Vol. 2: Lie Groups, Principal Bundles, and Characteristic Classes (Pure and Applied Mathematics Series; v. 47-II)
T h e Picard theorem asserts that for each EM there is a unique orbit of X through x. If X E X ( M ) and f E Y ( M ) , then X ( f ) E Y ( M ) is defined by ( X (f ) ) ( x ) = X ( x ) (f ) . ),X(x) = Y(v(x)),x E M . In this case we write X 7 Y. If g, is a diffeomorphism, v * ( X ) denotes the unique vector field on N which is prelated to X . A dzflerential form on M is a cross-section, 0,in A T & . If each @(x) E APT,(M)*, then @hasdegreep. T h e differential forms are a graded 4. Summary of volume I 17 algebra, A ( M ) = C, A p ( M ) ,with multiplication given by (@ A Y)(x) = A Y ( x ) .
I n particular if M is any R-module, the tensor, exterior, and symmetric algebras over M are written ORM , A, M and, V RM . If M is finitely generated and projective, there are isomorphisms, (8: M ) * 0 sM * , (A: M ) * A: M * , (V: M ) * V: M*, defined in exactly the same way as in sec. 5. 7. Differential spaces. A dzflerential space is a vector space X together with a linear map 6: X -+ X satisfying a2 = 0. 6 is called the dzflerential operator in X . T h e elements of the subspaces Z ( X ) = ker 6 and B ( X ) = Im 8.
14, volume I, and Proposition VIII, sec. 13, volume I, that 0 = [Z'RX,iLY] 7[ X , Y ] . Since p is surjective, [ X , Y ] = 0. D. Finally, consider the inversion map v: x t-t x-l of G. Since v2 a diffeomorphism. +. restricts to an isomorphism -- TdG) TAG) of Lie algebras. In view of Lemma I (2) sec. 3. Lie algebra of a Lie group. T h e Lie algebra of a Lie group G is the vector space, T,(G), together with the Lie algebra structure induced from S L ( G )by the isomorphism of Corollary I to Proposition I, sec.
Connections, Curvature, and Cohomology. Vol. 2: Lie Groups, Principal Bundles, and Characteristic Classes (Pure and Applied Mathematics Series; v. 47-II) by Werner Hildbert Greub