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 resources 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 theory extends to vector-valued functions (integrate component by component). Finally, we shall use the Picard existence and uniqueness theorem for ordinary differential equations as given in [6, p. 221. 10. The exponential map. Let E be an n-dimensional real or complex vector space and let u: E + E be a linear transformation. It follows from the standard existence theorems of differential equations that there is a unique smooth map T : [w +LE satisfying the linear differential equation +=a07 and the initial condition ~ ( 0= ) L .
T h e correspondence x tt $(x) defines a smooth map $: B + B‘. If $ = L , then y is called a strong bundle map. The Cartesian product of 5 and 4‘ is the vector bundle f x 5‘ = ( E x E , n x n’, B x B‘, F @ F‘). If F ‘ = 0 (so that E’ = B’, n’ = L ) , we write simply 5 x B’. A vector bundle 4 determines vector bundles f * , o p t , A 5, V q f , whose F, , A F, and Vq F,. If is a second vector fibres at x are the spacesF2, bundle with the same base and with typical fibre H , then 5 @ q , 4 8 q , and L(5;q ) denote the vector bundles with fibres F, 0H , ,F, O H , , and L(F, ; H,).
An open cover of X is a family 0 of open sets whose union is X . 0 is called a rejinement of an open cover @ if each 0 E 0 is a subset of some U E a. X is called paracompact if every open cover of X has a locally finite refinement. A basis for the topology of X is a family 0 of open sets such that each open subset of X is the union of elements of 0. If 0 is closed under finite intersections, it is called an i-basis. If X has a countable basis, it is called second countable. s4. 12. Manifolds and vector bundles.
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