Download Differential Geometry, Lie Groups and Symmetric Spaces over by By (author) Wolfgang Bertram PDF

By By (author) Wolfgang Bertram

Differential Geometry, Lie teams and Symmetric areas Over normal Base Fields and earrings

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4. Vector bundles. ) and the gij (x) respect this structure, then each fiber also carries this structure. In particular, if N is a K-module and the transition maps respect this structure, then F is called a vector bundle. In this case, the bundle atlas is of the form A = (p−1 (Ui ), ϕi )i∈I with ϕi : p−1 (Ui ) → V × W, [i, x, w] → (x, w). Change of charts is given by the transition functions gij (x, w) = (ϕij (x), gij (x)w). Homomorphisms are required to respect the extra structure on the fibers.

1. The ring of dual numbers over K. The ring of dual numbers over K is the truncated polynomial ring K[x]/(x2 ) ; it can also be defined in a similar way as complex numbers, replacing the condition i2 = −1 by ε2 = 0 : K[ε] := K ⊕ εK, (a + εb)(a + εb ) = aa + ε(ab + ba ). 1) In analogy with the terminology for complex numbers, let us call z = x + εy a dual number, x its spacial part, y its infinitesimal part and ε the infinitesimal or dual number unit. A dual number is invertible iff its spacial part is invertible, and inversion is given by x − εy .

14) where a “hat” means “omit this coefficient”. 7) is, for any choice 1 ≤ j1 < . . < j ≤ k , 0 → i=1 εji R → R pji −→ i=1 K[ε1 , . . , εji , . . , εk ] → 0. 15) In particular, for the “maximal choice” ji = i , i = 1, 2, . . , k , we get 0 → ε1 . . εk R → R → k i=1 K[ε1 , . . , εj , . . , εk ] → 0. 16) corresponds to the “most vertical” bundle ε1 . . εk T M → T kM → k i=1 T k−1 M. There are also various projections T k K → T K for all = 0, . . , k −1 ; their kernels are certain sums of the ideals considered so far.

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