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Additional resources for Connections, Curvature, and Cohomology Volume 2: Lie Groups, Principal Bundles, and Characteristic Classes

Sample text

D. Proposition VIII: A continuous group homomorphism p: G --f H between Lie groups is smooth. 2. The exponential map 31 Proof: Consider first the case that G = R. It has to be shown that a continuous map a : R -+H which satisfies a(s + t ) = a(s) a@), s, t € R, is smooth. In view of Corollary I to Proposition VI, sec. 6, there is a neighbourhood V of 0 in T,(H) which exp,, maps diffeomorphically onto a neighbourhood U of e in H . Without loss of generality we may assume that a@)€ Define a continuous map u, It I < 1.

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 linear transformation ~ ( 1 )is called the exponential of u and is denoted by exp u. In this way we obtain a (nonlinear) map exp: LE +LE. I t has the following properties: (0) (1) (2) (3) e x p o = L.

An orientation of M is an orientation of T~ ; thus it is an equivalence class of nowhere vanishing n-forms. A smooth map v: M --t N (dim M = dim N ) is called orientation preserving (respectively, orientation reversing) if v*d (respectively, -v*d) represents the orientation of M when d represents that of N . , 4. Summary of volume I 19 in A ( M ) . Assume M oriented and of dimension n. Then the integral is defined; it is a linear map J M : A t ( M ) -+ R, natural with respect to orientation preserving diffeomorphisms, and satisfying where A , is the positive normed determinant function of an oriented Euclidean space E.