By Werner Hildbert Greub, Stephen Halperin, Ray Vanstone

ISBN-10: 0123027020

ISBN-13: 9780123027023

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**Extra resources for Connections, Curvature, and Cohomology: 2 **

**Example text**

2. Multilinear representations: Representations @P, A P and V P of G i n @W, AW, VWaregiven by (@P)(X) = (AP)(x) = A @P(x), W and ( V P ) ( X )= VP(X), x E G, (cf. sec. 5). e(h)wi ... P Bo(h)(wl 0 ... 0w p ) = @wp, i=l ev(h)(wl 1 P ... v w,) = W1 v ... e(hlwi ... p 3 I, w p , i=l and e,(h)h = 0, o,(~)A= 0, e,,(h)A = 0, A E R. Evidently, (@P)’ = (P’)@, (AP)’ = (P’),, (VP)’ = ( P ’ ) ” . and 3. Recall that Tp(W) denotes the space of p-linear functions in W. , P(x-’)w,), w iE W , X E G, @E TP(W). , w P ) , h E E.

7. Homomorphisms. Proposition VII: Let T: G + H be a homomorphism of Lie groups. Then the induced homomorphism, T', of Lie algebras satisfies cp o exp, = exp, o cp'. Proof: Fix h E T,(G). Then a: t H T(exp,(th)) and /3: t are 1-parameter subgroups of H. Moreover, i(0) = v'(h)= j(O), ++ expH(tT'(h)) 36 I. Lie Groups and hence (Proposition V, sec. 5) 01 = 8. D. Corollary I: Assume +: G --f H is a second homomorphism of Lie groups and that cp' = #'. If G is connected, then cp = +. Proof: Proposition VII implies that cp and + agree in exp,(T,(G)).

Assume B is oriented. , ak} -+ M such that 7r 0 u = i. ) Using the local product structure we obtain, from a, smooth maps + ui:U i- { a i }4 S , where U, is a neighbourhood of ai . T h e orientation of U, determines an orientation in a 'sphere' Siabout a, . Let T , be the restriction of u, to Si; then the degree of ri is independent of the various choices. It is called the index of u at a, and is written j J u ) or simplyj,(o). T h e sum j(u) = x,ji(u) is called the index sum of u. It satisfies the relation -4t J - x, =j(u).

### Connections, Curvature, and Cohomology: 2 by Werner Hildbert Greub, Stephen Halperin, Ray Vanstone

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