By Werner Hildbert Greub

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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)

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Lie Groups 32 are both orbits of xh (use the left invariance of Hence (cf. Proposition X, sec. 15, volume I) a(s xh), and agree at t = 0. (t). (3) o (1): Same proof as (2) o (1). D. Proposition V: T o every vector h E T,(G) corresponds a unique I-parameter subgroup, a , such that i ( 0 ) = h. Proof: T h e uniqueness is immediate from Proposition IV. Now we prove existence. According to Proposition X, sec. 15, volume I, for some E > 0 there is an orbit 010 : (-6, 6) -+ G, for X , , satisfying ao(0) = e.

Sphere bundles. An r-sphere bundle is a smooth bundle with fibre the r-sphere. If 4 = (E, T , B, F) is a vector bundle with a Riemannian metric, then the unit spheres S, C F, are the fibres of a sphere bundle 4, = ( E , , T ~B ,, S ) called the associated sphere bundle. An orientation in 8 defines an orientation in the fibres F, ; the induced orientations in the spheres S, (cf. sec. 13) define an orientation in es. 4. Summary of volume I 23 Suppose 28 = ( M , 7r, B, S ) is an oriented r-sphere bundle.

A h, ... exp h, , hi E Ei neighbourhood of 0 diffeomorphically 2. The exponential map 35 ) ~ to the identity in each Ei; hence it is Proof: Clearly ( d ~restricts the identity in E . D. Corollary 111: If G is connected, then exp(E) generates G. Proof: By Corollary I, exp(E) contains a neighbourhood of e. Thus the corollary follows from Lemma I11 below. D. Lemma 111: If G is connected, and U C G is a neighbourhood of e, then U generates G. Proof: U generates an open subgroup H of G. Thus each coset Ha (a E G) is open and G=HuUHa &H partitions G into two disjoint open sets.

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