By Nomizu K., Sasaki T.

ISBN-10: 0521441773

ISBN-13: 9780521441773

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It turns out that the groups O(n, m) and SO(n, m) have the same dimension as O(n + m). The dimensions of the other classical groups are SL(n, R) : n2 − 1, SL(n, C) : 2n2 − 2, U(n, m) : (n + m)2 , SU(n, m) : (n + m)2 − 1, Sp(n, R) : n(2n + 1), Sp(n, C) : 2n(2n + 1), Sp(n, m) : (n + m) 2(n + m) + 1 . 7 1. Let G ⊂ GL(n, K) denote one of the above classical groups. With respect to the smooth structures provided by the Level Set Theorem, the mapping G × G → G, given by (a, b) → ab−1 , is smooth. 3).

2/1. This will be needed for the Transversal Mapping Theorem in the next section and for the discussion of distributions and foliations in Sect. 5. The question under consideration may be rephrased as follows. Let M ⊂ N and let ι : M → N denote the natural inclusion mapping. Under which conditions does there exist a C k -structure on M such that (M, ι) is a submanifold (initial submanifold, embedded submanifold) of N ? Here, by a C k structure on M we mean both a topology on M and a maximal atlas whose charts are local homeomorphisms with respect to this topology.

For n = 2 and n = 3, how is this fact related to the definition of polar and spherical coordinates? 2 Show that the mapping Φ : R3 → R3 , defined by Φ(x) := x1 sin(x3 ) + x2 cos(x3 ), x1 cos(x3 ) − x2 sin(x3 ), x3 , maps the unit sphere S2 ⊂ R3 diffeomorphically onto itself. 5/(b). Φ(α, β) := (a1 cos α cos β, a2 cos α sin β, a3 sin α), Ψ (α, β) := (−a1 cos α cos β, a2 sin α, a3 cos α sin β). Show that (a) the images of Φ and Ψ are open subsets of M and hence smooth manifolds, (b) Φ and Ψ are diffeomorphisms onto their images.

### Affine differential geometry. Geometry of affine immersions by Nomizu K., Sasaki T.

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